Stochastic Differential - 213.230.96.51:8090

Stochastic DifferentialEquations: Theory and

Appllications

INTERDISCIPLINARY MATHEMATICAL SCIENCES

Series Editor: Jinqiao Duan (Illinois Inst. of Tech., USA)

Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin,Charles Doering, Paul Fischer, Andrei V. Fursikov, Fred R. McMorris,Daniel Schertzer, Bjorn Schmalfuss, Xiangdong Ye, andJerzy Zabczyk

Published

Vol. 1: Global Attractors of Nonautonomous Dissipative Dynamical SystemsDavid N. Cheban

Vol. 2: Stochastic Differential Equations: Theory and ApplicationsA Volume in Honor of Professor Boris L. Rozovskiieds. Peter H. Baxendale & Sergey V. Lototsky

Vol. 3: Amplitude Equations for Stochastic Partial Differential EquationsDirk Blömker

Vol. 4: Mathematical Theory of Adaptive ControlVladimir G. Sragovich

Vol. 5: The Hilbert–Huang Transform and Its ApplicationsNorden E. Huang & Samuel S. P. Shen

Vol. 6: Meshfree Approximation Methods with MATLABGregory E. Fasshauer

EH - Stochastic Diff Eqns.pmd 3/26/2007, 9:59 AM2

Interdiscliplinary Mathematical Sciences - Vol. 2

A Volume in Honor of

Professor Boris L. Rozovskii

Stochastic DifferentialEquations: Theory and

ApplicationsEditors

Peter H. Baxendale

Sergey V. LototskyUniversity of Southern California, USA

World Scientific

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Interdisciplinary Mathematical Sciences — Vol. 2STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONSA Volume in Honor of Professor Boris L. Rozovskii

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Professor Boris L. Rozovskii

February 26, 2007 9:27 World Scientific Review Volume - 9.75in x 6.5in RozVol

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Preface

This festschrift volume is dedicated to Boris Rozovskii on his 60th birthday.

The first paper in the volume, Stochastic Evolution Equations by N. V. Krylov

and B. L. Rozovskii, was originally published in Russian in 1979 (Itogi Nauki i

Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol, 14, pp. 71–146). The

English translation was first published in the Journal of Soviet Mathematics, Vol.

14, pp. 1233–1277, 1981, c©Plenum Publishing Co. We are very grateful to the

current copyright holder, Springer, for the permission to include the paper in the

volume. After more than a quarter-century, this paper remains a standard reference

in the field of stochastic partial differential equations (SPDEs) and continues to

attract attention of mathematicians of all generations, because, together with a

short but thorough introduction to SPDEs, it presents a number of optimal and

essentially non-improvable results about solvability for a large class of both linear

and non-linear equations.

The other papers in this volume were specially written for the occasion. The

14 contributions deal with a wide range of topics in the theory and applications of

stochastic differential equations, both ordinary and with partial derivatives.

Eight of the contributions are related to stochastic partial differential equations.

D. Blomker and J. Duan investigate behavior of the mean energy and correla-

tion function for the Burgers equation with various types of random perturbation.

L. Borcea, G. Papanicolaou, and C. Tsogka study asymptotics of the space-time

Wigner transform for the stochastic Schrodinger equation and apply the results to

broadband array imaging in random media. A. de Bouard and A. Debussche estab-

lish existence and uniqueness of a global square-integrable solution of the stochastic

Korteweg-de Vries equation with multiplicative noise. Z. Brzezniak and L. Debbi

establish global existence and uniqueness of a mild solution for a fractional Burgers

equation with multiplicative space-time white noise. F. Flandoli and M. Romito

study regularity properties of a transition semigroup associated with the Navier-

Stokes equation driven by a non-degenerate additive space-time noise. I. Gyongy

and A. Millet establish the rate of convergence of the implicit Euler scheme for a

class of nonlinear SPDEs. N.V. Krylov proves the maximum principle for a large

class of linear stochastic parabolic equations and uses the result to study spatial

regularity of the solution for equations on the half-line. G. Da Prato investigates

vii


viii Preface

the Kolmogorov equation associated with a class of non-linear stochastic parabolic

equations.

Six of the contributions are related to stochastic ordinary differential equations.

A. Cadenillas, J. Cvitanic, and F. Zapatero study a stochastic control problem that

models the use of stock options for executive compensation. P. Chigansky and R.

Liptser establish the large deviations principle for a class of stochastic equations

with rapidly growing coefficients and a possibly degenerate diffusion. D. Crisan

and S. Ghazali study numerical approximation of the probability distribution of

the solution at a fixed time for a class of stochastic equations in the Stratonovich

form and apply the results to the nonlinear filtering problem. L. Decreusefond and

D. Nualart show that the solutions of stochastic equation driven by a fractional

Brownian motion with Hurst parameter bigger than 1/2 generate a flow of home-

omorphisms. Yu. A. Kutoyants reviews recent results on statistical inference for a

stochastic equation with delay when the delay parameter is unknown. R. Mikulevi-

cius and H. Pragarauskas investigate a linear integro-differential equation connected

with jump-diffusion processes.

Preparation of this volume was a joint effort by a number of people. We are

very grateful to J. Duan, who initiated the project; to Rok Ting Tan and Yubing

Zhai, who managed the project at World Scientific; to Inge Weijman and Berendina

van Straalen, who processed our copyright clearance at Springer; to Abhinav Guru,

who helped to prepare the TEX file of the paper by Krylov and Rozovskii. Our

special gratitude goes to all the contributors to this volume and all the referees

who carefully reviewed the submitted papers and assured the highest quality of all

contributions.

P. Baxendale

S. Lototsky


Preface ix

Boris Rozovskii

In the life of every mathematician there are usually several key events that facilitate

a successful career. For Boris Rozovkii, one such event happened in 1962, when,

shortly after entering the Odessa State University, he got a special invitation to

transfer to the Faculty of Mechanics and Mathematics (the famous MechMat) at

the Moscow State University, and immediately found himself at the center of the

Soviet and, to some extent, the world mathematics. The undergraduate program of

study at MechMat leads to an analog of Master of Science degree and requires the

students to engage in serious research. Boris started to work with I. Girsanov, of

the Girsanov Theorem fame. After the tragic death of I. Girsanov in an avalanche,

Boris became a student of A. N. Shiryev, and thus an “academic grandson” of A. N.

Kolmogorov. While Kolmogorov did not participate directly in the upbringing of his

“grandson,” his influence through lectures and seminars was strong and beneficial.

From the start, Boris worked at the junction of stochastic analysis and statistics,

and with considerable success. His undergraduate work on change detection in a

Poisson process was praised by A. N. Kolmogorov, which is probably the highest

reward a young mathematician could have received at the time. His Ph.D. disser-

tation, On stochastic equations arising in filtering of Markov processes, published

in 1972, became one of the three works that formed the foundation of the modern

theory of stochastic partial differential equations (SPDEs); the other two, by E.

Pardoux and M. Viot, appeared in 1975 and were also Ph.D. dissertations.

In 1972 Boris became a faculty member at the Moscow Institute of Advanced

Studies for Engineers and Managers in Chemical Industry (MIASCME). This insti-

tution had a strong statistical group, which Boris joined. Working with chemical

engineers was a valuable experience, as it taught him how to effectively communi-

cate mathematics to non-mathematicians and provided ample opportunity to learn

about various applications of mathematics. This experience came in very handy

later, when Boris became the director of the Center for Applied Mathematical Sci-

ences at the University of Southern California (USC).

Meanwhile, Boris continued to work, both independently and with N. V. Krylov,

on nonlinear filtering of diffusion processes and the general theory of SPDEs. The

result was a complete L2 theory for linear SPDEs, as well as several key results

for nonlinear equations. The linear theory was summarized in the book Stochastic

Evolution Systems (1983 in Russian, 1990 in English). While the Russian edition

looked more like a brochure than a book, the English translation, with extra ma-

terial on Malliavin Calculus, was in hard cover and had a much more respectable

appearance. The inside of the English translation was a different story; because

of a technical glitch, the text contained an enormous number of typos, some of

them rather serious. In the end, though, this proved to be a blessing in disguise,

as reading the book and correcting the typos became a rite of passage for every

student interested in SPDEs; correcting all those typos makes the reading much


x Preface

more challenging and rewarding.

In 1984 Boris was granted the second doctoral degree, Doctor of Science in

Physics and Mathematics, by the Vilnius State University. This degree was an

important benchmark for a scholar in the former Soviet Union.

In 1988 Boris emigrated to the USA. After a short tenure at the University of

North Caroline, Charlotte, he moved to Los Angeles, CA, to become director of the

Center for Applied Mathematical Sciences (CAMS) at the University of Southern

California. Nonlinear filtering and SPDEs are connected with a large number of

mathematical problems, both applied and theoretical, and, while at USC, Boris

explored these connections to the fullest. As a Director of CAMS, Boris was very

successful in attracting extensive external research funds and became involved in

numerous projects in computational fluid dynamics, finance, network security, phys-

ical oceanography, target recognition and tracking, and many other areas.

Applied problem not only help to write a successful grant proposal, but can

also lead to new and unexpected theoretical developments. For example, physical

oceanography, the subject of the first big grant from the Office of Naval Research

Boris secured as the director of CAMS, lead to a new estimation theory for SPDEs,

developed in his joint papers with M. Huebner, R. Khasminskii, S. Lototsky, and L.

Piterbarg. Similarly, numerical computation of the optimal nonlinear filter, the key

component of many successful proposals, lead to a new way of analyzing linear and

nonlinear SPDEs using Wiener chaos decomposition. Yet another direction of his

research at USC was a comprehensive theory of absolute continuity and singularity

of measures generated by solutions of nonlinear SPDEs, developed jointly with R.

Mikulevicius.

B. L. Rozovskii has published in over 25 different journals with over 40 collabora-

tors. His Erdos number is 4, via several connections, such as A. N. Shiryaev→L. A.

Shepp→P. Frankl; R. Z. Khasminskii→O. Zeitouni→P. Diaconis; P. Baxendale→D.

Stroock→P. Diaconis.

Not long ago Boris became a grandfather (his granddaughter Mia was born in

2005), and shortly after that, an “academic grandfather.” Still, he shows no signs

of slowing down, publishing more papers in the past year than in the previous five.

In fact, his recent move to the East Coast suggests that he is at the peak of his

mathematical career.

Main landmarks of B. L. Rozovskii’s scientific career

Boris L’vovich Rozovskii was born on June 8, 1945, in Odessa, Ukraine. Currently,

he is Professor at the Division of Applied Mathematics, Brown University.

1968 Graduated from the Department of Mechanics and Mathematics, Moscow

State University, with Master of Science degree in Probability and Statistics.

1972 Ph.D. in Physical and Mathematical Sciences, Moscow State University.

1984 Doctor of Science in Physics and Mathematics.


Preface xi

1985 Professor and Head of the Informatics Laboratory, Moscow Institute of Ad-

vanced Studies for Engineers and Managers in Chemical Industry.

1991 Professor, Department of Mathematics, USC.

1992 Director, Center for Applied Mathematical Sciences, University of Southern

California.

2006 Professor, Division of Applied Mathematics, Brown University.

Graduate students of B. L. Rozovskii

M. Huebner (USC, 1993)

K. Owens (USC, 1994)

A. Fung (USC, 1995)

S. Lototsky (USC, 1996)

C. Rao (USC, 1998)

S. Kligys (USC, 1998)

A. Petrov (USC, 2000)

G. Yaralov (USC, 2000)

B. L. Rozovskii: Honors and Awards

Fellow of the Institute of Mathematical Statistics (1997)

Peter-the-Great Medal (International Academy of Natural and Social Sciences, 1997)

Kolmogorov Medal (Kolmogorov Centennial Conference, 2003)

Publications of B. L. Rozovskii

Books

1. Stochastic evolution systems. Linear theory and applications to the statistics of

random processes (in Russian). Moscow: Nauka, 1983.

2. Data analysis in chemical research. Statistical foundations (in Russian).

Moscow: Khimija, 1984.

3. Stochastic evolution systems. Linear theory with applications to non-linear fil-

tering. Mathematics and its Applications (Soviet Series), vol. 35. Dordrecht:

Kluwer Academic Publishers, 1990.

Papers

1. Wiener chaos solutions of linear stochastic evolution equations (with S. Lotot-

sky). Ann. Probab., 34 (2006), no. 2, 638–662.

2. Wiener chaos expansions and numerical solutions of randomly forced equations

of fluid mechanics (with T. How et al.), J. Comput. Phys. 216 (2006), no. 2,

687–706.

3. Stochastic differential equations: A Wiener chaos approach (with S. Lototsky).

In From Stochastic Calculus to Mathematical Finance, ed. Yu. Kabanov et al.,


xii Preface

pp. 433–506. Berlin: Springer, 2006.

4. Strong Solutions of Stochastic Generalized Porous Media Equations: Existence,

Uniqueness and Ergodicity. (with G. Da Prato et al.) Comm. Partial Dif. Eq.,

31 (2006), no. 1-3, 277–291.

5. A novel approach to detection of intrusions in computer networks via adaptive

sequential and batch-sequential change-point detection methods (with R. Blazek

et al.), IEEE Transactions on Signal Processing, to appear.

6. Detection of intrusions in information systems by sequential change-point meth-

ods (with A. Tartakovsky et al.). Stat. Methodol., 3 (2006), no. 3, 252–293.

7. Detection of intrusions in information systems by sequential change-point meth-

ods. Authors’ response (with A. Tartakovsky et al.) Stat. Methodol. 3 (2006),

no. 3, 329–340.

8. A filtering approach to tracking volatility from prices observed at random times

(with J. Cvitanic et al). Ann. Appl. Probab., 16 (2006), no. 3, 1633–1652.

9. Numerical estimation of volatility values from discretely observed diffusion data

(with J. Cvitanic et al), Journal of Computational Finance (to appear).

10. A novel approach to detection of intrusions in computer networks via adaptive

sequential and batch-sequential change-point detection methods (with R. Blazek

et al.) International Journal of Computing and Information Sciences.

11. Global L2-solutions of stochastic Navier-Stokes equations (with R. Mikulevi-

cius). Ann. Probab., 33 (2005), No. 1, 137–176.

12. Passive Scalar Equation in a Turbulent Incompressible Gaussian Velocity Field

(with S. Lototsky), Russian. Math. Surveys, 59 (2004), no.2, 297–312.

13. Stochastic Navier-Stokes equations for turbulent flows (with R. Mikulevicius).

SIAM J. Math. Anal. 35 (2004), no. 5, 1250–1310.

14. A diffusion model of roundtrip time (with S. Bohacek). Computational Statis-

tics and Data Analysis, Comput. Statist. Data Anal., 45 (2004) no. 1, 25–50.

15. On martingale problem solutions for stochastic Navier-Stokes equations (with

R. Mikulevicius). In Stochastic partial differential equations and applications,

ed. G. Da Prato and L. Tubaro. Lecture Notes in Pure and Applied Mathe-

matics Series 227. New York: Marcel Dekker, 2002.

16. A note on Krylov’s Lp-theory for systems of SPDEs (with R. Mikulevicius).

Electron. J. Probab. 6 (2001), no. 12, 1–35.

17. On equations of stochastic fluid mechanics (with R. Mikulevicius). In Stochas-

tics in finite and infinite dimensions: in honor of Gopinath Kallianpur, ed. T.

Hida et al., 285–302. Trends Math. Boston: Birkhauser, 2001.

18. Stochastic Navier-Stokes equations: propagation of chaos and statistical mo-

ments (with R. Mikulevicius). In Optimal control and partial differential equa-

tions: in honor of Professor Alain Bensoussan, ed. J. L. Menaldi et al., 258–267.

Amsterdam: IOS Press, 2001.

19. Approximation of the Kushner equation of nonlinear filtering (with K. Ito).

SIAM J. Control Optim. 38 (2000), no. 3, 893–915.


Preface xiii

20. Parameter estimation for stochastic evolution equations with non-commuting

operators (with S. Lototsky). In Skorokhod’s ideas in probability theory, ed. V.

Korolyuk et al., pp. 271–280. Kiev: Institute of Mathematics of the National

Academy of Sciences of Ukraine, 2000.

21. Fourier-Hermite expansions for nonlinear filtering (with R. Mikulevicius). Teor.

Veroyatnost. i Primenen. 44 (1999), no. 3, 675–680. English translation

Theory Probab. Appl. 44 (2000), no. 3, 606–612.

22. Spectral asymptotics of some functionals arising in statistical inference for

SPDE’s (with S. Lototsky). Stochastic Process. Appl. 79 (1999), no. 1, 69–94.

23. Recursive nonlinear filter for a continuous-discrete time model (with S. Lotot-

sky). IEEE Trans. Automatic Cont. 48 (1998), no. 8. 1154–1158.

24. Martingale problems for stochastic PDE’s (with R. Mikulevicius). In Stochas-

tic partial differential equations: six perspectives, ed. R. Carmona and B. L.

Rozovskii, pp. 243–325. Math. Surveys Monogr., vol. 64. Providence, RI:

American Mathematical Society, 1998.

25. Normalized stochastic integrals in topological vector spaces (with R. Mikule-

vicius). In Seminaire de Probabilites XXXII, pp. 137–165. Lecture Notes in

Math, vol. 1686. Berlin: Springer, 1998.

26. Linear parabolic stochastic PDE’s and Wiener chaos (with R. Mikulevicius).

SIAM J. Math. Anal. 29 (1998), no. 2, 452–480.

27. Weighted stochastic Sobolev spaces and bilinear SPDE’s driven by space-time

white noise (with D. Nualart). J. Funct. Anal. 149 (1997), no. 1, 200–225.

28. On asymptotic problems of parameter estimation in stochastic PDE’s: discrete

time sampling (with L. Piterbarg). Math. Methods Statist. 6 (1997), no. 2,

200–223.

29. Nonlinear filtering revisited: a spectral approach (with S. Lototsky and R.

Mikulevicius). SIAM J. Control Optim. 35 (1997), no. 2, 435–461.

30. On asymptotic properties of an approximate maximum likelihood estimator for

stochastic PDEs (with M. Huebner and S. Lototsky). In Statistics and control

of stochastic processes, ed. Yu. M. Kabanov et al. pp. 139–155. River Edge,

NJ: World Scientific, 1997.

31. Recursive multiple Wiener integral expansion for nonlinear filtering of diffusion

processes (with S. Lototsky). In Stochastic processes and functional analysis,

ed. J. Goldstein et al., pp. 199–208. Lecture Notes in Pure and App. Math.,

vol. 186. New York: Marcel Dekker, 1997.

32. Maximum likelihood estimators in the equations of physical oceanography (with

L. Piterbarg). In Stochastic modelling in oceanography, ed. R. Adler et al., pp.

397–421. Progr. Probab., vol. 39. Boston: Birkhauser, 1996.

33. On asymptotic properties of maximum likelihood estimators for parabolic

stochastic PDE’s (with M. Huebner). Probab. Theory Related Fields 103 (1995),

no. 2, 143–163.

34. On stochastic integrals in topological vector spaces (with R. Mikulevicius).


xiv Preface

Stochastic analysis (Ithaca, NY, 1993), pp. 593–602. Proc. Sympos. Pure

Math., vol. 57. Providence, RI: American Mathematics Society, 1995.

35. Estimates of turbulent parameters from Lagrangian data using a stochastic

particle model (with A. Griffa et al.). Journal of Mar. Res. 53 (1995), no. 3,

371–401.

36. Statistics and physical oceanography (with D. B. Chelton et al.). Stat. Sci. 9

(1994), no. 2, 167–201.

37. Uniqueness and absolute continuity of weak solutions for parabolic SPDE’s

(with R. Mikulevicius). Acta Appl. Math. 35 (1994), no. 1–2, 179–192.

38. Soft solutions of linear parabolic SPDE’s and the Wiener chaos expansion (with

R. Mikulevicius). In Stochastic analysis on infinite-dimensional spaces, ed. H.

Kunita and H.-H. Kuo, pp. 211–220. Pitman Res. Notes Math. Ser., vol. 310.

Baton Rouge, LA: Longman Sci. Tech, Harlow, 1994.

39. Kinematic dynamo and intermittence in a turbulent flow. (with P. Baxendale).

Geophys. Astrophys. Fluid Dynam. 73 (1993), no. 1-4, 33–60.

40. Two examples of parameter estimation for stochastic partial differential equa-

tions (with M. Huebner and R. Khasminskii). In Stochastic processes. A

festschrift in honor of Gopinath Kallianpur, pp. 149–160. New York: Springer,

1993.

41. Some results on a diffusion approximation to the induction equation. In Stochas-

tic partial differential equations and applications (Trento, 1990), ed. G. Da

Prato and L. Tubaro, pp. 268–281. Pitman Res. Notes in Math. Ser., vol. 268.

Baton Rouge, LA: Longman Sci. Tech, Harlow, 1992.

42. A simple proof of uniqueness for Kushner and Zakai equations. In Stochastic

analysis, ed. E. Mayer-Wolf, pp. 449–58. Boston: Academic Press, 1991.

43. Measure-valued solutions of second-order stochastic parabolic equations (with

O.G. Purtukhiya, in Russian). In Statistics and control of random processes,

ed. A. N. Shiryaev, pp. 177–79. Moscow: Nauka, 1989.

44. On the mathematical theory of a hydromagnetic dynamo in a random flow (in

Russian). Dokl. Akad. Nauk SSSR 293 (1987), no. 6, 1311–1314.

45. On the statistic estimation of reliability of determining aqueous solution pH by

acid-base indicator paper (with V.M. Ostrovskaja et al., in Russian). J. Analit.

Chem. USSR Acad. Sci. 42 (1987), no. 9, Part 2, 1369–1371.

46. Nonnegative L1-solutions of second order stochastic parabolic equations with

random coefficients. In Statistics and control of stochastic processes (Moscow,

1984), ed. N. V. Krylov et al. pp. 410–427. Transl. Ser. Math. Engrg., New

York: Optimization Software, 1985.

47. Filtering interpolation and extrapolation of degenerate diffusion processes.

Backward equations (in Russian). Teor. Veroyatnost. i Primenen. 28 (1983),

no. 4, 725–737.

48. Stochastic partial differential equations and diffusion processes (with N. V.

Krylov, in Russian). Uspekhi Mat. Nauk 37 (1982), no. 6, 75–95.


Preface xv

49. Characteristics of second-order degenerate parabolic Ito equations (with N. V.

Krylov, in Russian). Trudy Sem. Petrovsk. 8 (1982), 153–168.

50. Smoothness of solutions of stochastic evolution equations and the existence of

a filtering transition density (with A. Shimizu). Nagoya Math. J. 84 (1981),

195–208.

51. On the first integrals and Liouville equations for diffusion processes (with N.

V. Krylov). In Stochastic differential systems (Visegrad, 1980), pp. 117–125.

Lecture Notes in Control and Information Sci., vol. 36. New York: Springer,

1981.

52. On complete integrals of Ito equations (with N.V. Krylov, in Russian). Uspekhi

Mat. Nauk, 35 (1980), no. 4, 147.

53. A note on the strong solutions of stochastic differential equations with random

coefficients. In Stochastic differential systems. (Proc. IFIP-WG 7/1 Work-

ing Conference, Vilnius, 1978), pp. 287–296. Lecture Notes in Control and

Information Sci., vol. 25. New York: Springer, 1980.

54. Conditional distributions of degenerate diffusion processes (in Russian). Teor.

Veroyatnost. i Primenen. 25 (1980), no. 1, 149–154.

55. Ito equations in Banach spaces and strongly parabolic stochastic partial differ-

ential equations (with N. V. Krylov, in Russian). Dokl. Akad. Nauk SSSR 249

(1979), no. 2, 285–289.

56. Stochastic evolution equations (with N. V. Krylov, in Russian), Current Prob-

lems in Mathematics, vol. 14, pp. 71–147. Moscow: Akad. Nauk SSSR,

Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, 1979. English translation J.

Soviet Math. 16 (1981), no 4, 1233–1277.

57. Fundamental solutions of stochastic partial differential equations and the fil-

tering of diffusion processes (with L. G. Margulis, in Russian). Uspekhi Mat.

Nauk 33, no. 2 (1978), 197.

58. Conditional distributions of diffusion processes (with N. V. Krylov, in Russian).

Izv. Akad. Nauk SSR Ser. Mat. 42 (1978), no. 2, 356–378.

59. The Cauchy problem for linear stochastic partial differential equations (with N.

V. Krylov, in Russian). Izv. Akad. Nauk SSR Ser. Mat. 41 (1977), no. 6,

1329–1347.

60. Stochastic partial differential equations (in Russian). Mat. Sb. (N.S.) 96

(1975), no. 138, 314–341.

61. Stochastic differential equations in infinite-dimensional spaces and filtering

problems (in Russian). In Proceedings of the School and Seminar on the Theory

of Random Processes (Druskininkai, 1974), Part II, pp. 147–194. Vilnius: Inst.

Fiz. i Mat. Akad. Nauk Litovsk. SSR, 1975.

62. The Ito-Wentzell formula (in Russian). Vestnik Moskov. Univ. Ser. I Mat.

Meh. 28 (1973), no. 1, 26–32.

63. On infinite systems of stochastic differential equations that arise in the theory of

optimal nonlinear filtering (with A. N. Shiryaev, in Russian). Teor. Verojatnost.


xvi Preface

i Primenen. 17 (1972), 228–237.

64. Stochastic partial differential equations that arise in nonlinear filtering problems

(in Russian). Uspekhi Mat. Nauk 27 (1972), no. 3, 213–214.

65. The problem of “disorder” for a Poisson process (with L. I. Galtchuk, in Rus-

sian). Teor. Verojatnost. i Primenen. 16 (1971), 729–734.

Edited Volumes

1. Applied Mathematics & Optimization. Special issue on Approximation in

Stochastic Partial Differential Equations, Guest editor B. Rozovskii, Springer,

2006.

2. Stochastic partial differential equations: six perspectives. Ed. R. Carmona and

B. L. Rozovskii. Mathematical Surveys and Monographs, vol. 64. Providence,

RI: American Mathematical Society, 1998.

3. Statistics and control of stochastic processes. The Liptser festschrift: papers

from the Steklov Seminar (Moscow, 1995/1996). Ed. Yu. M. Kabanov, B. L.

Rozovskii, and A. N. Shiryaev. River Edge, NJ: World Scientific, 1997.

4. Stochastic modelling in oceanography. Ed. R. Adler, P. Muller, and B. L.

Rozovskii. Progress in Probability 39. Boston: Birkhauser, 1996.

5. Stochastic partial differential equations and their applications. Proceedings of

the IFIP WG 7/1 International Conference (Charlotte, NC, 1991). Ed. B. L.

Rozovskii and R. B. Sowers. Lecture Notes in Control and Information Sci.,

vol. 176. Berlin: Springer, 1992.

Selected Conference Proceedings

1. Chaos expansions and numerical solutions of randomly forced equations of fluid

dynamics (with T. How et al.), Proceedings of the Sixth Helenic-European

Conference on Computer Mathematics and its Applications, HERCMA 2003,

Vol. 1, ed. E. A. Lipitakis, pp. 12–22.

2. Novel Approach to Detection of “Denial-of-Service” Attacks via Adaptive

Sequential and Batch-Sequential Change-Point Detection Methods (with R.

Blazek et al.). In Proceedings of the 2nd Annual IEEE Systems, Man, and

Cybernetics Information Assurance Workshop (West Point, NY, 2004). New

York: Institute of Electrical and Electronics Engineers, 2004.

3. A New Adaptive Batch and Sequential Methods for Rapid Detection of Net-

work Traffic Changes with Emphasis on Detection of “Denial-of-Service” At-

tacks, (with R. Blazek and H. Kim). In Proceedings of the 53rd Session of

the International Statistical Institute (Seoul, 2001). New York: Physica-Verlag,

2001.

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Preface xix

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A.Tartakovsky). Technical Report CAMS-98.9.2, Center for Applied Mathe-

matical Sciences, University of Southern California, 1998.

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al., in Russian). MIASCME, 1984.

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Adler et al., in Russian). MIASCME, 1983.

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al., in Russian). MIASCME, 1981.

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in Russian). MIASCME, 1978.

14. Optimal design of experiments. Methods (with Yu. P. Alder et al., in Russian).

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16. On stochastic equations arising in filtering of Markov processes (in Russian).

Ph. D. Dissertation, Moscow State (Lomonosov) University, 1972.




Contents

Preface vii

Boris Rozovskii ix

Publications of B. L. Rozovskii xi

1. Stochastic Evolution Equations 1

N. V. Krylov and B. L. Rozovskii

2. Predictability of the Burgers Dynamics Under Model Uncertainty 71

D. Blomker and J. Duan

3. Asymptotics for the Space-Time Wigner Transform with

Applications to Imaging 91

L. Borcea, G. Papanicolaou, and C. Tsogka

4. The Korteweg-de Vries Equation with Multiplicative

Homogeneous Noise 113

A. de Bouard and A. Debussche

5. On Stochastic Burgers Equation Driven by a Fractional

Laplacian and Space-Time White Noise 135

Z. Brzezniak and L. Debbi

6. Stochastic Control Methods for the Problem of Optimal

Compensation of Executives 169

A. Cadenillas, J. Cvitanic, and F. Zapatero

xxi


xxii Contents

7. The Freidlin-Wentzell LDP with Rapidly Growing Coefficients 197

P. Chigansky and R. Liptser

8. On the Convergence Rates of a General Class of Weak

Approximations of SDEs 221

D. Crisan and S. Ghazali

9. Flow Properties of Differential Equations Driven by Fractional

Brownian Motion 249

L. Decreusefond and D. Nualart

10. Regularity of Transition Semigroups Associated to a 3D

Stochastic Navier-Stokes Equation 263

F. Flandoli and M. Romito

11. Rate of Convergence of Implicit Approximations for Stochastic

Evolution Equations 281

I. Gyongy and A. Millet

12. Maximum Principle for SPDEs and Its Applications 311

N. V. Krylov

13. On Delay Estimation and Testing for Diffusion Type Processes 339

Yu. A. Kutoyants

14. On Cauchy-Dirichlet Problem for Linear Integro-Differential

Equation in Weighted Sobolev Spaces 357

R. Mikulevicius and H. Pragarauskas

15. Strict Solutions of Kolmogorov Equations in Hilbert Spaces and

Applications 375

G. Da Prato

Author Index 391

Subject Index 393


Chapter 1

Stochastic Evolution Equations

Nicolai V. Krylov and Boris L. Rozovskii∗

The theory of strong solutions of Ito equations in Banach spaces is expounded.The results of this theory are applied to the investigation of strongly parabolicIto partial differential equations.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1 Ito Equations in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Examples of Stochastic Evolution Equations . . . . . . . . . . . . . . . . . . . . 2

1.3 Stochastic Evolution Equations with Bounded Coefficients and Linear StochasticEvolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Nonlinear Stochastic Evolution Equations . . . . . . . . . . . . . . . . . . . . . 7

1.5 Content and Organization of the Work . . . . . . . . . . . . . . . . . . . . . . . 8

2 Stochastic Integration in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Stochastic Integrals in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Ito’s Formula for the Square of the Norm . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Proof of Theorem 2.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Ito Stochastic Equations in Banach Spaces and the Method of Monotonicity . . . . . . 27

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Assumptions and the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Ito Equations in Rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Uniqueness Theorem: A Priori Estimates and Finite-DimensionalApproximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Existence of Solution and the Markov Property: Passing to the Limit by theMethod of Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Ito Stochastic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 First Boundary-Value Problem for Nonlinear Stochastic Parabolic Equations . . 57

4.3 Cauchy Problem for Linear Second-Order Equations . . . . . . . . . . . . . . . 61

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

∗This paper was originally published in Russian in 1979 (Itogi Naukt i Tekhniki, Seriya Sovre-

mennye Problemy Matematiki, Vol, 14, pp. 71–146) The English translation was first publishedin the Journal of Soviet Mathematics, Vol. 14, pp. 1233–1277, 1981, c© Plenum Publishing Co.The permission of the current copyright holder, Springer, to include the paper in this volume isgratefully acknowledged.

1


2 N. V. Krylov and B. L. Rozovskii

1. Introduction

1.1. Ito Equations in Banach Spaces

The theory of Ito stochastic differential equations is one of the most beautiful and

most useful areas of the theory of stochastic processes. However, until recently the

range of investigations in this theory had been, in our view, unjustifiably restricted:

only equations were studied which could, in analogy with the deterministic case,

be called ordinary stochastic equations. The situation has began to change in the

last 10–15 years. The necessity of considering equations combining the features of

partial differential equations and Ito equations has appeared both in the theory of

stochastic processes and in related areas.

Such equations have appeared in the statistics of stochastic processes (filtering of

diffusion process), statistical hydromechanics, population genetics, Euclidean field

theory, classical statistical field theory, and other areas. Concrete examples of

equations of this type are presented below. These equations describe the evolution

in time of processes with values in function spaces or in other words, random fields

in which one coordinate — the “time” — is distinguished.

The objective of the present work is to show how to create a unified theory which

include both ordinary Ito equations and a rather broad class of stochastic partial

differential equations.

We realize our program by considering equations of the Ito type in Banach

spaces. More precisely we consider the equation

du (t, ω) = A (u (t, ω) , t, ω) dt+B (u (t, ω)) dw (t) , (1.1)

where A(·, t, ω) and B(·, t, ω) are families of unbound operators in Banach spaces,

depending on the elementary outcome ω in a non-anticipating fashion, and w(t)

is a process with values in some Hilbert space and with independent (in time)

increments. Such equations are usually called stochastic evaluation equations.

1.2. Examples of Stochastic Evolution Equations

1.2.1. Linear Equation of Filtring of Diffusion Processes

Filtering is one of the most important problems in statics of random processes. In

essence, it consists of the following [1]. Consider a two-component process Z =

(x, y), e.g., the (n+m)-dimensional diffusion process

dx(t) = a(x(t), y(t), t)dt + b(x(t), y(t), t)dw(t),

dy(t) = g(x(t), y(t), t)dt + σ(y(t), t)dw(t),

x (0) = x0, y(0) = y0,

where dimx = n and w(t) is a standard m + n-dimensional Wiener process. It is

assumed that the component x of the process z is nonobservable. Given a function

January 29, 2007 10:48 World Scientific Review Volume - 9.75in x 6.5in RozVol

Stochastic Evolution Equations 3

f = f(x), it is required to find the best mean-square estimate of f(x(t)) on the

basis of observations of the trajectory of the observable component y up to time t.

In other words, this estimate is to be sought as a functional of the component y. It

is well known that such an estimate is the conditional expectation of f(x(t)) given

the σ-algebra Fyt generated by the values of y up to time t, i.e., E[f(x(t))|Fy

t ]. The

filtering problem consists in computing this conditional expectation.

In Ref. [2] we showed that under broad assumptions

E[f(x(t))|Fyt ] =

∫

Rn

f(x)ϕt(x)dx

(∫

Rn

ϕt(x)dx

)−1

, (1.2)

where ϕt(x) is the solution of the Cauchy problem

dϕt(x) = 1

2trDxx(bb∗ϕt(x)) −Dx(aϕt(x))dt+ [(σσ∗)−1/2gϕt(x)

+Dx((σσ∗)−1/2σb∗ϕt(x))](σσ∗)−1/2dy(t), ‡

ϕ0(x) = P (x0 ∈ dx)/dx,

Dxx is the matrix of the second derivatives, and Dx is the vector of the first deriva-

tives. This is a linear stochastic differential equation with unbounded operators of

“drift” and “diffusion”.

1.2.2. Equations of Population Genetics

One of the most important types of models in population genetics are the models

with geographic structure. These are models in which the structure of population

change not only in time but also in space (geographically). Various probabilistic

models of this sort have been proposed by Bailey [3], Crow and Kinmura [4], Malecot

[5], and others. All these models are discrete. Dawson [6] and Fleming [7] proposed

continuous (in time and space) models which are limits of the corresponding discrete

models. These works of Dawson and Fleming continue conceptually the well-known

work of Feller [8].

The equation proposed by Dawson for the mass distribution p(t, x) of the pop-

ulation has the form

dp(t, x) = a∆p(t, x)dt+ c√p(t, x) dw(t, x), (1.3)

while the equations of Fleming has the form

dp(t, x) =(∆p(t, x) + ap(t, x) − β

)dt+

√p(t, x)(1 − p(t, x))+

2dw(t, x) (1.4)

In both cases ∆ is the Laplace operator, α, β, c are constants, (α)+ = max(a, 0),

and w (t, x) is a Wiener process with values in L2

(Rd), where d is the dimension of

‡ Here ∗ is the symbol for the conjugate; the arguments x, t, and y(t) of the coefficients have beendropped.



x, and with a nuclear covariance operator. This means that w(t, x) is a stochastic

process with values in L2(Rd), such that, for every function e ∈ L2(Rd),

we(t) =

∫

Rd

w(t, x)e(x)dx

is a one dimensional Wiener process and

E(we1 (t) − we1(s))(we2 (t) −We2(s)) = (t− s)(e1, Qe2)L2(Rd),

where Q is a nuclear operator on L2(Rd) and (·, ·)L2(Rd) is the inner product in

L2(Rd); for more details, see Ref. [9]).

Wiener processes with values in Hilbert spaces are discussed in more detail in

Section 2.2 (see, in particular, Definition 2.15).

1.2.3. System of Navier-Stokes Equations with Random External Forces

In the physics literature on the theory of turbulence (see, e.g. Novikov [10], Monin

and Yagiom [11], Klyatskin [12] and the literature cited there) a model of the motion

of an incompressible fluid is considered under the action of random external forces;

the model is described by the following system of Navier-Stokes equations:

dui(t, x) =

(ν∆ui(t, x) −

3∑

k=1

uk∂ui(t, x)

∂xk− ∂p

∂xi

)dt+ dwi(t, x),

3∑

k=1

∂uk∂xk

= 0.

(1.5)

Here u = (u1, u2, u3) is the velocity vector, p is the pressure, ν, the viscosity, and

wi(t, x) are independent Wiener processes with values in a function space. For

equation (1.5) in a cylinder (0, T ) ×G, where G is a domain in R3 with boundary

Γ, the first boundary value problem has been considered:

u (t, x) |[0,T ]×Γ = 0, u(0, x) = u0 (x) .

1.2.4. Equation of the Free Field

Let L = L(Rd+1

)be the space of rapidly decreasing function on Rd+1, and let L′

be the dual space of L, that is, the Schwartz space of slowly increasing generalized

functions. We denote by G the σ-algebra in L′, generated by the cylinder sets. On

the measurable space (L′,G), it is possible to construct a probability measure ν

with the characteristic functional

Cν(η) =

∫

L′e√−1 ηων (dω) = exp

−(η,

1

2

(−∆t,x +m2

)−1η

)

L2(Rd+1)

,

where η ∈ L, ∆t,x =∑d

i=1∂2

∂x2i

+ ∂2

∂t2 , m is a number, and ηω is the value of the

functional ω ∈ L′ on η ∈ L.



It is known (see, e.g., the monograph of Simon [13]) that the free field ξ is one

of the simplest objects of relativistic quantum mechanics; in the Euclidean model it

can be interpreted as a canonical generalized random field on the probability space

(L′,G, ν): ξ (ω, t, x) = ω(t, x) for each ω ∈ L′.Further, let w be a generalized white noise, i.e., the canonical generalized random

field on the probability space (L′,G, µ), where µ is the Gaussian measure with the

characteristic operator

Cµ(η) = exp

(−1

2‖η‖2

L2(Rd+1)

);

see e.g. Ref. [9].

Hida and Strett [14] showed that the Euclidean free field ξ(t, x) is a stationary

in t solution of the equation

∂ξ(t, x)

∂t= −

√−∆x +m2 ξ(t, x) + w(t, x), (1.6)

where

∆x =

d∑

i=1

∂2

∂x2i

and the operator√−∆x +m2 is understood in the sense of the theory of generalized

functions. Regarding this equation, see also the survey of Dawson [15]. The work of

Albeverio and Hoegh-Krohn [16] is a good example of stochastic evolution equations

in Euclidean field theory.

The above examples present only a small fraction of stochastic evolution equa-

tions considered in recent years. We have selected these examples because they

have been studied in detail at a mathematical level of rigor.

Modern physics journals are an inexhaustible source of stochastic evolution equa-

tions of the most varied type which are studied only at a physical level of rigor.

1.3. Stochastic Evolution Equations with Bounded Coefficients and

Linear Stochastic Evolution Equations

The impetus for the first mathematical investigations in the area of stochastic evo-

lution equations were not, however, the demands of physics or biology but rather

the inner requirements of mathematics, viz., of the theory of differential equations

with variational derivatives. In the mid-sixties, Daletskii and Baklan [17–19] stud-

ied stochastic evolution equations with the goal of constructing a solution of the

Cauchy problem for the Komogorov equation in variational derivatives

−∂F (x, t)

∂t=

1

2tr[B∗(x, t)F ′′(x, t)B(x, t)] +A(x, t)F ′(x, t), t ≤ T, F (x, t) = φ(x).

In these works, a probabilistic method of constructing solutions of the Kolmogorov

equation, which is well known in the finite-dimensional case, was used; the method



consists in writing the solution in the form F (x, t) = E [Φ(u(T ))|u(t) = x], where

u(t) is a solution of the stochastic evolution equation (1.1). To carry out this

program, it was necessary to study stochastic equations, and Refs. [17–19] were

among the first to address this issue. In these works,B is assumed to be bounded (as

an operator) and to satisfy a Lipschitz condition, while A is either the infinitesimal

generator of some contractive semigroup or is also bounded and Lipschitz. This

direction was further developed by the authors themselves and their students in

Refs. [20–24].

Moreover, a number of works [25–32] appeared in which linear stochastic evo-

lution equations were studied for other applications, such as control, filtering, and

extrapolation of linear stochastic Ito partial differential equations. From the point of

view of proving the existence of a solution, these works are very similar to Ref. [18]:

the operator A(s) is always assumed to generate a semigroup Ts,t, and the operator

B is bounded and satisfies a Lipschitz condition (we remark that these conditions

are often not satisfied for the filtering equations). In the above works, the following

equations was considered:

u (t) = T0,tu0 +

∫ t

0

Ts,tB(u(s))dw(s). (1.7)

The proof of existence and uniqueness of solution of this equation is accomplished

simply by the method of contraction mappings, since Ts,t is a bounded operator. It

is then proved, or simply taken for granted, that, under additional conditions the

solution of equation (1.7) belongs to the domain of the operator A, and equation

(1.1) is equivalent to (1.7). It should be noted that the conditions for the equivalence

of equations (1.1) and (1.7) obtained in these works are rather burdensome to be

useful in application to stochastic Ito partial differential equations, and the reason

is similar to the deterministic case. Indeed, the theorems on the solvability of

inhomogeneous parabolic equations obtained e.g., by methods of potential theory

(see Chapter 1 in Ref. [33]) are much stronger than their counterparts obtained by

the theory of inhomogeneous semigroups for linear operator equations [34].

Methods of potential theory in application to linear equations of type (1.1)

were used by Rozovskii and Margulis [35, 36]. In these works it is assumed that

the “diffusion” operator has order zero, while the “principal part” of the “drift”

operator does not depend on the elementary outcome. In the first work, the unique

solvability of the Cauchy problem is proved, while in the second a fundamental

solution of the equation is constructed, and precise estimates of it in the Holder

classes are obtained. By modifying the method of Ref. [35], Shimizu [37] proved the

unique solvability of the second boundary-value problem for the same equation as

in Ref. [35].

We mention several works of qualitative character. Markus [38] investigated

stationary solutions of linear equations, Shimizu [39] proved some comparison the-

orems, and Mahno [40] studied the averaging principle for equations of type (1.7).

Bensoussan [41] used a completely different idea to construct solutions of a linear



equations of type (1.1) for B ≡ 1. In contrast to all the works cited above, he did

not assumed that A generates a semigroup; this assumption was replaced by a

coercivity condition. To construct the solution, Bensoussan utilized the method of

time discretization. The coercivity condition ensures that the corresponding discrete

equation is easily solvable and the solution satisfies the necessary estimates; a weak

limiting procedure is then carried out. In the deterministic case, this method was

applied earlier by Lions [42].

1.4. Nonlinear Stochastic Evolution Equations

The important program started by Bensoussan in Ref. [41] was continued in his

joint paper with Temam [43], where the method of discretization was applied to an

equation with a nonlinear “drift” operator. It was assumed that B = 1, while the

operator A satisfies a monotonicity condition.

Theory of monotone operators is one of the most beautiful areas of modern

nonlinear analysis. The foundations of this theory were laid in the works of Vain-

berg, Kachurovskii, Minty, and Browder [44–47]. Further development is reflected

in the monographs [42, 48–50]. Application of this theory to nonlinear elliptic

and parabolic equations are described in Ref. [51]. The work of Bensoussan and

Temam [43] generalizes one of the methods from the monograph of Lions [42].

The method of monotonicity in application to stochastic evolution equations

was further developed in the works of Pardoux [52, 53]. In these works a general

stochastic evolution equation is considered with unbounded nonlinear operators of

“drift” and “diffusion.” As a special case, the result of Pardoux contains the results

of [41, 43, 54], as well as the deterministic situation described by Lions [42, Section

2.1]. The results of [25–27, 29–32] are also essentially covered, although here a

region where they are formally distinct can be indicated: the situation is analogous

to the difference between the theory of differential equations in divergence and non-

divergence forms.

The solution obtained in Ref. [53] belongs to the domains of the operators A and

B and is measurable with respect to the σ-algebra generated by the Wiener process

and the initial condition. The solution is constructed on a prescribed probability

space; i.e., in correspondence with the terminology of stochastic ordinary differential

equations, it is a strong solution.

It should be noted that equations (1.3), (1.4), (1.5), and (1.6) do not satisfy

the assumptions of Paradox; in fact, solutions of the equations (1.3), (1.4), (1.5)

are constructed in the sense of (probabilistic) distributions. The works [55–61] are

devoted to an investigation of the Navier-Stokes equation (1.5), while Eqs. (1.3) and

(1.4) are investigated in Refs. [15, 61]. In these works, a measure is sought which is

supported on the trajectories of solutions and is a solution of either the Kolmogorov

or the Hopf equation (formally) associated with the equation in question.



1.5. Content and Organization of the Work

In the present work, we generalize the results of Pardoux. In considering the same

situation as in Ref. [53], we admit dependence of the coefficients on the elementary

outcome in a nonanticipating way. We show that certain conditions of Pardoux

are superfluous; in particular, this is the case of the local Lipschitz condition on

the “diffusion coefficient.” The Markov property of the solution (in t) is proved for

equations with deterministic coefficients. On the basis of general results regarding

the solvability of stochastic evolution equations, we investigate quasilinear stochastic

Ito partial differential equations (of any order) which satisfy the so-called condition

of strong probability. In the deterministic case (B ≡ 0) this condition coincides

with the well-known condition of strong parabolicity of Vishik [62]. In the linear

case, we introduced the condition of strong parabolicity [63] and systematically

investigated it [64]. An analogous condition in the linear situation was introduced

also by Pardoux [53].

We point out that in the finite-dimensional case our results generalize somewhat

Ito’s classical theorem on the strong solvability of stochastic equations with random

coefficient satisfying Lipschitz conditions [34, 65] (see Example 3.15).

The paper is organized as follows. Aside from the introduction, it contains three

big sections, each with its own introduction outlining the content and organization.

Section 2 is devoted to the theory of stochastic integration in Hilbert spaces. In

Section 2.2, the concept of a martingale and of a Wiener process in a Hilbert space is

introduced, and stochastic integrals over these processes are described. In Sections

2.3 and 2.4, Ito’s formula for the square of the norm of a semimartingale in a rigged

Hilbert space is proved. This result plays an extremely important role in the entire

theory.

Section 3 is devoted to the proof of the main theorems about solvability and

regularity of solutions of stochastic evolution equations.

In Section 4, the results of Section 3 are applied to the Ito stochastic partial

differential equations.

Sections 2.3 and 2.4, where Ito’s formula for the square of the norm of a semi-

martinagle in a Hilbert space is establishes, and Section 3.4, where the finite-

dimensional case is considered, may be of independent interest for some readers.

The exposition and notations in these sections are independent of the remainder of

the work. On the other hand, we go much further here than is required by the rest

of the text, so that a reader interested in a rapid acquaintance with the ideas of

Ito’s stochastic partial differential equations should concentrate on the statements

of the main results in Sections 2.3, 2.4, and 3.4, while skipping most of the proofs.



2. Stochastic Integration in Hilbert Spaces

2.1. Introduction

The theory of stochastic integration in infinite-dimensional spaces is a broard and

rapidly developing area of the theory of stochastic processes. A complete survey

of this area is beyond the purpose of the present work, and we will discuss only

“selected” questions of the theory of stochastic integration in Hilbert spaces which

we use directly in the sequel.

There are two groups of these “selected” questions:

a) construction of a stochastic integral over a square-integrable martingale with

values in a Hilbert space;

b) derivation of Ito’s formula for the square of the norm of a semimartinagale

is a rigged Hilbert space.

In Section 2.2 we consider questions of group a) in a very simple situation —

the integration with respect to a continuous martingale. This section is entirely of

survey character and there are practically no proofs given. It is based on Refs. [32,

66, 67].

The problem formulated in part b) occupies a central spot and is solved in

Sections 2.3 and 2.4. It is assumed that a Hilbert space H is rigged by a pair of

Banach spaces V and V ′, i.e. there are the following dense embeddings: V ⊂ H ⊂V ′), and we consider a “semimartingale” of the form

v (t) =

∫ t

0

v′ (s) ds+m (t) ,

where v ∈ V, v′ ∈ V ′, and m(t) is a martingale in H . Ito’s formula is derived for

‖v(t)‖2H .

The derivation of Ito’s formula for the square of the norm of a semimartingale

with all components take value in a single Hilbert space differs little from the finite-

dimensional case. However, in passing from this situation to the one described

above, the jump in complexity is comparable to the jump in complexity in passing

from bounded to unbounded operators. This comparison is more apt than is appar-

ent at a first glance. We will see below that evolution equations with unbounded

operators cannot be considered in a single space — it is necessary to separate the

domain of the operators (the space V ) from the range of the operators (the space

V ′ or H). This situation was first considered by Pardoux [53]. Using entirely dif-

ferent methods, we generalize his result to non-reflexive and non-separable spaces

and provide a much sorter proof.

Ito’s formula for the square of the norm is essentially used in Section 3 to derive

a priory estimates and to prove uniqueness and continuity of the solution.

To conclude the section we give a brief and, of course, incomplete survey of

existing results related to stochastic integrals in Hilbert spaces.



The first important work in this direction was apparently the work of Daletskii

[17], where he constructed a Wiener process with an identity covariance operator

in a Hilbert space, or, more precisely, in a certain nuclear extension of this space,

and defined a stochastic integral. Later, on the basis of the work of Gross [68, 69],

Kuo [70] studied a stochastic integral with respect to an abstract Wiener process in

a Banach space. The results of Daletskii were also extended in Refs. [71–73], and

others.

Kunita [66] initiated the study of the problem of integrability with respect to

a square-integrable martingale in a Hilbert space. Metivier and his student sub-

sequently achieved considerable progress in this direction [31, 32, 40, 73–75]. The

important work of Meyer [67] should also be mentioned.

The interested reader can systematically study the subject using Refs. [40, 67,

70, 74, 76].

2.2. Stochastic Integrals in Hilbert Spaces

We begin this section by recalling and discussing some very basic concepts and

results about mappings of measurable spaces into Banach spaces.

Let (S,Σ, µ) be a complete measure space, and let (X,X ) be a Banach space

with the σ -algebra X of Borel sets relative to the strong topology. We denote by

X∗ the Banach space dual to X and by xx∗ the value of a functional x∗ ∈ X∗ on

x ∈ X .

A mapping (function) x : S → X is called measurable if, for each Γ ∈ X ,

s : x(s) ∈ Γ ∈ Σ.

A mapping x : S → X is called weakly measurable if for each x∗ ∈ X∗, x(s)x∗

is a measurable mapping of S into R.

A mapping x : S → X is called strongly measurable if there exists a sequence of

measurable simple functions converging to x(s), µ-almost surely.

It is clear that the concept of strong measurability coincides with the concept

of measurability if X is separable.

Theorem 2.1 (Pettis). A mapping x : S → X is strongly measurable if and only

if it is weakly measurable and there exists a set B ⊂ Σ such that µ(B) = 0, while

the set of values of x(s) on S \B is separable. In particular, if X is separable, then

the concept of weak and strong measurability coincide.

Proof. See Yosida [51].

Let (Ω,F , P ) be a probability space with an expanding system of σ-algebras

Ft ⊂ F , t ∈ R+ = [0,∞). We assume that F0 is complete with respect to the

measure P .

Definition 2.2. A random variable in X is a measurable mapping of (Ω, F, P ) into

X .



Definition 2.3. We say that x = x(t, ω) is an Ft-consistent stochastic process in

X if, for every t ≥ 0, x(t, ·) is a measurable mapping of (Ω,Ft, P ) into X .

Definition 2.4. Completely measurable sets are the subsets of [0,∞) × Ω that

are elements of the smallest σ-algebra relative to which all real Ft-consistent right

continuous process without discontinuities of the second kind are measurable in

(t, ω). A mapping (process) x : [0,∞) × Ω → X is called completely measurable if

for any Borel set Γ ⊂ X the set (t, ω) : x(t, ω) ∈ Γ is completely measurable.

In this paper we must sometimes consider random variables with values in em-

bedded Banach spaces. Let V and H be two separable Banach spaces, where V

is a subset of H and the embedding operator assigning to an element v ∈ V the

corresponding element v ∈ H is a continuous operator from V to H.

Lemma 2.5. a) If x is a random variable with values in V (relative to the Borel

σ-algebra in V ), then x is a random variable with values in H.

b) If x is a random variable with values in H, then ω : x(ω) ∈ V ∈ F .

Proof. Assertion a) of the lemma is a consequence of the measurability of a

superposition of measurable mappings. Assertion b) follows from the completeness

of F and the fact that the continuous image of a Borel (in V ) set of V is analytic

and is hence universally measurable in H ; cf. Ref. [77].

We now define a martingale with values in a real Hilbert space H and the

stochastic integral over such a martingale. We will consider only martingales m =

m(t) that are strongly continuous in t and have a separable range. We temporarily

denote by H1 the closed linear hull of the range of m(t, ω), t ≥ 0, ω ∈ Ω.

If h(t, ω) ⊥ H1, then, for all (t, ω), it is natural to set∫ t0 h (s) dm (s) = 0.

Therefore, in view of the orthogonal decomposition of H into H1 and HL1 , it suffices

to study integration of functions with values in H1. These arguments make the

following assumption natural, and we adopt it to the end of the section: H is a

separable Hilbert space, identified with its dual in the natural way.

For h1,h2 ∈ H we denote by h1h2 the scalar product of h1, h2; h21 = h1h1, |h1| =

(h21)1/2.

For random variables in H with finite expectation of the norm, it is possible

to define the conditional expectation in complete analogy to the finite-dimensional

case. Namely, let G be a sub-σ-algebra of F and let x be a random variable in H

with E|x| <∞.

Definition 2.6. The conditional expectation of x given G is the random variable

E(x|G) in H such that, for every y ∈ H ,

yE(x|G) = E(yx|G) (a.s.).

It is clear that the random variable so defined in unique (a.s.).



Definition 2.7. A stochastic process x in H is a martingale relative to the family

Ft if

a) x is Ft-consistent,

b) E|x(t)| <∞ for all t ≥ 0,

c) E[x(t)|Fs] = x(s) (a.s.) for all 0 ≤ s ≤ t.

The next theorem follows immediately from Definitions 2.6 and 2.7, and the

equivalence of the three concepts of measurability in separable spaces.

Theorem 2.8. A stochastic process x(t) in H with E|x(t)| < ∞ for all t ≥ 0 is a

martingale relative to the family Ft if and only if, for every y ∈ H, the stochastic

process yx(t) is a one-dimensional martingale relative to Ft.

Definition 2.9. As in the finite-dimensional case, we say that a stochastic process

x(t) in H is a local martingalea and write x ∈ Mloc(R+,H) if there exists a sequence

of stopping times τn ↑ ∞ (a.s.) such that the process x(t ∧ τn) is a martingale for

each n. We call the sequence τn a localizing sequence.

For simplicity we henceforth consider only (strongly) continuous martingales

and local martingales.

We denote by Mcloc(R+,H) the collection of all continuous local martingales in

H issuing from zero. It is easy to see that the following results holds.

Theorem 2.10. If x ∈ Mcloc(R+,H), then there exists a sequence τ ′n of stopping

times localizing x for which E supt≥0 |x(t ∧ τ ′n)|2 < ∞. If E|x(t)|2 < ∞ for some

t ≥ 0, then E sups<t |xs|2 ≤ 4E|xt|2.

We assume from now on that, for every local martingale, the localizing sequence

τn from Definition 2.9 coincides with the sequence τ ′n from Theorem 2.10.

The next theorem plays an important role in the construction of a stochastic

integral with respect to x ∈ Mcloc(R+, H).

Theorem 2.11. If x ∈ Mcloc(R+, H), then |x(t)|2 is a local submartingale.

Proof. This follows from the equality

E(|x(t ∧ τn) − x(s ∧ τn)|2|Fs

)= E

[(|x(t ∧ τn)|2|Fs

)− |x(s ∧ τn)|2,

which, in turn, follows immediately from the martingale property of the Fourier

coefficient of x in a basis in H .

Definition 2.12. An increasing process 〈x〉t for x ∈ Mcloc(R+, H) is the increasing

process in the Doob-Meyer decomposition of |x(t)|2.

From the Doob-Meyer theorem it follows that 〈x〉t is a.s. uniquely defined and

continuous in t.aHere and below we consider martingales and local martingales only relative to the family Ft.



As in the finite-dimensional case, if x, y ∈ Mcloc(R+, H), then we set

〈x, y〉t =1

4

(〈x+ y〉t − 〈x− y〉t

).

It is easy to verify that if the first assertion of Theorem 2.10 holds with the same

sequence τn for both x(t) and y(t), then, for each t, s ≥ 0, t ≥ s, and each n, we

have

E((x(t ∧ τn) − x(s ∧ τn))(y(t ∧ τn) − y(s ∧ τn))|Fs

)

= E(〈x, y〉t∧τn − 〈x, y〉s∧τn |Fs

)(a.s.)

Theorem 2.13 (Burkholder’s Inequality). If x ∈ Mcloc(R+, H) and τ is an a.s.

finite stopping time, then

E supt≤τ

|x(t)| ≤ 3E〈x〉1/2τ .

A proof of this theorem can be found, e.g., in Ref. [77].

In H we fix an orthogonal basis hi, i ≥ 1 and set xi(t) = hix(t). It is known

that, almost surely for each i, j, the measure on [0,∞) generated by the process

〈xi, xj〉t is absolutely continuous with respect to the measure generated by 〈x〉t.For further exposition of the theory of martingales, we need the following con-

struction. Let E be a separable Hilbert space which is naturally identified with

its dual, let ei i ≥ 1 be an orthonormal basis in E, let L(H,E) be the space of

continuous linear operators from H to E, and let L2(H,E) be subspace of L(H,E)

consisting of all Hilbert-Schmidt operators. It is known that L2(H,E) is a separable

Hilbert space with the norm

‖B‖ =

(∑

i

|Bhi|2)1/2

=

∑

i,j

(ej , Bhi)2

1/2

,

and ‖B‖ does not depend on the choice of bases in H and E.

Given a symmetric nonnegative nuclear operator Q in L(H,H), we denote by

LQ(H,E) the set of all linear (bounded or unbounded) operators B defined on

Q1/2H , taking Q1/2H into E and having the property BQ1/2 ∈ L2(H,E).

For B ∈ LQ(H,E) we define |B|Q = ‖BQ1/2‖. It is known that if B ∈ L2(H,E),

then |B| ≤ ‖B‖,

B ∈ LQ(H,E), and |B|Q ≤ |B|(trQ)1/2.

We return to x ∈ Mcloc(R+, H). It can be shown that there exists a completely

measurable process Qx = Qx(t, ω) with values in L2(H,H) such that, for all (t, ω),

the operator Qx(t, ω) is symmetric, nonnegative, and nuclear, trQx(t, ω) = 1, and

hiQx(t, ω)hj =d〈xi, xj〉td〈x〉t

(dP × d〈x〉t a.s.)



for all i, j and any basis hi, where dP × d〈x〉t is the differential of the measure

µ(A) = E

∫ ∞

0

χA

(t, ω)d〈x〉t,

defined on the product of F and the Borel σ-algebra on [0,∞). We call the process

Qx the correlation operator of x.

If B(t) is a completely measurable process in L2(H,E) and

E

∫ t

0

‖B(s)‖2d〈x〉s <∞ (a.s.)

for any t ≥ 0, then there exists a square-integrable martingale y(t) in E which is

strongly continuous in t and such that, for every orthonormal basis hi and every

v ∈ E, T ≥ 0,

limn→∞

E supt≤T

∣∣∣∣∣v y(t) −n∑

i=1

∫ t

0

vB(s)hid(hix(s))

∣∣∣∣∣

2

= 0.

Two processes possessing this property obviously coincide a.s. for all t. It is

therefore correct to write

y(t) =

∫ t

0

B(s)dx(s). (2.1)

To compute y(t) we fix bases in H,E and set

y(t) =

∞∑

i=1

eiyi(t), yi(t) =

∞∑

j=1

∫ t

0

eiB(s)hjd(hjx(s)),

where the series converges in L2(Ω, C([0, T ], E)), i.e. in the mean-square, uniformly

on each finite time interval. By direct computation,

〈y〉t =

∫ t

0

|B(s)|2Qs(s)d〈x〉s, (2.2)

and this together with the inequality |B|Qx ≤ ‖B‖ leads to an extension of the

stochastic integrals (2.1) to completely measurable functions B(s) such that, for all

t ≥ 0,∫ t

0

‖B(s)‖2d〈x〉s <∞.

This extension is done in the usual way and preserves the property (2.2). The re-

sulting stochastic integral is continuous in t and is a local martingale. The following

theorem shows that it is possible to extend the concept of a stochastic integral to

an even larger class of processes B(s).

Theorem 2.14. Let B = B(s, ω) ∈ LQx(s,ω)(H,E) be such that BQ1/2x is a com-

pletely measurable process in L2(H,E), and for each t the right-hand side of (2.2)



is a.s. finite. Then the sequence

yn(t) =

∫ t

0

B(s)Q1/2x (s)

(1

n+Q1/2

x (s)

)−1

dx(s)

converges in probability, uniformly with respect to t, to a limit denoted by y(t).

Moreover, y(t) ∈ Mcloc(R+, E) and equality (2.2) holds.

Proof. Note that Bn(s) = B(s)Q1/2x (s)(1/n + Q

1/2x (s))−1 is a completely mea-

surable process with values in L2(H,E) and

‖Bn(s)‖ ≤ n|B(s)|Qx(s),

|Bn(s) −Bm(s)|2Q =∑

i≥1

|B(s)Q1/2x (s)ei(s)|2

(ai(s)

1n + ai(s)

− ai(s)1m + ai(s)

)2

,

where ei(s) are the eigenvectors of Qx(s) and a2i (s) are the corresponding eigenval-

ues. It is evident from this that |Bn(s)−Bm(s)|Q ≤ |B(s)|Q and |Bn(s)−Bm(s)|Q →0 as n,m→ ∞. Hence,

〈yn − ym〉t =

∫ t

0

|Bn(s) −Bm(s)|2Qx(s)d〈x〉s → 0,

and the assertions of the theorem are deduced in the well-known way.

Stochastic integrals for functions in LQx(H,E) have previously been defined in

Ref. [32], but the construction presented here is somewhat different.

By definition, the process y(t) in Theorem 2.14 is taken equal to the right-hand

side of (2.1). If X is a separable Hilbert space and A ∈ L(E,X), then for all t (a.s.)

Ay(t) =

∫ t

0

AB(s)dx(s). (2.3)

We choose an element e ∈ E and by means of it define an operator e ∈ L(E,R)

by the formula ey = ey, where ey is the scalar product in E. From (2.3) we then

have

ey(t) =

∫ t

0

eB(s)dx(s).

We observe that the operator eB(s) acts on Q1/2x H by the formula h→ eB(s)h,

while the latter is equal to (B∗e, h) if B ∈ L(H,E). Finally, if h(s) ∈ H is com-

pletely measurable, and for all t ≥ 0,∫ t

0

|h(s)|2Qx(s)d〈x〉s =

∫ t

0

|Q1/2x (s)h(s)|2d〈x〉s <∞ (a.s.),

then we write∫ t

0

h(s)dx(s) =

∫ t

0

h(s)dx(s).

We now introduce the concept of a Wiener process in H .



Definition 2.15. Let Q be a nuclear symmetric nonnegative operator on H with

trQ <∞. A Wiener process relative to Ft in H with covariance operator Q is a

continuous martingale w(t) with values in H , correlation operator (trQ)−1Q, and

such that w(0) = 0, 〈w〉t = trQ · t.

It is known that, for every suitable probability space and every nuclear symmet-

ric nonnegative operator Q on H with trQ > 0, there exists a Wiener process with

covariance operator Q. Clearly, Ew2(t) = trQ · t. Stochastic integrals with respect

to a Wiener process possess especially good properties. For example, they are de-

fined not only for completely measurable B(s) but also for operators measurable in

(s, ω) that are Fs-consistent and satisfy∫ t0 |B(s)|2Qds <∞ (a.s.) for all t ≥ 0.

We conclude this section with a remark that, in place of an infinite time interval

above, we could consider a segment of the form [0, T ]. To have the possibility of

doing this, it suffices to extend the processes in question to t ≥ T by setting them

equal to the value they assume at t = T .

2.3. Ito’s Formula for the Square of the Norm

Let V be a Banach space, let V ∗ be the dual space of V , and let H be a Hilbert

space (we assume that all three are real spaces). If v ∈ V (h ∈ H, v∗ ∈ V ∗), then

|v| (resp. |h|, |v∗|) denotes the norm of v (h, v∗) in V (H,V ∗); if h1, h2 ∈ H , then

h1h2 denotes the scalar product of h1, h2; the result of the action of a functional

v∗ ∈ V ∗ on an element v ∈ V is written as either vv∗ or v∗v. Let Λ be a bounded,

linear operator acting from V to H such that ΛV is dense in H .

We consider three processes v(t, ω) ∈ V, h(t, ω) ∈ H, v∗(t, ω) ∈ V ∗ defined

for t ≥ 0 on some complete probability space (Ω,F , P ) and connected with some

expanding family of complete σ-algebras Ft ⊂ F , t ≥ 0, in the following way:

(1) v(t, ω) is strongly measurable in (t, ω) and is weakly Ft-measurable in ω for

almost all t;

(2) for every g ∈ V the quantity gv∗(t, ω) is Ft- measurable in ω for almost every

t and is measurable in (t, ω);

(3) h(t, ω) is strongly continuous in t, is strongly measurable in ω relative to Ftfor each t, and is a local semimartingale: there exist strongly Ft-measurable

continuous processes A(t),m(t) in H such that m(t) is a local martingale, the

trajectories A(t, ω) for each ω have a finite variation on bounded time intervals,

and h(t) = A(t) +m(t).

We fix p ∈ (1,∞) and set q = p/(p−1). We assume that |v(t)| ∈ Lp ([0, T ]) (a.s.)

for all T ≥ 0 and there exists a function f(t, ω) measurable in (t, ω) such that f(t) ∈Lq ([0, T ]) (a.s.) for all T ≥ 0 and |v∗(t)| ≤ f(t) for all (t, ω). Regarding the last

condition, it is useful to note that |v∗(t)| is, generally speaking, not measurable. This

norm is measurable, e.g., if V is separable, and in this case |v∗(t)| ∈ Lp([0, T ])(a.s.)

for all T ≥ 0.



We formulate the main result regarding Ito’s formula.

Theorem 2.16. Let τ be a stopping time and suppose that, for every g ∈ V almost

everywhere on the set (t, ω) : t < τ(ω),

(Λg)(Λv(t)) =

∫ t

0

gv∗(s)ds+ (Λg)h(t). (2.4)

Then there exist a set Ω′ ⊂ Ω and a function h(t) with values in H such that

a) P (Ω′ = 1), h(t) is strongly Ft- measurable on the set ω : t < τ(ω) for any

t, h(t) is continuous in t on (0, τ(ω)) for every ω, and Λv(t) = h(t) (a.s. on

(τ, ω) : t < τ(ω);b) for ω ∈ Ω′ and t < τ(ω)

h2(t) = h2(0) + 2

∫ t

0

v(s)v∗(s)ds+ 2

∫ t

0

h(s)dh(s) + 〈m〉t; (2.5)

c) if V is separable, then, for ω ∈ Ω′, t < τ(ω), g ∈ V

(Λg)h(t) =

∫ t

0

gv∗(s)ds+ (Λg)h(t); (2.6)

d) if V is separable and (2.4) is satisfied for some t ≥ 0 and each g ∈ V (a.s.)

on ω : t < τ(ω), then Λv(t) = h(t) (a.s.) on ω : t < τ(ω).

We take up the proof of this theorem after discussing its hypotheses and asser-

tions.

The stochastic integral in (2.5) exists if h(s) is completely measurable and, for

t < τ(ω),∫ t

0

|h(s)|d‖A‖s +

∫ t

0

|h(s)|2d〈m〉s <∞,

where

‖A‖s = limn→∞

∞∑

k=0

∣∣∣∣A(s ∧ k + 1

2n

)−A

(s ∧ k

2n

)∣∣∣∣ .

Both these conditions are satisfied because h(s) is continuous in s and is Fs-consistent, while 〈m〉t + ‖A‖t < ∞. We point out that by a stochastic integral

we always understand a continuous (for all ω) process.

Further, since v(s) is strongly measurable, while v∗(s) is weakly measurable,

v(s)v∗(s) is measurable in (s, ω), and by our assumptions it is locally integrable in

s (a.s.) All expressions in (2.5) are therefore meaningful.

Assertion d) is a simple corollary of the preceding assertions. Indeed, from (2.4)

and (2.6) we have (Λg)h(t) = (Λg)Λv(t) (a.s.) on ω : t < τ(ω) for every g ∈ V .

But V is separable, and therefore (Λg)h(t) = (Λg)Λv(t) for all vg ∈ V , a.s. on

ω : τ < τ(ω). Since ΛV is dense in H , assertion d) follows.



We now show that it is enough to prove the theorem for separable H and V

Indeed, since process v(t) is strongly measurable in (t, ω), there exists a process v′(t)which coincides with v(t) for almost all (t, ω) and has its range in some separable

subspace V ′ ⊂ V . It is shown similarly that on a set of full probability Ω′′ all

the values of h(t, ω), t ≥ 0, ω ∈ Ω′′ lie in some separable subspace H ′ ⊂ H . We

may assume with no loss of generality that Ω′′ = Ω. By hypothesis, ΛV is dense

in H . Hence, there exists a separable subspace V ′′ ⊂ V such that H ′ ⊂ ΛV ′′.Suppose now that V1 is the closed space spanned by V ′ ∪ V ′′ and H1 is the closed

space spanned by ΛV1. The spaces V1, H1 are separable, and ΛV1 is dense in H1.

Further, v′(t) ∈ V1, h(t) ∈ H1, while the functionals v∗(t) on V are also functionals

on V1. It may therefore be assumed that v∗(t) ∈ V ∗1 and the norm of v∗(t) in

V ∗1 is no larger than |v∗(t)|V ∗ . Relation (2.4) is preserved for every g ∈ V1 (even

for g ∈ V ), because v(t) = v′(t) a.s. in (t, ω). Hence, if Theorem 2.16 is true for

separable V and H , then we obtain the fist assertion in the general case by applying

it to V1, H1.

We may thus assume with no loss of generality until the end of the section that

V and H are separable.

We will now explain why (2.5) is called Ito’s formula for the square of the norm.

For this purpose we place all processes v(t), h(t), v∗(t) in a single space.

In those cases where the same vector belongs to various spaces we equip its norm

with the symbol of the space in which it is considered.

Suppose that the space V is a (possibly, non-closed) subspace of H , is dense in

H in the norm of H , and |ϕ|H ≤ N |ϕ|V for all ϕ ∈ V , where N does not depend

on v. Suppose that H is, in turn, a subspace of some Banach space V ′ and that H

is dense in V ′. Then

V ⊂ H ⊂ V ′. (2.7)

We assume that the scalar product in H possesses the following property: if

ϕ ∈ V , ψ ∈ H , then |ϕψ| ≤ |ϕ|V |ψ|V ′ . Since the embeddings in (2.7) are dense, it

is possible to uniquely define ϕψ for ϕ ∈ V , ψ ∈ V ′ as limn→∞ ϕψn, where ψn ∈ H

and |ψ − ψn|V ′ → 0. Obviously, for a fixed ψ ∈ V ′, the mapping ϕ 7→ ϕψ is a

bounded linear functional on V . We suppose that for ψ ∈ V ′ the equality ϕψ = 0

for all ϕ ∈ V implies that ψ = 0. Then the mapping which, to every ψ ∈ V ′ assigns

the corresponding functional ϕ 7→ ϕψ on V , is a one-to-one mapping of V ′ into

some subset of the dual space V ∗ of V . We now make an additional assumption

that every bounded linear functional on V has the form ϕ 7→ ϕψ for some ψ ∈ V ′,and then the mapping V ′ → V ∗ mentioned above becomes both one-to-one and

onto so that V ′ can be identified with V ∗ if desired. We note that under this

identification the norms of ψ as an element of V ′ and as an element of V ∗ are, in

general, different, and in this section we will not identify V ′ with V ∗b. Finally, we

suppose that that there is a function v′(t, ω) with values in V ′ such that, for every

bOn the contrary, in the following two sections we will make this identification.



g ∈ V , gv′(t) is measurable in (t, ω) and, for every t ≥ 0, is Ft-measurable in ω,

and also |v′(t)|V ′ ≤ f(t) for all (t, ω), where the function f is the same as on page

16. Then, for every g ∈ V , t ≥ 0, we have∣∣∣∣∫ t

0

gv′(s)ds

∣∣∣∣ ≤ |g|V∫ t

0

f(s)ds.

Therefore, from the properties of f(t), it follows that there exists a set Ω′ ⊂ Ω such

that P (Ω′) = 1 and for ω ∈ Ω′, t ≥ 0, the mapping g 7→t∫0

gv′(s)ds is a bounded

linear functional on V . Under our assumptions, this functional can be written in

the form g 7→ gψ(t), where ψ(t) ∈ V ′, and, for ω ∈ Ω′ we have

ψ(t) =

t∫

0

v′(s)ds.

Theorem 2.16 then takes the following form, where, for simplicity of the formu-

lation, we take τ = ∞; generalization to the case of arbitrary τ is obvious.

Theorem 2.17. For ω ∈ Ω′ and t ≥ 0 we define

h(t) =

∫ t

0

v′(s)ds+ h(t), (2.8)

and suppose that h(t) = v(t) for almost all (t, ω). Then there exists a set Ω′′ ⊂ Ω′

such that P (Ω′′) = 1 and, for ω ∈ Ω′′, the function h(t) takes values in H, is

continuous in H with respect to t, is strongly Ft-measurable with respect to ω for

each t (as a function with values in H), and

h2(t) = h2(0) + 2

∫ t

0

v(s)v′(s)ds+ 2

∫ t

0

h(s)dh(s) + 〈m〉t . (2.9)

We note that relation (2.9) is obtained if the rules for computing the stochastic

differential dh2(t) for the process h(t) defined in (2.8) are applied. One of the

difficulties in justifying (2.9) is that it is generally not clear why h(t) ∈ H (in other

words, why h2(t) exists), because equation (2.8) only defines h(t) as a process with

values in V ′, and it is generally not true that h(t) ∈ V for all (t, ω).

Proof. [of Theorem 2.17 using Theorem 2.16] We take for Λ the identify operator

and use the fact that the formulation of Theorem 2.17 does not contain the process

v∗(t). We construct a process v∗(t) on the basis of v′(t) as the process corresponding

to v′(t) under the mapping V ′ → V ∗. Then v∗(t) satisfies the assumptions preceding

Theorem 2.16, and (2.4) follows from (2.8). On the basis of Theorem 2.16, we

construct a set Ω′′ ⊂ Ω′ and a process h1 = h1(t) such that P (Ω′′) =1, h1 is

continuous in H with respect to t, is an Ft-consistent process, and h1(t) = v(t)

for almost all (t, ω); for ω ∈ Ω′′ formula (2.9) holds if h is replaced by h1, and for



ω ∈ Ω′′, g ∈ V , t ≥ 0,

gh1(t) =

t∫

0

gv′(s)ds+ gh(t) = gh(t).

From the equality of the extreme terms, it follows that g(h1(t) − h(t)) = 0 for

all g ∈ V , and hence h1 = h for t ≥ 0, ω ∈ Ω′′; from the properties of h1 we now

obtain all the required properties of h. The proof of the theorem is complete.

We proceed to prepare for the proof of Theorem 2.16. We have already agreed

to consider sparable V and H . Further, if in place of τ in Theorem 2.16 we take

τ ∧n, prove the theorem for τ ∧ n and then let n→ ∞, then we obtain the proof of

the theorem for τ . It may therefore be assumed that τ is a bounded stopping time,

and then a non-random change of time reduces everything to the case τ ≤ 1.

We further note that both |v(s)| and |v∗(s)| are Fs-measurable for almost all s

and are measurable in (s, ω). Hence, the process

r(t) = h2(0) + ‖A‖t + 〈m〉t +

∫ t

0

|v(s)|p ds+

∫ t

0

|v∗(s)|q ds

is Ft-measurable for each t and is continuous in t (a.s). This implies that for every

n ≥ 0 τn = inf t ≥ 0 : r(t) ≥ n ∧ τ is a stopping time. Since τn ↑ τ , it suffices

to prove Theorem 2.16 with τ in its formulation replaced by τn. Moreover, process

r(t) is bounded in (t, ω) on (t, ω) : t ≤ τn(ω), and it may therefore be assumed in

the proof of Theorem 2.16 that the process r(t) is bounded on (t, ω) : t ≤ τ(ω).

For t > τ , we set v(t) = 0, v∗(t) = 0, h(t) = h(τ), thus ensuring that the process

r(t) is bounded on [0,∞) × Ω. After this, multiplying (2.4) by a suitable constant

ε and replacing v, v∗, h by εv, εv∗, εh, we reduce the matter to the case where

r(t) ≤ 1.

These arguments show that it suffices to prove Theorem 2.16 in the special case

considered below.

2.4. Proof of Theorem 2.16

In this section we prove Theorem 2.16 under the additional assumptions that V and

H are separable, τ ≤ 1, and, for all ω,

1∫

0

|v(s)|p ds+

1∫

0

|v∗(s)|q ds+ ‖A‖1 + 〈m〉1 + h2(0) ≤ 1 (2.10)

It was shown above that these additional assumptions do not lead to any loss of

generality.

Lemma 2.18. There exist (a) a sequence of nested partitions

0 = tn0 < tn1 < . . . < tnk(n)+1 = 1



of the interval [0, 1] with diameter tending to zero, and (b) a set Ω′ ⊂ Ω with the

following properties:

1) P (Ω′) = 1 and, if ω ∈ Ω′ and t ∈ I = tni ; i = 1, . . . , k(n), n ≥ 1 are such

that t < τ(ω), then equality (2.4) is satisfied for all g ∈ V and, for every

s ∈ I, the quantity v(s) is Fs-measurable;

2) Define the processes v1n(t) and v2

n(t) by v(1)n (t) = v(tni ) for t ∈

[tni , t

ni+1

),

i = 1, . . . , k(n), v(1)n (t) = 0 for t ∈ [0, tn1 ); v

(2)n (t) = v(tni+1) for t ∈

[tni , t

ni+1

),

i = 1, . . . , k(n) − 1, v(2)n (t) = 0 for t ∈

[tnk(n), 1

). Then, for j = 1, 2

E supt∈[0,1]

∣∣∣v(j)n (t)

∣∣∣p

<∞, limn→∞

E

∫ 1

0

∣∣∣v(t) − v(j)n (t)

∣∣∣p

dt = 0. (2.11)

Proof. Let [a] be the integer part of the number a, (for a > 0, [a] is the largest

integer smaller than a) and let κ1(n, t) = 2−n [2nt], κ2(n, t) = 2−n [2nt] + 2−n,

v(t) = 0 for t ∈ [0, 1]. Standard arguments of Doob show that there exists a

sequence of integers rn → ∞ such that, for j = 1, 2 and almost all s ∈ [0, 1],

limrn→∞

E

∫ 1

0

∣∣v(t) − v(κj(rn, t+ s) − s)∣∣p dt = 0 (2.12)

Further, it follows from Fubini’s theorem and separability of V that there exists a

set T ⊂ [0, 1] of unit Lebesgue measure such that, for all t ∈ T and all g ∈ V , equality

(2.4) is satisfied (a.s) on ω : t < τ(ω), and the quantity v(t) is Ft-measurable. It is

clear that, for every s ∈ [0, 1], all values of the functions κj(rn, t+s)−s for t ∈ [0, 1],

j = 1, 2, n ≥ 1, lying in [0, 1] also belong to T . We fix a suitable s so that (2.12) is

also satisfied; we define tni as the set of values of κ1(rn, t+s)−s for t ∈ [0, 1] which

lie in [0, 1], to which we add the points 0 and 1, and we denote by Ω′ the set of ω for

which equation (2.4) is satisfied for all g ∈ V, t = tni < τ(ω), i = 1, . . . , k(n), n ≥ 1.

All assertions of the lemma are then valid except possibly for the first inequality

in (2.11). We note, however, that by virtue of the second inequality in (2.11), for

sufficiently large n,

E

∫ 1

0

∣∣∣v(j)n (t)

∣∣∣p

dt <∞

This inequality is equivalent to the first inequality in (2.11), which is thus valid

for large n. For small n it is clearly valid since our partitions are nested. The proof

of the lemma is complete.

Lemma 2.19. For ω ∈ Ω′, t, s ∈ I, s ≤ t ≤ τ(ω)

|Λv(t)|2 − |Λv(s)|2 = 2

∫ t

s

v(t)v∗(u)du+ 2Λv(s)(h(t) − h(s)) + |(h(t) − h(s)|2

− |Λ(v(t) − v(s)) − (h(t) − h(s))|2 ,(2.13)



|Λv(t)|2 = 2

∫ t

s

v(t)v∗(u)du+ h2(t) − |Λ(v(t) − (h(t)|2 . (2.14)

The proof of this lemma is based on using (2.4) with g = v(t) or g = v(s)

and simple algebraic transformations, which we leave to the reader while suggesting

that (2.14) be derived first and (2.13) then proved by subtracting the appropriate

equalities (2.14).

Lemma 2.20.

E supt∈I, t<τ

|Λv(t)|2 <∞.

Proof. From (2.14) and (2.11), with t ∈ I ,

Eχt<τ

|Λv(t)|2 ≤ Eh2(t) +2

ptE |v(t)|p +

2

qE

∫ t

0

|v∗(s)|q ds <∞. (2.15)

Further, by (2.13) and (2.14), with t = tni < τ(ω), ω ∈ Ω′

∣∣Λv1n(t)

∣∣2 = h2(0) + 2

∫ t

0

v2n(s)v∗(s)ds+ 2

∫ t

0

Λv1n(s)dh(s) + 2h(0)(h(tni ) − h(0))

+

i−1∑

j=1

∣∣h(tnj+1) − h(tnj )∣∣2 −

i−1∑

j=0

∣∣Λ(v(tnj+1) − v(tnj )

)− (h(tnj+1) − h(tnj ))

∣∣2 ,

(2.16)

where in the last sum the term corresponding to j = 0 is taken equal to

|Λ(v(tn1 ) − h(tn1 )|2. To estimate the second therm on the right side of (2.16), we

apply (2.10) and the Burkholder inequality. We then obtain

E supi≥1

χtn1 <τ

∣∣∣∣∣

∫ tn1

0

Λv1n(u)dh(u)

∣∣∣∣∣ ≤ E supt≤1

∣∣∣∣∫ t

0

χu<τ

Λv1n(u)dh(u)

∣∣∣∣

≤ E supi≥1

|Λv(tni |χtni

<τ‖A‖1 + 3E

(∫ 1

0

χu<τ

∣∣Λv1n(u)

∣∣2 d〈m〉u)1/2

≤ 4E supi≥1

|Λv(tni |χtni

<τ≤ 1

4E supi≥1

|Λv(tni |2 χtni

<τ+ 16.

We note that it follows from (2.15) that the last expression is finite. Moreover, from

(2.10)

E

k(n)∑

j=0

∣∣h(tnj+1) − h(tnj )∣∣2 ≤ 4E ‖A‖2

1 + 2Em2(1) ≤ 6.

From this and from (2.16)

E supi≥1

|Λ(v(tni )|χtni

<τ≤ 4

pE

∫ 1

0

∣∣∣v(2)n (t)

∣∣∣p

dt+4

q+ 100.



Letting here n → ∞ and using the fact that by (2.11) the right side is bounded in

n, we obtain the assertion of the lemma. The proof of the lemma is complete.

Lemma 2.21. Let

Ω′′ = Ω′ ∩ω : sup

t∈I, t<τ|Λv(t)| <∞

.

Then there exists a function h = h(t) with values in H, which, for ω ∈ Ω′′ and

t ∈ I, t < τ(ω), is weakly continuous in H with respect to t and satisfies (2.6) for

all g ∈ V Moreover, h(t) is Ft-measurable on ω : t < τ(ω), h(t) = Λv(t) (a.s.)

on (t, ω) : t < τ(ω), h(t) = Λv(t) for ω ∈ Ω′′, t ∈ I, t < τ(ω).

Proof. For ω ∈ Ω′′, t ∈ I , t < τ , g ∈ V the function (Λg)Λv(t) coincides with

the right side of (2.6), which is continuous in t. Hence, for s < τ there exists

limI 3t→st<τ

(Λg)Λv(t) < τ .

Since |Λv(t)| is bounded on I ∩ t < τ, this means that for s < τ , I 3 t → s

there exists the weak limit of Λv(t), which we denote by h(s). It is clear that h(t)

satisfies (2.6) for all ω ∈ Ω′′, t < τ(ω) g ∈ V . The assertions of the lemma follow

from this in an almost obvious way. The proof of the lemma is complete.

We set h(t) = h(τ) for t ≥ τ , h(t) = 0 for ω /∈ Ω′′, and we note that for ω ∈ Ω′′

supt≤1

∣∣∣h(t)∣∣∣ = sup

t∈I, t<τ|Λv(t)| <∞. (2.17)

Moreover, for t ≤ 1, the integral∫ t0

∣∣∣h(s)∣∣∣ ds is well defined, because h(s) is

completely measurable due to separability of H and continuity of πh(s) for ω ∈ Ω′′,where π is the operator of projecting onto some finite-dimensional subspace of H .

Lemma 2.22. Define hn(t) = h(tni ) for t ∈ [tni , tni+1), i = 1, . . . , k(n). Then

limn→∞

supt≤1

∣∣∣∣∫ t

0

hn(s)dh(s) −∫ t

0

h(s)dh(s)

∣∣∣∣ = 0 (2.18)

in probability.

Proof. Let h1, . . . , hr, . . . be an orthonormal basis in H , and let πr be the operator

projecting H onto the space spanned by h1, . . . , hr. Since πrh(s) is continuous on

Ω′′, it follows that (a.s.)

limn→∞

∫ t

0

∣∣∣πrhn(s) − πrh(s)∣∣∣ d 〈m〉s +

∫ t

0

∣∣∣πrhn(s) − πrh(s)∣∣∣ d ‖A‖s

= 0.

It therefore suffices to prove that, for every ε > 0,

limr→∞

supnP

supt ≤ 1

∣∣∣∣∫ t

0

(1 − πr)hn(s)dh(s)

∣∣∣∣ > 2ε

= 0 (2.19)



and

limr→∞

P

supt≤1

∣∣∣∣∫ t

0

(1 − πr)h(s)dh(s)

∣∣∣∣ > 2ε

= 0.

We shall prove only the first equality. The second is proved similarly. Noting that

πr is a self-adjoint operator, for every N , δ > 0 we estimate the probability in (2.19)

as follows:

P

supt≤1

∣∣∣∣∫ t

0

hn(s)d((1 − πr)h(s)

)∣∣∣∣ > 2ε

≤ δ

ε2+ P

∫ t

0

∣∣∣hn(s)∣∣∣ d ‖(1 − πr)A‖s > ε

+ P

∫ t

0

∣∣∣hn(s)∣∣∣2

d 〈(1 − πr)m〉s > δ

≤ δ

ε2+ 2P

sups≤1

∣∣∣h(s)∣∣∣ > N

+N

εE‖(1 − πr)A‖1 +

N

δE |(1 − πr)m(1)|2 .

(2.20)

We denote by hi the i-th coordinate of h ∈ H in the basis hi, and we set

ai(t) = dAi(t)/d ‖A‖t. Then as r → ∞

‖(1 − πr)A‖1 =

∫ 1

0

(∑

i>r

(ai(t)

)2)1/2

d ‖A‖t → 0,

|(1 − πr)m(1)|2 =∑

i>r

(mi(1)

)2 → 0.

From this and from (2.20) and (2.10) we see that the left side of (2.19) does not

exceed

δ

ε2+ 2P

sups≤1

∣∣∣h(s)∣∣∣ > N

.

Since δ, N are arbitrary and (2.17) is satisfied, the last expression can be made

arbitrarily small. The proof of the lemma is complete.

We now define the set Ω′′′ which will play the role of Ω′ in Theorem (2.16). It

is possible to find a sequence n′ along which the left side of (2.17) tends to zero

(a.s.). It may be assumed with no loss of generality that the original sequence has

this property. Moreover, we set π0 = 0; then, as is well know, in probability for

r > 0, t ∈ [0, 1],

limn→∞

∑

tnj+1≤t

∣∣(1 − πr)(h(tnj+1) − h(tnj )∣∣2 = 〈(1 − πr)m〉t. (2.21)

Therefore, there exists a subsequence along which the last equality, understood

in the sense of pointwise convergence, is true for all r > 0, t ∈ I almost surely.

To simplify the notation, we assume that this subsequence also coincides with the

original sequence. We set

Q1 =

ω : lim

n→∞supt≤1

∣∣∣∣∫ t

0

hn(s) − h(s))dh(s)

∣∣∣∣ = 0

,



Q2 =

∞⋂

r=0

⋂

t∈I

ω : lim

n→∞

∑

tnj+1≤t

∣∣(1 − πr)(h(tnj+1) − h(tnj )∣∣2 = 〈(1 − πr)m〉t

,

Q3 =

ω : lim

n→∞

∫ 1

0

∣∣∣v(i)n (s) − v(s)

∣∣∣p

ds = 0, i = 1, 2

,

Q4 =ω : lim

r→∞〈(1 − πr)m〉i = 0

,

Q′′′ = Q′′⋂ 4⋂

i=1

Qi,

and set h(t, ω) = 0 on the complement of Q′′′.From what has been said above, P (Qi) = 1, i = 1, 2. From Lemma 2.18 it follows

that it may be assumed that P (Q3) = 1. Since the sequence 〈(1 − πr)m〉1, r ≥ 0,

is decreasing (a.s.) and E〈(1 − πr)m〉1 = E |(1 − πr)m(1)|2 → 0, it follows that

P (Q4) = 1. Thus, P (Q′′′) = 1.

Lemma 2.23. For ω ∈ Q′′′, t, s ∈ I, s < t < τ(ω)

∣∣∣h(t) − h(s)∣∣∣2

= 2

∫ t

s

(v(u) − v(s)) v∗(u)du

+ 2

∫ t

s

(h(u) − h(s)

)dh(u) + 〈m〉t − 〈m〉s,

(2.22)

∣∣∣h(t)∣∣∣2

= h2(0) + 2

∫ t

0

v(u)v∗(u)du+ 2

∫ t

0

h(u)dh(u) + 〈m〉t. (2.23)

Proof. We first prove (2.23). We fix ω ∈ Q′′′, t ∈ I , t < τ(ω). From (2.16) and

Lemma 2.21 for n such that t is a point of the partition tn0 < tn1 < . . . tnk(n)+1, we

have

h2(t) = h2(0) + 2

∫ t

0

v(2)n (u)v∗(u)du+ 2

∫ t

0

hn(u)dh(u)

+∑

tnj+1≤t

∣∣h(tnj+1) − h(tnj )∣∣2 −

∑

tnj+1≤t

∣∣∣(h(tnj+1) − h(tnj )

)−(h(tnj+1) − h(tnj )

)∣∣∣2

.

Letting n→ ∞, we find

h2(t) = h2(0) + 2

∫ t

0

v(u)v∗(u)du+ 2

∫ t

0

h(u)dh(u) + 〈m〉t − J,

where

J = limn→∞

∑

tnj+1≤t

∣∣∣(h(tnj+1) − h(tnj )

)−(h(tnj+1) − h(tnj )

)∣∣∣2

,



and the last limit exists and is finite. For the proof of (2.23) it suffices to show that

J = 0. It is convenient to assume that the basis hi is formed from elements of

the set ΛV . Since the latter is dense in H , this last assumption involves no loss of

generality. Moreover, it is obvious that for every r ≥ 1 there exists a function vr(t)

continuous in the norm of V such that Λvr(t) = πrh(t) for all (t, ω). On the basis

of the function vr we construct functions v(i)r,n, i = 1, 2 just as in Lemma 2.18 the

functions v(i)r are constructed on the basis of v. Noting that by (2.4) and Lemma

2.21 for tnj+1 ≤ t and every ϕ ∈ V there is the equality

((h(tnj+1) − h(tnj )

)−(h(tnj+1) − h(tnj ))

)Λϕ =

tnj+1∫

tnj

ϕv∗(u)du,

we find easily for any r ≥ 0

J = limn→∞

∫ t

0

(v(2)n (u) − (v(1)

n (u))v∗(u)du− limn→∞

∫ t

0

(v(2)r,n(u) − (v(1)

r,n(u))v∗(u)du

− limn→∞

∑

tnj+1≤t

((h(tnj+1) − h(tnj )) − (h(tnj+1) − h(tnj ))

)(1 − πr)(h(tnj+1) − h(tnj )).

Here the first limit is equal to zero; the second is equal to zero, because ω ∈ Q3

and the continuity of vr(u) implies limn→∞∣∣∣v(2)r,n(u) − v

(1)r,n(u)

∣∣∣ = 0 uniformly in u.

Hence,

J ≤ limn→∞

∑

tnj+1≤t

∣∣∣(

(h(tnj+1) − h(tnj ))− (h(tnj+1) − h(tnj ))

∣∣∣2

×

∑

tnj+1≤t

∣∣(1 − πr)((h(tnj+1) − h(tnj )

)∣∣2

1/2

= J1/2〈(1 − πr)m〉1/2t .

As r → ∞ this implies that J = 0. Equality (2.23) is proved. Equality (2.22) is

deduced from (2.23) by means of the relations (a− b)2

= a2 − b2 − 2b(a− b) and

−2h(s)(h(t) − h(s)) = −2

∫ t

s

v(s)v∗(u)du− 2

∫ t

s

h(s)dh(u).

The proof of the lemma is complete.

We now finish the proof of Theorem 2.16. Because of (2.21) and (2.23) it remains

for us to prove the strong continuity of h(t, ω) in t for t < τ(ω), ω ∈ Ω′′′. Since

a weakly continuous function with a continuous norm is strongly continuous, it

suffices to prove (2.23) for t < τ(ω), ω ∈ Ω′′′. For t = 0 (2.23) is obvious. We fix

t > 0, t < τ(ω), ω ∈ Ω′′′.



For all sufficiently large n it is possible to define j = j(n) such that 0 < tnj ≤t < tnj+1. We set t(n) = tnj(n) and note that t(n) ↑ t,

limn→∞

∫ t

t(n)

|v(u) − v(t(n))| · |v∗(u)| du ≤ limn→∞

∫ 1

0

∣∣∣v(u) − v(1)n (u)

∣∣∣ · |v∗(u)| du = 0,

limn→∞

sups≤t

∣∣∣∣∣

∫ s

t(n)

h(u) − h(t(n))dh(u)

∣∣∣∣∣ ≤ 2 limn→∞

sups≤1

∣∣∣∣∫ s

0

h(u) − hn(u))dh(u)

∣∣∣∣ = 0,

limn→∞

(〈m〉t − 〈m〉t(n)

)= 0.

Therefore, there exists a subsequence n(k) such that for s(k) = t(n(k)) we have

∞∑

k=1

s(k+1)∫

s(k)

|v(u) − v(s(k))| · |v∗(u)| du

1/2

+(〈m〉s(k+1) − 〈m〉s(k)

)1/2

+

s(k+1)∫

s(k)

(h(u) − h(s(k))dh(u)

)1/2

<∞.

From (2.22) we then find that

∞∑

k=1

∣∣∣h(s(k + 1)) − h(s(k))∣∣∣ <∞.

Therefore, h(s(k)) for k → ∞ has a strong limit. Since s(k) → t, it follows that

h(s(k)) converges weakly to h(t). Thus, h(s(k)) → h(t) strongly in H , and, substi-

tuting the numbers s(k) in place of t in (2.23), and passing to the limit k → ∞, we

obtain (2.23) for the t chosen. This complete the proof of Theorem 2.16.

3. Ito Stochastic Equations in Banach Spaces and the Method of

Monotonicity

3.1. Introduction

In this chapter we consider the Ito equations

v(t) = u0 +

∫ t

0

A(v(s), s)ds +

∫ t

0

B(v(s), s)dw(s) (3.1)

in Banach spaces. The coefficients A(v, s), B(v, s) of “drift” and “diffusion” are

generally assumed to be unbounded non-linear operators. They may depend on the

elementary outcome in a nonanticipatory fashion. By w we understand a Wiener

process with values in some Hilbert space.



An existence and uniqueness theorem will be proved for the solution of an equa-

tion slightly more general than (3.1) and certain qualitative results on the solution

will be obtained.

A solution is understood to be a trajectory with values in the domain of the

operators A(·, t), B(·, t) (the domain of the operators A and B is assumed to be

independent of t) that satisfies (3.1) and is consistent with the same system of σ-

algebras as w(t), A(·, t), and B(·, t). This system is assumed to be given together

with the original probability space and the Wiener process. A solution is thus

understood in the “strong” sense.

The main conditions on A an B are the conditions of monotonicity and co-

erciveness; see (A2), (A3) in Section 3.2. The following equations satisfy these

assumptions in spaces of Sobolev type:

du(t, x) = a(t, ω)(−1)m+1 ∂m

∂xm

(∣∣∣∣∂m

∂xmu(t, x)

∣∣∣∣p−2

∂m

∂xmu(t, x)

)dt

+ b(t, ω)

∣∣∣∣∂m

∂xmu(t, x)

∣∣∣∣p/2

dw(t),

(3.2)

p > 1, x ∈ G ⊂ R, −2(p− 1)a+ 4−1p2b2 ≤ −ε, ε > 0, and w(t) is a Wiener process

with values in R;

du(t, x) =

n∑

i,j=1

∂

∂xi

(aij(t, x, ω)

∂

∂xju(t, x)

)dt

+

m∑

j=1

n∑

i=1

bij(t, x, ω)∂

∂xju(t, x)dwj(t),

(3.3)

x ∈ Rn, aij , bij are bounded measurable functions such that, for some λ > 0,

n∑

i,j=1

aijξiξj −

m∑

i,j=1

n∑

`=1

1

2bi`bj`ξ

iξj ≥ λ

n∑

i=1

(ξi)2

for all t, x, ω and every vector ξ ∈ Rn; wi(t) are independent Wiener process with

values in R, and ω is an elementary outcome.

These and other stochastic partial differential equations are considered in detail

in Section 4.

The results of the present section are a refinement of the results of Pardoux [52,

53] which, in turn, generalize the results of Bensoussan and Temam [43].

As already mentioned above, we have succeeded in showing that certain condi-

tions of Pardoux are superfluous, in particular, the local Lipschitz condition for the

operator B.

The method of proving the existence theorem (the most difficult and impor-

tant part of this section) has been borrowed from Pardoux and corresponds to a

Galerkin scheme: a finite-dimensional analog of equation 3.1 is considered (Section

3.3), estimates of the solution independent of the dimension are obtained (Section



3.4), and then (by the method of monotonicity) a passage to the limit is realized

(Section 3.5). The basic improvements which make it possible in the final analysis

to generalize the results of Pardoux are made at the first step in Section 3.3, where

a theorem is obtained generalizing a well-known theorem of Ito on the existence

and uniqueness of strong solutions of a stochastic equation with random coefficients

satisfying Lipschitz conditions.

3.2. Assumptions and the Main Results

Let (Ω,F , P ) be a complete probability space with an expanding system of σ-

algebras Ft (t ∈ [0, T ], T <∞) imbedded in F . We assume that the family Ftis complete with respect to the measure P .

Further, let H and E be real separable Hilbert spaces, naturally identified with

their duals H∗ and E∗; let w(t) be a Wiener process in E with nuclear covariance

operator Q (see Section 2.2), and let z(t) be a square-integrable martingale in H .

We also consider a real, separable, reflexive Banach space V and its dual space V ∗.

As in Section 2, if v is an element of V and v∗ is an element of V ∗, then vv∗

denotes the value of v∗ on v; |·|X and (·, ·)X denote, respectively, the norm in the

spaceX and the scalar product in the spaceX if X is a Hilbert space. In Section 3.3,

where finite-dimensional spaces are considered, this notation is simplified; special

mention is made of this.

As before, LQ(E,H) is the space of all linear operators Φ defined on Q1/2E and

taking Q1/2E into H such that ΦQ1/2 ∈ L2(E,H) (the space of Hilbert-Schmidt

operators from E to H). Recall that LQ(E,H) is a separable Hilbert space relative

to the scalar product (Φ,Ψ)Q = trΦQ1/2(ΨQ1/2

)∗; we write |·|Q to denote the

norm in this space.

The following assumptions are henceforth used:

a) V ⊂ H ≡ H∗ ⊂ V ∗;

b) V is dense in H (in the norm of H);

c) there exists a constant c such that, for all v ∈ V , |v|H ≤ c |v|V ;

d) vv∗ = (v, v∗)H if v∗ ∈ H .

An important example of spaces possessing properties a)– d) is the Sobolev

space Wmp (G)(= V ) and L2(G)(= H) where G is a bounded domain in Rd, and

dp ≥ 2(d−mp). These spaces are discussed in more detail in Section 4. Some other

triples of spaces possessing properties a)–d) are also presented there. We recall also

that in Section 2 triples of spaces V , H , V ′ connected by less rigid assumptions

have already been considered, and the “implication” of assumptions a) – d) was

discussed in some detail; in particular, the possibility of identifying V ′ with V ∗ by

means of (·, ·)H was discussed.

We fix numbers p and q, p ∈ (1,∞), q =p

p− 1.



Suppose that, for each (v, t, ω) ∈ V × [0, T ] × Ω,

A(v, t, ω) ∈ V ∗, B(v, t, ω) ∈ LQ(E,H).

We assume that, for each v ∈ V , the functions A(v, t, ω), B(v, t, ω) are measurable

in (t, ω) relative to the measures dt× dP and are Ft-consistent, i.e., for each v ∈ V ,

t ∈ [0, T ] they are Ft-measurable in ω.

We recall that, since V ∗ and LQ(E,H) are separable, the concepts of strong and

weak measurability coincide, and we will speak simply of measurability. Suppose

further that on Ω there is given an F0-measurable function u0 with values in H ,

while on [0, T ]×Ω there is given a nonnegative function f(t, ω) measurable in (t, ω)

and Ft-consistent.

We assume that, for some constants K, α > 0 and for all v, v1, v2 ∈ V , (t, ω) ∈[0, T ] × Ω, the following conditions are satisfied.

(A1) Semicontinuity of A: the function vA(v1 + λv2) is continuous in λ on R.

(A2) Monotonicity of (A, B):

2(v1 − v2)(A(v1) −A(v2)) + |B(v1) −B(v2)|2Q ≤ K |v1 − v2|2H .

(A3) Coercivity of (A, B):

2vA(v) + |B(v)|2Q + α |v|p ≤ f +K |v|2H .

(A4) Boundedness of the growth of A:

|A(v)|V ∗ ≤ f1/q +K |v|p−1V .

(A5)

E |u0|2H <∞, E

∫ T

0

f(t)dt <∞.

Under these assumptions we consider on [0, T ]×Ω the stochastic evolution equa-

tion

v(t, ω) = u0(ω)+

∫ t

0

A(v(s, ω), s, ω)ds+

∫ t

0

B(v(s, ω), s, ω)dw(s, ω)+z(t, ω). (3.4)

Definition 3.1. A solution (or a V-solution) of equation (3.4) is a function v(t, ω)

with values in V defined on [0, T ]×Ω, measurable in (t, ω), Ft-consistent, satisfying

the inequality

E

∫ T

0

(|v(t)|pV + |v(t)|2H

)dt <∞, (3.5)

and satisfying equation (3.4) in the sense of equality of elements of V ∗ for almost

all (t, ω) ∈ [0, T ] × Ω.



In this definition it is implicitly assumed that integrals in (3.4) and (3.5) are

meaningful. This assumption is, however, superfluous in view of what follows.

By Lemma 2.5 the function v is measurable not only as a function with values in

V but also as a function with values in H . Therefore, |v(t)|V , |v(t)|H are measurable

with respect to (t, ω) and condition (3.5) is meaningful.

We show further that the functions A(v(t), t), B(v(t), t) are measurable in (t, ω)

and Ft-measurable in ω. We first consider A(v(t), t) and use the same sort of lemma

as in the theory of monotone operators (see e.g. Ref. [42]).

Lemma 3.2. If a sequence vn converges strongly in V to v, then the sequence A(vn)

converges weakly to A(v) in V ∗.

Proof. Because of assumption (A4), for every subsequence µ of natural num-

bers, the sequence A(vµ) is bounded in V ∗, and therefore there exists a subsequence

η of the sequence µ along which A(vη) converges weakly to some A∞ ∈ V ∗.

We will now show that A∞ = A(v).

Let u be an arbitrary element of V . By assumption (A2)

(u− vη)(A(u) −A(vη)) −K |u− vη |2H ≤ 0.

Passing to the limit η → ∞ in this equality, we obtain

(u− v)(A(u) −A∞) −K |u− v|2H ≤ 0. (3.6)

We now suppose that u = v + λy where λ ∈ R+ and y is some element of V ;

from (3.6) it then follows that

y(A(v + λy) −A∞) −Kλ |y|2H ≤ 0.

Passing to the limit λ ↓ 0 in this equality and using the semicontinuity of A (as-

sumption (A1),) we obtain

y(A(v) −A∞) ≤ 0.

Since y is arbitrary, it follows that A∞ = A(v) and, since the subsequence µ was

arbitrary, the proof of the lemma is complete.

It follows from the lemma that the function v → uA(v, t, ω) is continuous for each

u ∈ V , (t, ω) ∈ [0, T ] × Ω. As a result, if v = v(t, ω) is measurable in (t, ω) and Ft-consistent, then so is A(v(t, ω), t, ω). Moreover, in view of (A4), (A5), A(v(t, ω), t, ω)

is a summable function of t for almost all ω if (3.5) is satisfied.

The problem of the measurability of B(v(t, ω), t, ω) is easier, since it is obvious

that by assumptions (A2) and (A4), B(·, t, ω) for all t, ω is a strongly continuous

function from V to LQ(E,H).

From assumptions (A3) and (A4) it follows that, for all u, t, ω,

|B(u, t, ω)|2Q ≤ c(f(t, ω) + |u|2H + |u|pV

). (3.7)



Therefore,

E

∫ T

0

|B(v(t), t)|2Q dt <∞,

and the stochastic integral in (3.4) is defined and is a square-integrable martingale

in H .

Remark 3.3. It is obvious (see the clarifications of Theorem 2.14) that v is a

solution of (3.4) if and only if, for some set Y ⊂ V dense in V (in the norm of V)

and for every u ∈ Y , the equality

(u, v(t))H = (u, u0)H + uA(v(s), s)ds+

∫ t

0

uB(v(s), s)dw(s) + (u, z(t))H

holds a.e. (t, ω).

Definition 3.4. An H-solution of (3.4) is a solution u(t, ω) with values in H defined

on [0, T ] × Ω, strongly continuous in H with respect to t, Ft-consistent, and such

that

(a) u ∈ V a.e. in (t, ω) and

E

∫ t

0

(|u(t)|pV + |u(t)|2H

)dt <∞.

(b) there exists a set Ω′ ⊂ Ω of probability one on which, for all t ∈ [0, T ],

u(t) = u0 +

∫ t

0

A(u(s), s)ds+

∫ t

0

B(u(s), s)dw(s) + z(t), (3.8)

where the equality is understood as an equality of elements of V ∗.

In clarification of (3.8) we note that, by Lemma 2.5, the function χV

(u(t)) is

measurable in (t, ω), Ft-measurable in ω, equal to one [a.e. in (t, ω)] by condition

(a), and the integrals in (3.8) are therefore understood as∫ t

0

χV

(u(s))A(u(s), s)ds,

∫ t

0

χV

(u(s))B(u(s), s)dw(s).

Definition 3.5. An H-solution u is called a continuous modification of a solution

v in H if u(t, ω) = v(t, ω) a.e. in (t, ω).

We now formulate the main results of this Section. All the assumptions above

are assumed to be satisfied.

Theorem 3.6. A solution v of equation (3.4) exists.

From this theorem and Theorem 2.17 we immediately obtain the following result.

Corollary 3.7. There exists a continuous modification of the solution v of equation

(3.4) in H.



Theorem 3.4 is proved in Section 3.5. The next theorem contains the uniqueness

assertion for the solution of equation (3.4).

Theorem 3.8. Let vn(t), n = 0, 1, 2, . . ., be solutions of equation (3.4) with initial

data u0 = un0 , where E |un0 |2H < ∞ and E∣∣un0 − u0

0

∣∣2H

→ 0 as n → ∞. Suppose that

un(t) are continuous modifications of vn(t) in H. Then, for every ε > 0,

limn→∞

supt≤T

E∣∣un(t) − u0(t)

∣∣2H

+ P

supt≤T

∣∣un(t) − u0(t)∣∣H

≥ ε

= 0.

Remark 3.9. We will see from the proof of the theorem that the result is valid if,

in the definition of the solution, condition (3.5) is dropped and only condition (A2)

is required of the coefficients of (3.4).

Theorem 3.10. If v is a solution of (3.4) and u is its continuous modification in

H, then

E supt≤T

|u(t)|2H + E

∫ T

0

|v(t)|pV dt ≤ c

(E |u0|2H + E

∫ T

0

f(t)dt+ E |z(t)|2H

),

where c depends only on K, p, T , and α.

Theorems 3.8 and 3.10 are proved in Section 3.4. The following theorem on

the Markov property of solutions of (3.4) also belongs to the basic results of this

section. This theorem is proved at the end of Section 3.5.

Theorem 3.11. Suppose that A and B do not depend on ω, z(t) ≡ 0, v = v(t) is a

solution of equation (3.4), and u = u(t) is its continuous modification in H. Then

u(t) is a Markov random variable.

Remark 3.12. With no loss of generality, we can take K = 0 in conditions (A2)

and (A3). Indeed, if v is a solution of (3.4), then v(t)e−Kt is a solution of an

equation of the type (3.4) with A replaced by e−Kt(A−KI), where I is the identity

operator, and with B replaced by e−KtB, and these new A and B satisfy conditions

(A2) and (A3) with K = 0.

Remark 3.13. Our assumption that the spaces in question are real is not essential

and may be relaxed if, in conditions (A2)–(A3), in place of

vA(v1 + λv2), (v1 − v2)(A(v1) −A(v2)), vA(v)

we write

Re(vA(v1 + λv2)

), Re

((v1 − v2)(A(v1) −A(v2))

), Re

(vA(v)).



3.3. Ito Equations in Rd

Let Rd be Euclidean space of dimension d with a fixed orthonormal basis, let xi be

the i−th coordinate of a point x ∈ Rd, let (Ω,F , P ) be a complete probability space,

and let Ft be an expanding family of complete σ-algebras Ft ⊂ F . Let m(t) be

a d1-dimensional continuous local martingale relative to Ft, with m(0) = 0, and

let A(t) be a continuous, real, nondecreasing Ft-consistent process with A(0) = 0.

Suppose that, for t ≥ 0, x ∈ Rd, ω ∈ Ω, a d× d1 matrix b(t, x) and a d-dimensional

vector a(t, x) are defined. We assume that, for each x ∈ Rd, a(t, x) and b(t, x) are

completely measurable relative to Ft and are continuous in x for each (t, ω). Let

x0 be a d-dimensional F0-measurable random vector.

We consider the following equation:

x(t) = x0 +

∫ t

0

a(s, x(s))dA(s) +

∫ t

0

b(s, x(s))dm(s). (3.9)

Equation (3.9) will be considered under certain additional conditions on a, b,

A, and m whose formulation requires the following notation. By the Doob-Meyer

theorem there exists a continuous increasing process 〈m〉t for which (m2(t)〉−〈m〉t)is a local martingale relative to Ft and 〈m〉0 = 0. For i, j = 1, . . . , d1 we further

define by means of the Doob-Meyer theorem continuous processes 〈mi, mj〉t having

locally bounded variation in t for which every process mi(t)mj(t) − 〈mi,mj〉t is

a local martingale and 〈mi, mj〉0 = 0. We recall that the matrix(〈mi,mj〉t

)is

nonnegative definite and

〈m〉t =

d1∑

i=1

〈mi,mi〉t

for all t (a.s.). We fix a continuous real nondecreasing Ft-consistent process Vt such

that, for each ω, the measures on the t axis generated by the functions A(t), 〈m〉tare absolutely continuous relative to the measure corresponding to Vt (for example,

we can take Vt = A(t) + 〈m〉t.)We define

cij(t) =d〈mi,mj〉t

dVt, C(t) =

(cij(t)

),

α(t, x) = a(t, x)dA(t)

dVt, β(t, x) = b(t, x)C

1/2t .

We assume that the following conditions are satisfied, in addition to those enu-

merated above: for each x ∈ Rd, T > 0

∫ T

0

|α(t, x)| dVt =

∫ T

0

|a(t, x)| dA(t) <∞ (a.s.); (3.10)



for every R > 0 there exists a nonnegative completely measurable process Kt(R)

such that,for all T > 0,∫ T

0

K(R)dVt <∞ (a.s.),

and, for each z, x, y ∈ Rd such that |x|, |y| ≤ R,

2(x− y)(α(t, x) − α(t, y)) + ‖β(t, x) − β(t, y)‖2 ≤ Kt(R)(x− y)2, (3.11)

2zα(t, z) + ‖β(t, z)‖2 ≤ Kt(1 + z2) (3.12)

for almost all t relative to the measure dVt, where Kt = Kt(1), ‖γ‖ for a matrix γ

means (tr (γγ∗))1/2

, and, for ε, δ ∈ Rd, δε denotes the scalar product of δ, ε.

Theorem 3.14. There exists a continuous, Ft-consistent process x(t) for which

(3.9) is satisfied for all t with probability one. If x(t), y(t) are two continuous

Ft-consistent processes satisfying (3.9) for each t (a.s.), then

P

supt≥0

|x(t) − y(t)| > 0

= 0.

The present theorem is a generalization of Ito’s classical theorem on the existence

of a strong solution of a stochastic equation of type (3.9) with random coefficients.

In the present theorem we have avoided the Lipschitz condition and replaced it by

condition (3.11) which we shall call the monotonicity condition.

Example 3.15. Let w(t) be a one-dimensional Wiener process, let p ∈ (1, 2), and

let a(t, ω) and b(t, ω) be completely measurable functions such that∫ T

0

a(t, ω)dt <∞ (a.s.)

and

−2(p− 1)a(t, ω) +p2

4b2(t, ω) ≤ 0 (a.s. in (t, ω)).

We consider the equation

dx(t) = − |x(t)|p−1sgnx(t)a(t)dt + |x(t)|p/2 b(t)dw(t).

It is clear that for p < 2 the coefficients of this equation do not satisfy Lips-

chitz conditions, but satisfy the monotonicity condition. Indeed, by the formula of

Hadamard,

−2a · (|x|p−1sgnx− |y|p−1

sgn y)(x− y) + b2 · (|x|p/2 − |y|p/2)2

= −2a · (x − y)2∫ 1

0

(p− 1) |x+ τ(y − x)|p−2dτ

+b2 · (x− y)2(∫ 1

0

p

2|x+ τ(y − x)|

p2−1 sgn (x + τ(y − x))dτ

)2

≤ (x− y)2[p2

4b2 − 2(p− 1)a

] ∫ 1

0

|x+ τ(y − x)|p−2 dτ ≤ 0.



If we take Vt = t then from this inequality we obtain the monotonicity condition

with Kt = 0 and also condition (3.12), since in the present case α(t, 0) = β(t, 0) = 0.

Thus, condition (3.11) is weaker than the Lipschitz condition. However, ex-

tremely “bad” functions may not satisfy (3.11). We remark without proof that

(3.11) implies differentiability of α(t, x) with respect to x for almost all x and

that the first generalized derivatives of β(t, x) with respect to x are locally square-

summable.

We note that in contrast to Ito’s work we consider the equation over a semi-

martingale. Generalizations of Ito’s results to stochastic equations over a semi-

martingale have been obtained by Kazamaki [78], Doleans-Dade [79], Protter [80],

Gal’chuk [81], Lebedev [54] and others. The results of these works are a special case

of our assertion as far as continuous semimartingales are concerned. It seems to us

that the extension of Theorem 3.14 to the case of discontinuous semimartingales is

an accessible problem, but is beyond our present interest.

As always in similar situations, the uniqueness assertion in Theorem 3.14 is very

easy to prove. Indeed, let x(t), y(t) be two solutions of (3.9). By the continuity

with respect to t of x(t), y(t), and the integrals in (3.9), the processes x(t), y(t)

satisfy (3.9) for all t on the same set of probability one. Next, define

Ψt(R) = exp

(∫ t

0

Ks(R)dVs

),

apply Ito’s formula to |x(t) − y(t)|2 Ψt(R) and use (3.11). We then find that, (a.s.)

for all t,

|x(t ∧ τ(R)) − y(t ∧ τ(R))|2 Ψt∧τ (R)

≤ 2

∫ t∧τ(R)

0

Ψs(R) (x(s) − y(s)) (b(s, x(s) − b(s, y(s)))dm(s) ≡ m′t(R),

where τ(R) is the first exit time of max(|x(t)| , |y(t)|

)from [0, R). We see that the

local martingalem′t(R) is nonnegative, and hence m′

t(R) is a supermartingale. Since

m′0(R) = 0, it follows that m′

t(R) = 0 (a.s.), so that |x(t ∧ τ(R)) − y(t ∧ τ(R))| = 0

(a.s.) for all t, and, since |x(t) − y(t)| is continuous in t and R is arbitrary, it follows

that supt |x(t) − y(t)| = 0 (a.s.) as required.

The existence assertion in Theorem 3.14 will be proved after a number of auxil-

iary propositions. In Theorem 3.14 it is asserted, in particular, that the right side

of (3.9) exists for some process x(t). This fact for any Ft-consistent continuous

process follows immediately from the following lemma.

Lemma 3.16. For every T,R > 0 (a.s.)

∫ T

0

sup|x|≤R

|α(t, x)|dV (t) <∞,

∫ T

0

supx≤R

‖β(t, x)‖2dVt <∞.



Proof. Let yi be a countable dense subset of x : |x| ≤ R+ 2. We define

ϕn(x, z) = max((x− yi)z, i = 1, . . . , n, x, z ∈ Rd. It is clear that each ϕn(x, z) is

a continuous function and ϕn(x, z) ↑ xz+ |z| (R+2) as n→ ∞. The last function is

continuous and greater than two if |x| ≤ R, |z| = 1. By Dini’s theorem there exists

an n0 such that ϕn0(x, z) ≥ 1 for |x| ≤ R, |z| = 1. This implies that ϕn0(x, z) ≥ |z|for |x| ≤ R and all z ∈ Rd. Substituting the points yi into (3.11) in place of y and

computing the upper bounds over i for |x| ≤ R, we find (with α(t, x) in the place

of z) that

2 |α(t, x)| ≤ 4(R+ 2)2Kt(R + 2) + 2(R+ 2)

n0∑

i=1

α(t, yi).

Because of (3.10) this proves the first assertion of the lemma. The second assertion

follows in an obvious way from the first and from (3.12). The proof of the lemma

is complete.

Lemma 3.17. Let f(x) be a real locally bounded function on Rd, let n > 0, and let

N = sup |f(x)| : |x| ≤ n. Then on Rd there exists a real function g(x) such that

g(x) = f(x) for |x| ≤ n and g(x) = 0 for |x| ≥ n+ 1, |g(x)| ≤ |f(x)|, and

|g(x) − g(y)|2 ≤ |f(x) − f(y)|2 +N2 (x− y)2

(3.13)

for all x, y ∈ Rd. Moreover, if f(x) is continuous in x and depends in a measurable

way on several parameters, then g(x) is continuous in x and is measurable with

respect to these parameters.

Proof. Definec h(x) = (n+ 1)N −N |x| and

g(+)(x) = max(min(h(x), f+(x)), 0),

g(−)(x) = max(min(h(x), f−(x)), 0),

g(x) = g(+)(x) + g(−)(x).

We will prove that this function g satisfies all the requirements. The last asser-

tion of the lemma is obvious, since N is a measurable function of those parameters

on which f depends, while the continuity of g follows from the continuity of f and

(3.13).

Further, g(+) ≤ f+, g(−) ≤ f−. Therefore, g± = g(±), |g| ≤ |f |, and g(x) has the

same sign as f(x). Moreover, it is obvious that h(x) ≤ 0 for |x| ≥ n+ 1. Therefore,

g(x) = 0 for |x| ≥ n + 1. Since h(x) ≥ N ≥ |f(x)| for |x| ≤ n, it follows that

g(x) = f(x) for |x| ≤ n.

It remains to prove (3.13). We fix points x, y. If f(x) and f(y) have different

signs, then the same is true of g(x) and g(y), and

|g(x) − g(y)| =∣∣ |g(x)| + |g(y)|

∣∣ ≤∣∣ |f(x)| + |f(y)|

∣∣ = |f(x) − f(y)| .

cThis method of constructing g was suggested to us by A. D. Wentzell.



We now consider the case where f(x) and f(y) have the same sign. Suppose,

to be specific, that f(x) ≥ 0, g(x) ≥ 0; then g(x) = g(+)(x), g(y) = g(+)(y). Since

the absolute value of the difference of the upper (lower) bounds does not exceed the

upper (lower) bound of the absolute value of the difference, it follows that

|g(x) − g(y)| ≤ |min(h(x), f(x)) − min(h(y), f(y))|≤ max (|h(x) − h(y)| , |f(x) − f(y)|) ,

which implies (3.13). The proof of the lemma is complete.

Lemma 3.18. For every n > 0, there exist processes a(t, x), b(t, x), Nt such that

a(t, x) ∈ Rd, b(t, x) is a d × d1 matrix, Nt is a real process, a, b, N are defined

for all x ∈ Rd, t ≥ 0, ω ∈ Ω, are continuous in x and completely measurable,

a(t, x) = a(t, x), b(t, x) = b(t, x) for |x| ≤ n, a(t, x) = 0, b(t, x) = 0 for |x| ≥ n+ 3

for all t,

∫ t

0

NsdVs <∞ (a.s.), (3.14)

and, for all x, y ∈ Rd,

|a(t, x)| +∥∥∥β(t, x)

∥∥∥2

≤ Nt, 2(x− y)(α(t, x) − (α(t, y))

+∥∥∥β(t, x) − β(t, y)

∥∥∥2

≤ Nt(x− y)2 (a.e. dP × dVt),

(3.15)

where

α(t) = a(t)dA(t)

dVt, β(t) = b(t)C1/2(t).

Proof. We fix n ≥ 1. Let Tt be an orthogonal d1 × d1 matrix, and let Λt be

a diagonal matrix of the same dimension so that Ct = TtΛT∗t . It is well known

that such matrices exist, and they can be chosen to be completely measurable.

For every element of the matrix b(t, x)Tt, we construct by means of Lemma 3.17

the corresponding truncated elements which we assign to the matrix b′(t, x), and

then define b(t, x) = b′(t, x)T ∗t . Consider the function η ∈ C∞

0 (Rd), η(x) = 1 for

|x| ≤ n + 2, η(x) = 0 for |x| ≥ n + 3, 0 ≤ η ≤ 1, and define a(t, x) = a(t, x)η(x).

We shall prove that there exists a process Nt such that the assertions of the lemma

hold for a, b, N .

It is not hard to see that by Lemmas 3.16 and 3.17 there exists a process N(1)t

satisfying condition (3.14) and such that

∥∥∥β(t, x) − β(t, y)∥∥∥

2

=∥∥∥b(t, x)Tt − b(t, y)TtΛ

1/2t

∥∥∥2

≤ N(1)t (x − y)2 +

∥∥∥(b(t, x)Tt − b(t, y))TtΛ1/2t

∥∥∥2

= N(1)t (x − y)2 + ‖β(t, x) − β(t, y)‖ .

(3.16)



Moreover, by Lemma 3.16 there exists a process N(2)t satisfying (3.14) and such

that

2(x− y)(α(t, x) − (α(t, y)) = 2(x− y)(α(t, x))(η(x) − η(y))

+2(x− y)η(y)(α(t, x) − α(t, y))

≤ 2(η(x) ∧ η(y))(x − y)(α(t, x) − α(t, y)) +N(2)t (x− y)2.

(3.17)

Further, by Lemma 3.16 there exists a process N(3)t satisfying (3.14) and such that

2 |α(t, x)| +∥∥∥β(t, x)

∥∥∥2

≤ N(3)t . (3.18)

Finally, if |x|, |y| ≤ n + 2, then η(y) = 1 and the second inequality in (3.15) is

satisfied by (3.16), (3.17), and (3.11), with N = N (2) + N (3) + K(n + 3). If |x|,|y| ≥ n+ 1, then β(x) = β(y) = 0 and (3.15) is satisfied with N = N (2) +K(n+ 3).

If one of the values of |x|, |y| is less than n + 1 and the other is greater than

n + 2, then |x− y| ≥ 1, and (3.15) is satisfied by (3.17), (3.18), and (3.11), with

N = N (2) +N (3) +K(n+ 3), since one of the values of β(t, x), β(t, y) is zero. Thus,

inequalities (3.15) are satisfied with N = N (1) + N (2) + N (3) + K(n + 3) and the

proof of the lemma is complete.

Lemma 3.19. Suppose there exists a completely measurable process Nt ≥ 0 such

that∫ T0 NsdVs <∞ (a.s.) for all T > 0 and, for all x, y,

|α(t, x)| + |β(t, x)|2 ≤ Nt (a.e. dP × dVt),

2 |x− y| |α(t, x) − α(t, y)| + ‖β(t, x) − β(t, y)‖2 ≤ Nt |x− y|2 (a.e. dP × dVt).

Then equation (3.9) has a unique solution.

Proof. We have already proved the uniqueness of the solution of (3.9). Define

ψ(t) = exp

(−3

∫ t

0

NsdVs − |x0|),

x0t = x0, x

r+1 = x0 +

∫ t

0

a(s, xr(s))dA(s) +

∫ t

0

b(s, xr(s))dm(s), r > 0. (3.19)

Using Ito’s formula for(xr+1(t) − xr(t)

)2ψ(t), it is easy to show that, for some



local martingale mr(t), r ≥ 1,

(xr+1(t) − xr(t)

)2ψ(t) =

∫ t

0

2(xr+1(s) − xr(s)

) (α(s, xr(s)) − α(s, xr−1(s))

)

+∥∥β(s, xr(s)) − β(s, xr−1(s))

∥∥2 − 3Ns(xr+1(s) − xr(s)

)2ψs

dVs +mr(t)

≤∫ t

0

∣∣xr+1(s) − xr(s)∣∣Ns

∣∣xr(s) − xr−1(s)∣∣+Ns

(xr(s) − xr−1(s)

)2

− 3Ns(xr+1(s) − xr(s)

)2 ψsdVs +mr(t)

≤∫ t

0

3

2Ns(xr(s) − xr−1(s)

)2 − 5

2Ns(xr+1(s) − xr(s)

)2ψsdVs +mr(t)

or

(xr+1(t) − xr(t)

)2ψ(t) +

∫ t

0

5

2Ns(xr+1(s) − xr(s)

)2ψsdVs

≤∫ t

0

3

2Ns(xr(s) − xr−1(s)

)2+mr(t).

In the last inequality, we put t = τ i ∧ τ , where τ i are stopping times such that

τ i ↑ ∞ and mr(t ∧ τ i

)is a martingale, and the stopping time τ will be specified

later. Computing the expectations and letting τi ↑ ∞ we conclude that

E

[5

∫ τ

0

(xr+1(s) − xr(s)

)2NsψsdVs + 2

(xr+1(τ) − xr(τ)

)2ψτ

]

≤ 3E

∫ τ

0

(xr(s) − xr−1(s)

)2NsψsdVs, r ≥ 1,

where (xr(τ + 1) − xr(τ))2ψτ is taken equal to zero on the set where τ = ∞.

Similarly,

E

(x1(τ)

)2ψτ + 2

∫ τ

0

(x1(s)

)2NsψsdVs

≤ Ex20e

−|x0| + E

∫ τ

0

NsψsdVs ≤ Ex20e

−|x0| +1

3<∞,

where(x1(τ)

)2ψτ is also taken equal to zero on the set where τ = ∞. From these

inequalities it follow that, for some constant N ′ and for all r, τ,

E

∫ τ

0

(xr+1(s) − xr(s)

)2NsψsdVs ≤ N ′

(3

5

)r,

E(xr+1(τ) − xr(τ)

)2ψτ ≤ N ′

(3

5

)r.

We now take τ = τ(r) = inft :(xr+1(t) − xr(t)

)2ψt ≥ r−4

and conclude that

r−4P τ(r) <∞ ≤ N ′(

3

5

)r,



P

supt≥0

(xr+1(t) − xr(t)

)2ψt ≥ r−4

≤ N ′r4

(3

5

)r.

Hence, by the Borel-Cantelli lemma the series

∞∑

r=1

∣∣xr+1(t) − xr(t)2∣∣ψ1/2

t

converges uniformly for almost all ω on each finite time segment to some continuous

process x(t). Passing to the limit in (3.19), we complete the proof of the lemma.

Lemma 3.20. We choose n > 0, a, b, N from Lemma 3.18. If in equation (3.9)

a, b are replaced by a, b, then the equation has a unique solution.

Proof. For r = 1, 2, . . . we define functions ar, br by convolution of a, b on x

with a δ-type sequence of infinitely differentiable, compactly supported, nonnegative

functions of the form ζ(jrx)jdr , where jr ≥ 0, jr → ∞ as r → ∞. Since for any r

the first derivatives with respect to x and the functions ar, br can be bounded in

terms of the maxima of |a|, |b|, by the first inequality in (3.15) the conditions of

Lemma 3.19 are satisfied for a, b. Therefore, for every r > 0, there exists a unique

solution of the equation

xr(t) = x0 +

∫ t

0

ar(s, xr(s))dA(s) +

∫ t

0

br(s, xr(s))dm(s). (3.20)

Since a(t, x) = 0, b(t, x) = 0 for |x| ≥ n + 3 and ζ is a compactly supported

function, it can be assumed with no loss of generality that ar(t, x) = 0, br(t, x) = 0

for |x| ≥ n+ 4. In this case if |x0| ≤ n+ 4, then the process xr(t) never leaves the

set x : |x| ≤ n+ 4. If |x0| ≥ n+ 4, then xr(t) = x0 for all t. This implies that

supt≥0

supr

|xr(t) − x0| ≤ 2n+ 8 (a.s.) (3.21)

Further, we choose Nt from Lemma 3.18. We note that by (3.15) for αr =

ar(dA/dV ), βr = brC1/2, we have

∥∥∥βr(t, xr(t))∥∥∥

2

≤ supx

∥∥∥βr(t, x)∥∥∥

2

≤ supx

∥∥∥β(t, x)∥∥∥

2

≤ Nt, |αr(t, xr(t))| ≤ Nt.

(3.22)

We define ψt = exp(−∫ t0 NsdVs

)and show next that

limr,p→∞

supτ

E |xr(τ) − xp(τ)|2 ψt = 0, (3.23)

where the supremum is taken over all stopping times τ and (xr(τ) − xp(τ))2ψt is

set equal to zero on when τ = ∞.



By Ito’s formula, for some local martingale mr,p(t),

d |xr(t) − xp(t)|2 ψt =

2 (xr(t) − xp(t)) (αr(t, xr(t)) − αr(t, xp(t)))

+∥∥∥βr(t, xr(t)) − βr(t, xp(t))

∥∥∥2

−Nt (xr(t) − xp(t))2ψtdVt

+

2 (xr(t) − xp(t)) (αr(t, xp(t)) − αp(t, xp(t)))

+∥∥∥βr(t, xp(t)) − βp(t, xp(t))

∥∥∥2 ψtdVt

+ 2[βr(t, xp(t)) − βp(t, xp(t)), βr(t, xr(t)) − βr(t, xp(t))

]ψtdVt + dmr,p(t),

(3.24)

where, for two d× d1 matrices, [σ1,σ2] is the sum of all products of the form σij1 σij2 .

Application of the second inequality in (3.15) and the Cauchy-Schwartz inequality

shows that

2 (xr(t) − xp(t)) (αr(t, xr(t)) − αr(t, xp(t)))

+∥∥∥βr(t, xr(t)) − βr(t, xp(t))

∥∥∥2

−Nt (xr(t) − xp(t))2 ≤ 0

(a.e. dP × dVt). Moreover, the functions

(xr(t) − xp(t)) (αr(t, xp(t)) − αp(t, xp(t)))ψt

by (3.21), (3.22) are bounded in absolute value by 4(2n+8)Ntψt, which is summable

with respect to dP × dVt, and for allt, ω. Also, these function tend to zero as r,

p→ ∞ by continuity in x of the compactly supported α(t, x). For all t, ω, we have

|αr(t, x) − α(t, x)| as r → ∞ uniformly with respect to x ∈ Rd. Therefore,

limp,r→∞

E

∫ ∞

0

|(xr(s) − xp(s)) (αr(s, xp(s)) − αp(s, xp(s)))|ψsdVs = 0.

Similarly,

limr,p→∞

E

∫ ∞

0

∥∥∥βr(s, xp(s)) − βp(s, xp(s))∥∥∥

2

ψsdVs = 0,

limr,p→∞

E

∫ ∞

0

∣∣∣[βr(s, xp(s)) − βp(s, xp(s)), βr(s, xr(s)) − βr(s, xp(s))]∣∣∣ψsdVs.

It is now clear that (3.23) follows from (3.24). From (3.23), in turn, we obtain the

existence of a subsequence r(i) such that, for all stopping times τ ,

E

∣∣∣xr(i+1)(τ) − xr(i)(τ)∣∣∣ψτ ≤ 2−i.

Finally, as in Lemma 3.19, from this we obtain the uniform convergence of xr(i)(t)

with respect to t on any finite time interval, and by passing to the limit in (3.19)

along this subsequence r(i) we complete the proof of the lemma.



Proof of Theorem 3.14. Denote by an, bn the functions a, b constructed in

Lemma 3.18 for a given n > 0. By Lemma 3.20, for every n > 0, there exists a

solution of the equation

xn(t) = x0 +

∫ t

0

an(s, xn(s))dA(s) +

∫ t

0

bn(s, xn(s))dm(s).

Let τn = inf t : |xn(t)| ≥ n, τn,m = τn ∧ τm. Since an(t, x) = a(t, x) and

bn(t, x) = b(t, x) for |x| ≤ n, the processes xn (t ∧ τn,m), xm (t ∧ τn,m) satisfy the

same equation

dz(t) = χt<τn,ma(t, z(t))dA(t) + χ

t<τn,m b(t, z(t))dm(t).

Hence these processes coincide and xn(t) = xm(t) on t ≤ τn,m (a.s.). Therefore,

τn ≤ τm (a.s.) for n ≤ m and there exists τ = limn→∞ τn (a.s.). Also, for almost

all ω and t < τ , the process x(t) = limn→∞ xn(t) is defined, and for every n and all

t (a.s.)

x (t ∧ τn) = xn (t ∧ τn) = x0 +

∫ t∧τn

0

a(s, x(s))dA(s) +

∫ t∧τn

0

b(s, x(s))dm(s).

It remains to prove that τ = ∞. Define

ψt

= exp

(−∫ t

0

KsdVs − |x0|).

As we have done repeatedly, by means of (3.12) and Ito’s formula we find that

E(xn(τn))2ψτnχτn<∞ ≤ Ex2

0ψ0 + E

∫ ∞

0

KtψtdVt <∞.

Therefore

n2Eψτnχτn<∞ ≤ N, or Eψτnχ

τn<∞ → 0 and τn → ∞ (a.s.)

as n→ ∞. The proof of the theorem is complete.

Remark 3.21. The assumption of complete measurability of a(t, x), b(t, x), Kt(R)

ensures that the corresponding stochastic integrals with respect to dm(t) are defined

and the integrals with respect to dA(t) and dVt are Ft-consistent. Theorem 3.14 can

hold without this assumption. For example, we can instead assume that dA(t) dt,

a(t, x) is measurable in (t, ω) and is Ft-consistent, as are those elements of the matrix

b which are multiplied by dmi(t) with d〈mi〉t dt, while the remaining elements

of b are completely measurable, and Kt(R) is progressively measurable.

3.4. Uniqueness Theorem: A Priori Estimates and

Finite-Dimensional Approximations

In this section we prove Theorems 3.8 and 3.10 and prepare for the proof of Theorem

3.6.



Proof of Theorem 3.8. By Theorem 2.17, for yn = un − u0 we find for all t

(a.s.)

|yn(t)|2H =

∫ t

0

2(vn(s) − v0(s))

(A(vn(s), s) −A(v0(s), s)

)

+∣∣B(vn(s), s) −B(v0(s), s)

∣∣2Q

ds+mn

t +∣∣un0 − u0

0

∣∣2H,

where mnt is a local martingale with mn

0 = 0. We use this equality and apply Ito’s

formula to compute |yn(t)|2H e−Kt. After this, we use the monotonicity of (A,B)

(condition (A2)) and the equality y = vn − v0 (a.e. in (t, ω)). Then, for some local

martingales mnt , we obtain

|yn(t)|2H e−Kt ≤∣∣un0 − u0

0

∣∣2H

+ mnt ≡ ξn(t) (a.s.)

We see that the local martingales ξn(t) are nonnegative. Hence ξn(t) is a super-

martingale and, by assumption, Eξn(0) → 0. Therefore, for every ε > 0,

limn→∞

supt≤τ

Eξn(t) + P

supt≤τ

ξn(t) ≥ ε

= 0.

Application of the inequality |yn(t)|2H ≤ ξn(t)eKt completes the proof of Theorem

3.8.

Proof of Theorem 3.10. Let v be some solution of equation (3.4) and let u

be its continuous modification in H . By Theorem 2.17, for all t ∈ [0, T ] on a single

set of full probability,

|u(t)|2H = |u0|2H + 2

∫ t

0

v(s)A(v(s))ds + 2

∫ t

0

u(s)(B(v(s))dw(s) + dz(s)

)

+

⟨∫ t

0

B(v(s))dw(s) + z

⟩

t

.

(3.25)

Let

τn = inft : |u(t)|2H ≥ n

∧ T.

It is obvious that the stochastic integral on the right side of (3.26) is a local mar-

tingale with localizing sequence τn. Therefore,

E |u(t ∧ τn)|2H = E |u0|2H + 2E

∫ t∧τn

0

v(s)A(v(s))ds + E

∫ t∧τn

0

|B(vn(s))|2Q

+ E |z(t ∧ τn|2H + 2E

(∫ t∧τn

0

B(v(s))dw(s), z(t ∧ τn)

)

H

.

Hence, by assumption (A3)

E |u(t ∧ τn)|2H ≤ E |u0|2H − αE

∫ t∧τn

0

|v(s)|pV ds+

∫ t∧τn

0

(f(s) +K |u(s)|2H ds

+ E |z(t ∧ τn|2H + 2E

(∫ t∧τn

0

B(v(s))dw(s), z(t ∧ τn)

)

H

.

(3.26)



We now make use of the elementary inequality

2ab ≤ εa2 +1

εb2, ε > 0. (3.27)

Applying this inequality to the last term in (3.26), we obtain

E |u(t ∧ τn)|2H ≤ E |u0|2H − αE

∫ t∧τn

0

|v(s)|pV ds+

∫ t∧τn

0

(f(s) +K |u(s)|2H ds

+

(1 +

1

ε

)E〈z〉T + εE

∫ t∧τn

0

|B(v(s))|2Q ds.

(3.28)

On the other hand, in view of inequality (3.7),

E

∫ t∧τn

0

|B(v(s))|2Q ds ≤ cE

∫ t∧τn

0

(f(s) + |v(s)|pV + |u(s)|2H

)ds. (3.29)

Combining (3.28) and (3.29), and choosing ε sufficiently small, for some α1 we

obtain

E |u(t ∧ τn)|2H ≤ E |u0|2H − α1E

∫ t∧τn

0

|v(s)|pV ds

+ c

(E

∫ t

0

f(s)ds+ E〈z〉T +

∫ t

0

E |u(s ∧ τn)|2H ds).

(3.30)

From this by the Gronwall lemma

supt≤T

E |u(t ∧ τn)|2H ≤ c

(E |u0|2H + E

∫ T

0

f(s)ds+ E〈z〉T). (3.31)

Thus, from (3.30) and (3.31) it follows that

supt≤T

E |u(t ∧ τn)|2H + E

∫ t∧τn

0

|v(s)|pV ds ≤ c

(E |u0|2H + E

∫ τ

0

f(s)ds+ E〈z〉T).

(3.32)

Continuity of |u(t)|2H in t implies limn→∞ τn = T . Therefore, passing to the limit

as n → ∞ in (3.32) (by Fatou’s lemma and the monotone convergence theorem),

we obtain

supt≤T

E |u(t)|2H + E

∫ T

0

|v(s)|pV ds ≤ c

(E |u0|2H + E

∫ τ

0

f(s)ds+ E〈z〉T). (3.33)

To complete the proof of Theorem 3.10 it remains to show that an analogous

estimate also holds for E supt≤τ |u(t)|2H . For this we note that from (3.25) because



of (A3) and (3.27) it follows that

E supt≤T

|u(t ∧ τn)|2H ≤ E |u0|2H + E

∫ T

0

(f(s) +K|u(s)|2H

)ds

+ 2E supt≤T

∣∣∣∣∫ t∧τn

0


)∣∣∣∣2

+ 2E

∣∣∣∣∣

∫ T

0

B(v(s))dw(s)

∣∣∣∣∣

2

H

+ 2E〈z〉T .

(3.34)

Define

M(t) =

∫ t

0

B(v(s))dw(s) + z(t).

Using the Burkholder inequality (Theorem 2.13) and (3.27), we obtain

E supt≤T

∣∣∣∣∫ t∧τn

0


)∣∣∣∣2

≤ 3E

(∫ t∧τn

0

|u(s)|2H d〈M〉s)1/2

≤ 3

2εE sup

t≤T|u(t ∧ τn)|2H +

3

2εE |M(T )|2H

≤ 3

2εE sup

t≤T|u(t ∧ τn)|2H +

3

εE

∫ τ

0

|B(v(s))|2Q ds+3

εE〈z〉T .

(3.35)

Combining (3.34) and (3.35), choosing ε small, and using (3.29), (3.33), we obtain

E supt≤T

|u(t ∧ τn)|2H ≤ c

(E |u0|2H + E

∫ T

0

f(s)ds+ E〈z〉T).

Because of the continuity of |u(t)|H , the assertion of Theorem 3.10 follows from this

on the basis of Fatou’s lemma. Theorem 3.10 is proved.

We will now prepare for the proof of Theorem 3.6. For this we approximate

equation (3.4) by equations in finite-dimensional spaces.

Fix orthonormal bases hi, ei in the spaces H and E, respectively, and

assume that hi ∈ V , i = 1, 2, . . ., while ei are the eigenvectors of the covariance

operator Q of the Wiener process with the corresponding eigenvalues λi > 0: Qei =

λie1; this is possible because Q is a completely continuous (compact) operator.

Then, for each i, λ− 1

2i (w(t), ei)E is a one-dimensional standard Wiener process, and

for different i these process are independent.

For n ≥ 1 we consider the following system of stochastic equations for

(u1n(t), . . . , unn(t)) ∈ Rn



uin(t) = (hi, u0)H +

∫ t

0

hiA

n∑

j=1

ujn(s)hj , s

ds

+

n∑

j=1

∫ t

0

(hi, B

(n∑

k=1

ukn(s)hk, s

)ej

)

H

d(w(s), ej )E

+ (hi, z(t))H , i = 1, . . . , n, t ∈ [0, T ].

(3.36)

Lemma 3.22. System (3.36) has a unique Ft-measurable solution which is contin-

uous in t.

Proof. It suffices to show that system (3.36) can be written in the form (3.9) and

to verify assumptions of Theorem 3.14. We note immediately that the measurability

conditions are verified by using Remark 3.21 at the end of Section 3.3, and we leave

this verification to the reader.

We choose d = n, d1 = 2n and we set

a(t, x) = (ai(t, x), i = 1, . . . , n),

b(t, x) = (bij(t, x), i = 1, . . . , n, j = 1, . . . , 2n),

m(t) = (mi(t), i = 1, . . . , 2n), A(t) = t,

mi(t) = (w(t), ei)E , mn+i(t) = (z(t), hi)H , 1 ≤ i ≤ n,

ai(t, x) = hiA

n∑

j=1

xjhj , t

, 1 ≤ i ≤ n,

bij(t, x) =

(hi, B

(∑nk=1 x

khk, t)ej)H, 1 ≤ i, j ≤ n,

δn+i,j , 1 ≤ i ≤ n, n+ 1 ≤ j ≤ 2n,

where δk,j is the Kronecker symbol. With these notations it is obvious that (3.36)

can be written in form (3.9). As in Section 3.3, we introduce the functions cij(t),

α(t, x), β(t, x), corresponding to Vt = t+ 〈w〉t + 〈z〉t. It is clear that (3.10) follows

from (A4) and (A5).

Further, we choose x, y ∈ Rn and set u =∑n

i=1 xihi, v =

∑ni=1 y

ihi. It is no

hard to see that

‖β(t, x) − β(t, y)‖2dVt =

n∑

i,j=1

(hi,(B (u, t) −B(v, t)

)ej

)2

Hλjdt

≤∞∑

j=1

λj |(B (u, t) −B(v, t)) ej |2H dt = |B (u, t) −B(v, t)|2Q dt.(3.37)



Hence, by condition (A2)

2(x− y)(a(t, x) − a(t, y)) + ‖β(t, x) − β(t, y)‖2dVt

≤ 2(u− v)big(A(u, t) −A(v, t))dt+ |B (u, t) −B(v, t)|2Q dt

≤ K |u− v|2H dt = K (u− v)2dt.

Therefore, condition (3.11) is satisfied. Moreover, continuity of a(t, x) in x can be

derived from Lemma 3.2, and continuity of b(t, x), from the continuity of B (u, t)

which, in turn, follows from (A2). It remains to check condition (3.12).

Define

ξ(t) =

∫ t

0

b(s, x)dm(s).

For every ε > 0 we have by (3.7) in analogy to (3.37)

d〈ξ〉t ≤ (1 + ε)

n∑

i,j=1

λj (hi, B (u, t) ej)2H dt+

(1 +

1

ε

) n∑

i=1

d〈mn+i〉t

≤ |B (u, t)|2Q dt+

(1 +

1

ε

)d〈z〉t + εc

(f(t) + |u|2H + |u|pV

)dt,

(3.38)

where c does not depend on ε. We note further that

2(xα(t, x) + ‖β(t, x)‖2 dVt = 2uA(u, t)dt+ d〈ξ〉t. (3.39)

Combining (3.38) and (3.39) with condition (A3) and then choosing ε sufficiently

small, we conclude that(

2xα(t, x) + ‖β(t, x)‖2)dVt ≤ c

(x2 + f(t)

)dt+ cd〈z〉t

≤ c(1 + x2)

(1 + f(t)

dt

dVt

)dVt,

where c is a constant not depending on t, x, ω. Hence, condition (3.12) is satisfied,

and the proof of the lemma is complete.

Denote by Πn the projection operator of V ∗ onto the span of h1, . . . , hn, by

πn the projection operator of E onto the span of e1, . . . , en, and set un(t) =∑ni=1 u

inhi. It is then obvious that (3.36) is equivalent to the equation

un(t) = u0 +

∫ t

0

ΠnA(un(s), s)ds +

∫ t

0

ΠnB(un(s), s)πndw(s) + Πnz(t). (3.40)

To complete the preparations, we need the following result.

Theorem 3.23. 1) There exists a constant C0 such that, for all n at once,

E supt≤τ

|un(t)|2H + E

∫ T

0

|un(t)|pV dt ≤ C0

(E |u0|2H + E

∫ T

0

f(t)dt+ E〈z〉T)

;



2) With

Mn(t) =

∫ t

0

ΠnB(un(s))πndw(s), (3.41)

the function un satisfies

Ee−ct |un(t)|2H = E |Πnu0|2H

+ E

∫ t

0

e−cs

2un(s)A(un(s)) + |ΠnB(un(s))πn|2H − c |un(s)|2Hds

+ 2E

∫ t

0

e−csd〈z,Mn〉s + E

∫ t

0

e−csd〈Πnz〉s

for all t ≤ T and for every constant c.

The proof of this theorem is almost a word-for-word repetition of the proof of

Theorem 3.10, and we leave it to the reader. We only point out that

unΠnA(un) = unA(un), |ΠnB(un)πn|Q = |B(un)|Q ,and that assertion 2) is proved by applying the usual Ito formula, since un(t) lies

in a finite-dimensional subspace of H .

3.5. Existence of Solution and the Markov Property: Passing to the

Limit by the Method of Monotonicity

In this section we complete the proof of Theorem 3.6 and prove Theorem 3.11.

We denote by T the σ-algebra of progressively measurable sets on [0, T ] × Ω

and by T , its completion with respect to the measure ` × P , where ` is Lebesgue

measure on [0, T ]. We define S = ([0, T ] × Ω, T , ` × P ). From the results of the

preceding section (Theorem 3.23) it follows that there exists a subsequence κ of

the natural numbers along which, for some u, v, u∞(T ),

uκ → u weakly in L2(S;H), (3.42)

uκ → v weakly in Lp(S;V ), (3.43)

uκ(T ) → u∞(T ) weakly in L2(Ω,FT , P ;H). (3.44)

By Theorem 3.23 and conditions (A3), (A4) it may be assumed that

Auκ → A∞ weakly in Lq(S;V ∗), (3.45)

ΠκB(uκ)πκ → A∞ weakly in L2(S;LQ(E;H)). (3.46)

Strictly speaking, u, v, u∞(T ), A∞, and B∞ are all equivalence classes, but in each

class u, v, A∞, B∞ it is possible to choose a progressive measurable representative,

and in the class u∞(T ), a FT -measurable representative. We henceforth consider

only these representatives and preserve for them the notation of the corresponding



classes. We note also that since the embedding of V in H is dense, it follows that

u = v for a.a. (t, ω).

Further, it is well known that a strongly continuous linear operator is weakly

continuous. Therefore, because of (3.46),∫ t

0

ΠκB(uκ(s))πκdw(s) →∫ t

0

B∞(s)dw(s) (3.47)

weakly in L2(S;H) and in L2(Ω,Ft, P ;H) for each t

Let y be a bounded random variable, and let ψ(t) be a bounded function on

[0, T ]. It follows from (3.40) that, for each hi and κ ≥ 1

E

∫ T

0

yψ(t)(uκ(t), hi)Hdt

= E

∫ T

0

yψ(t)hi

(u0 +

∫ t

0

A(uκ(s))ds+

∫ t

0

ΠκB(uκ(s))πκ dw(s) + z(t)

)dt.

Passing to the limit in this equality and using (3.43), (3.45)–(3.47), we obtain

E

∫ T

0

yψ(t)(v(t), hi)Hdt

= E

∫ T

0

yψ(t)hi

(u0 +

∫ t

0

A∞(s)ds+

∫ t

0

B∞(s)dw(s) + z(t)

)dt.

From this it follows that for a.a. (t, ω)

v(t) = u0 +

∫ t

0

A∞(s)ds+

∫ t

0

B∞(s)dw(s) + z(t). (3.48)

In the same way, using (3.44)–(3.47), we find that (a.s.)

u∞(T ) = u0 +

∫ T

0

A∞(s)ds+

∫ T

0

B∞(s)dw(s) + z(T ). (3.49)

By Theorem 2.17, there exists an Ft-consistent function continuous in t with

values in H which coincides with v(t) for a.a. (t, ω) and is equal to the right side

of (3.48) for all t ∈ [0, T ] and ω ∈ Ω′, where P (Ω′) = 1. We identify v with u; this

is possible, since u = v a.e. in (t, ω). In view of (3.49) we then have

u∞(T ) = u(T ) (a.s.). (3.50)

By the same Theorem 2.17, for all (t, ω) ∈ [0, T ] × Ω′,

|u(t)|2H = |u0|2H + 2

∫ t

0

v(s)A∞(s)ds+ 2

∫ t

0

u(s)(B∞(s)dw(s) + dz(s)

)

+ 〈M∞ + z〉t,(3.51)

where

M∞(t) =

∫ t

0

B∞(s)dw(s).



The remaining argument in the proof of Theorem 3.6 are standard in the theory of

monotone operators and do not require special attention of the reader, because, on

the first reading, these arguments produce the impression of a collection of trivial

computations, mysteriously leading to the required result.

Let y(t, ω) be an Ft-consistent function, measurable in (t, ω) with values in V

and satisfying

E

∫ T

0

(|(y(t))|pV + |(y(t))|2H

)dt <∞. (3.52)

Define

Oκ = E

∫ T

0

e−ct

2(uκ(t) − y(t))(A(uκ(t) −A(y(t)) − c |uv(t) − y(t)|2H

+ |ΠκB(uκ(t))πκ − ΠκB(y(t))πκ|2Qdt.

From (A2) it follows that, with a suitable choice of c,

Oκ ≤ 0. (3.53)

Further, we represent Oκ in the form Oκ = O1κ + O2

κ where

O1κ = E

∫ T

0

e−ct

2(uκ(t)A(uκ(t)) − c |uκ(t)|2H + |ΠκB(uκ(t))πκ|2Qdt,

and O2κ = Oκ −O1

κ. By Theorem 3.23,

O1κ = Ee−ct |uκ(T )|2H − E |Πκu0|2H − 2 E

∫ T

0

e−ctd〈Mκ, z〉t − E

∫ T

0

e−ctd〈Πκz〉t,

with Mκ defined in (3.41). After integration by parts,

E

∫ T

0

e−ctd〈Mκ, z〉t = E e−cT(∫ T

0

ΠκB(uκ(s))πκdw(s), z(T )

)

H

+ cE

∫ T

0

e−ct(∫ t

0

ΠκB(uκ(s))πκdw(s), z(t)

)

H

dt.

From this and (3.47) it follows that

lim supκ→∞

O1κ = Ee−cT |u(T )|2H − E |u0|2H − E

∫ T

0

e−ctd〈z〉t

− 2 E

∫ T

0

e−ctd〈M∞, z〉t + δ e−cT ,

(3.54)

where δ = lim supκ→∞

E |uκ(T )|2H−E |u(T )|2H ≥ 0 by (3.44). On the other hand, because



of (3.51),

Ee−cT |u(T )|2H − E |u0|2H = E

∫ T

0

e−ct

2(v(t)A∞(t) − c |u(t)|2H + |B∞(t)|2Qdt

+ 2E

∫ T

0

e−ctd〈M∞, z〉t + E

∫ T

0

e−ctd〈z〉t.(3.55)

Comparing (3.54) and (3.55) we see that

lim supκ→∞

O1κ = E

∫ T

0

e−ct

2(v(t)A∞(t) − c |u(t)|2H + |B∞(t)|2Qdt+ δ e−cT . (3.56)

Further, from (3.42), (3.43), (3.45), and (3.47) it follows that

limκ→∞

O2κ = E

∫ T

0

e−ct(

2y(t)A(y(t)) − 2y(t)A∞(t) − 2v(t)A(y(t))

+ 2c(u(t), y(t))H − c |y(t)|2H − 2(B∞(t), B(y(t))

)Q

+ |B(y(t))|2Q)dt.

(3.57)

Combining (3.56) and (3.57) we find, in view of (3.53),

E

∫ T

0

e−ct

2(v(t) − y(t))(A∞(t) −A(y(t))) − c |u(t) − y(t)|2H

+ |B∞(t) −B(y(t))|2Qdt+ δ e−cT ≤ 0.

(3.58)

Setting y = v in (3.58), we see that B∞(t) = B(v(t)) (a.e. in (t, ω)), and

δ = lim supκ→∞

E |uκ(T )|2H − E |u(T )|2H = 0. (3.59)

On the other hand, it follows from (3.58) that

E

∫ T

0

e−ct

2(v(t) − y(t))(A∞(t) −A(y(t))

)− c |u(t) − y(t)|2H

dt ≤ 0. (3.60)

Suppose now that x(t, ω) is a process with values in V which satisfies an in-

equality analogous to (3.52), and let y = v − λx, λ ∈ R+. From (3.60) we then

obtain the inequality

E

∫ T

0

e−ctx(t)

(A∞(t) −A

(v(t) + λx(t)

))− cλ |x(t)|2H

dt ≤ 0.

Letting λ go to zero, we find from this by (A4) and the Lebesgue theorem that

E

∫ T

0

x(t)(A∞(t) −A(v(t)))dt ≤ 0.

Since x is arbitrary, the last inequality implies A∞(t) = A(v(t)) a.e. in (t, ω), which

together with (3.48) and an earlier proved equality B∞(t) = B(v(t)) a.e. in (t, ω)

complete the proof of Theorem 3.6.



Corollary 3.24. Let v be a solution of (3.4) and u, its continuous modification in

H, and let un be solutions of (3.40). Then for any t ≤ T

limn→∞

E |un(t) − u(t)|2H = 0.

Proof. Note that we proved equality (3.59) in which u is a continuous modifica-

tion of v in H . On the other hand, from (3.44) we have

E |u(T )|2H ≤ limκ→∞

E |uκ(T )|2Hand so

limκ→∞

E |uκ(T )|2H = E |u(T )|2H .

Returning again to (3.44) and recalling that a weakly convergent sequence converges

if the norm of the limit is equal to the limit of the norm, we conclude that

limκ→∞

E |uκ(T ) − u(T )|2H = 0.

Here u(T ) does not depend on the sequence κ by Theorem 2.3. We have thus

proved that any subsequence of the sequence un(T ), n = 1, 2, 3, . . . contains a

further subsequence converging strongly in L2(Ω,FT , P ;H) to u(T ). This implies

the convergence of the whole sequence: un(T ) → u(T ) as n → ∞. It remains to

note that, in place of segment [0, T ], we can consider any smaller segment.

Proof of Theorem 3.11. Together with (3.4), we consider the equation

v(t) = us +

∫ t

s

A(v(r), r)dr +

∫ t

s

B(v(r), r)dw(r), (3.61)

which we solve for t ∈ [s, T ] and E |u(s)|2H < ∞. The definitions of a V -solution

and H-solution are made for equation (3.61) in the usual way.

All the assertions proved for equation (3.4) carry over to (3.61) in a natural

way; in particular, (3.61) has an H-solution u(t) = u(t, s, us). We now assume that

us = x ∈ H is non-random and, for t ∈ [s, T ] and a bounded Borel function f(h)

on H , we define

Ms,xf(u(t)) = Ef(u(t, s, x)). (3.62)

To prove the theorem it obviously suffices to show that (3.62) defines a Borel

function of x, and for the H-solution u(t) of (3.4),

E f(u(t))|Fs = Ms,u(s)f(u(t) (a.s.) (3.63)

The required property of Borel measurability is easily proved by means of a cor-

responding analog of Theorem 3.8, which shows that for continuous f the function

Ms,xf(u(t)) is also continuous in x. It suffices to prove the relation (3.63) only for

continuous f . Moreover, for simplicity, we assume that s = 0. This can always be

achieved by changing the time variable.



We note, first of all, that u(t, 0, x) does not depend on F0; this follows from

Corollary 3.24 and the fact that in Section 3.3 solutions of stochastic equations

were constructed in the final analysis by passing to the limit in equations with coef-

ficients that satisfy a Lipschitz condition, and for such equations the independence

of a solution with nonrandom initial data from F0 is known [42, 81]. Further, we

approximate u0(ω) by step functions un0 (ω) and let Γn be the set of values of un0 (ω).

It is easy to see that u(t, 0, un0 ) and∑

x∈Γn

χx(un0 )u(t, 0, x)

satisfy the same equation and therefore coincide (a.s.) From this and the indepen-

dence of u(t, 0, x) and F0, it follows that

E f(u(t, 0, un0 ))|F0 = M0,un0f(u(t)) (a.s.)

Letting here n→ ∞ and using Theorem 3.8 we obtain (3.63) for continuous f . The

proof of Theorem 3.11 is complete.

4. Ito Stochastic Partial Differential Equations

4.1. Introduction

This section is devoted to applications of the results of Section 3 to stochastic partial

differential equations. We consider the first boundary-value problem for nonlinear

equations of second order. The latter merit special attention, since filtering of

diffusion process reduces to the investigation of equations of this type [1, 2, 82].

The results pertaining to nonlinear equations are related to Ref. [62], where

equations without stochastic terms are considered, while elements of the theory of

linear equations are presented following our work [64].

We start by recalling the definitions and some basic facts from the theory of

Sobolev spaces.

Let Rd be d-dimensional Euclidean space with a fixed basis. Denote by α, β,

γ, αi, βi, γi, (i = 1, 2, . . .) the coordinate vectors in this space and also the null

vector. If α is the null vector, then Dα is the identity operator, while if α is the

i− th basis vector, then Dα = ∂/∂xi. Suppose further that G is a domain in Rd, Γ

is the boundary of G, m ≥ 1 is an integer, p ∈ (1,∞).

Definition 4.1. The Sobolev space Wmp (G) is the space of real functions defined

on G with finite norm

‖u‖m,p ≡( ∑

α1...αm

‖Dα1 . . .Dαmu‖pp

)1/p

,

where Dα1 , . . . , Dαm are generalized derivatives, and

‖g‖p =

(∫

G

|g(x)|p dx)1/p

.



Definition 4.2. The Sobolev spaceWm

p (G) is the closure of C∞0 (G) (the space of

infinitely differentiable functions with compact supports in G) in the norm ‖·‖m,p.

Theorem 4.3. The spacesWm

p (G) are separable reflexive Banach spaces relative

to the norm ‖·‖m,p.

Theorem 4.4 (Sobolev Embedding Theorem). If the domain G is bounded,

its boundary Γ is regular, and 2(d − mp) ≤ pd, then Wmp (G) ⊂ L2(G) and this

embedding is dense and continuous. In particular, there exists a constant c such

that ‖u‖2 ≤ c ‖u‖m,p for all u ∈ Wmp (G).

Proofs of these theorems can be found in Refs. [83–85]; the conditions on the

boundary we call regular are also presented there.

The following assertion is obvious.

Theorem 4.5. The Sobolev space Wm2 (Rd) is a Hilbert space; it is continuously

and densely imbedded in L2(Rd).

We further need below the so-called Friederichs inequality [83, 85] which is given

in the following theorem.

Theorem 4.6. Suppose that the domain G is bounded and its boundary is regular.

Then there exists a constant c > 0 such that, for every u ∈Wm

p (G),

‖u‖m,p ≤ c

∑

|α1|+...+|αm|=m‖Dα1 . . .Dαmu‖p

.

The next fact is also well know; it follows easily by means of the Fourier trans-

form and the Parseval equality.

Theorem 4.7. There exists a constant c > 0 such that, for every u ∈Wm2 (Rd),

‖u‖m,2 ≤ c

∑

|α1|+...+|αm|=m‖Dα1 . . . Dαmu‖2 + ‖u‖2

.

We will also need the spaces W−mq (G) where q = p/(p− 1); Refs. [51, 85] present

a detailed account of these spaces. It is assumed below that either G is bounded

and its boundary is regular, or G = Rd and p = 2.

Definition 4.8. The negative norm of an element f ∈ Lq(G) is

‖f‖−m,q = sup(f, u)0,

where

(f, u)0 =

∫

G

f(x)u(x)dx



and the supremum is taken over the set of all functions u ∈Wm

p (G) for which

‖u‖m,p = 1.

Definition 4.9. The space with negative normW−mq (G) is the completion of Lq(G)

in the norm ‖·‖−m,p.

It is clear that, for f ∈ Lq(G) and u ∈Wm

p (G), we have

|(f, u)0| ≤ ‖f‖q · ‖u‖p ≤ ‖f‖q · ‖u‖m,p .Definition 4.9 is therefore correct.

From the above definitions it follows immediately that there are the natural

embeddings

Wm

p (G) ⊂ Lp(G) ⊂W−mp (G).

The duality betweenWm

p (G) and W−mq (G) is defined by means of the scalar

product in L2(G): if v ∈Wm

p (G), v∗ ∈ W−mq (G), vn ∈ C∞

0 , v∗n ∈ Lq(G), and

‖vn − v‖m,p → 0, then we define

〈〈v, v∗〉〉 = limn→∞

(vn, v∗n)0.

It is known that, for every continuous linear functional onWm

p (G), there exists

a v∗ ∈ W−mq (G) such that this functional is equal to 〈〈v, v∗〉〉. Similarly, every

continuous linear functional on W−mq (G) can be written as 〈〈v, v∗〉〉 for some v ∈

Wm

p (G). Moreover, for v ∈Wm

p (G) and v∗ ∈ W−mp (G),

‖v‖m,p = supw∗

〈〈v, v∗〉〉‖w∗‖−m,q

, ‖v∗‖ = supv∗

〈〈w, v∗〉〉‖w‖m,q

.

The relation 〈〈·, ·, 〉〉 makes it possible to identify the dual space ofWm

p (G) with

W−mq (G). We note that under this identification the duality 〈〈·, ·, 〉〉 plays a decisive

role. In particular, the spaceWm

2 (G) is a Hilbert space and, as any other Hilbert

space, can be identified in the well-known way with its dual space by means of

its own scalar product. At the same time, of course,Wm

2 (G) 6= W−m2 (G). Thus,

different bilinear forms make it possible to construct dual spaces ofWm

2 (G) in

different ways. We shall use these considerations in the proof of Theorem 4.20

below.

We identify the dual space ofWm

p (G) with W−mq (G), so that the result of

applying a functional v∗ ∈ W−mq (G) to an element v ∈

Wm

p (G) is written as 〈〈v,

v∗〉〉. This notation differs from the notation vv∗ used in Section 2 and 3 because

both v and v∗ can now be ordinary functions and then vv∗ can be misinterpreted

as the usual product of two functions.



Theorem 4.10. Let 2(d −mp) ≤ pd. ThenWm

p (G) ⊂ L2(G) ⊂ W−mq (G), where

each embedding is dense and continuous. If v∗ ∈ L2(G) and v ∈Wm

p then 〈〈v,v∗〉〉 = (v, v∗)0.

4.2. First Boundary-Value Problem for Nonlinear Stochastic

Parabolic Equations

Let (Ω,F , P ) be a complete probability space; let Ft be an increasing family

of complete σ-algebras imbedded in F ; let Gbe a bounded domain with a regular

boundary or G = Rd and p = 2. We assume that 2(d −mp) ≤ pd. Let z(t) be a

continuous square-integrable martingale (relative to Ft) with values in L2(G) and

let w(t) be a Wiener process with values in some separable Hilbert space E and

with covariance operator Q. By Theorem 4.10 the spaces V =Wm

p (G), H = L2(G),

V ∗ = W−mq (G) satisfy assumptions a) – d) of Section 3.

In the cylinder [0, T ]×G for fixed T > 0 we consider the problem

du(t, x, ω) = −(−1)|α1|+...+|αm|Dα1. . . DαmAα1...αm

(Dβ1. . . Dβmu(t, x, ω), t, x, ω

)dt

+B(Dβ1 . . . Dβmu(t, x, ω), t, x, ω

)dw(t, ω) + dz(t, x, ω),

(4.1)

u(0, x, ω) = u0(x, ω), x ∈ G, (4.2)

Dβ0 . . .Dβm−1u |s= 0 for all β0, . . . , βm−1 (4.3)

such that

|β0| + . . .+ |βm−1| ≤ m− 1,

where S is the lateral surface of the cylinder [0, T ] × G; summations over all the

values of the repeated indices αi is understood; the functions A, B depend on t, x, ω,

and all the derivatives of u with respect to x of order no greater than m; A and u

are real functions; B is a function with values in E; in the second term in (4.1) the

scalar product in E is understood.

We assume that for all collections of real numbers ξ = (ξβ1 ...βm

) and any e ∈ E

the functions A(ξ, t, x, ω), B(ξ, t, x, ω)e = (B(ξ, t, x, ω), e)E are

(1) measurable in (t, x, ω),

(2) measurable in (x, ω) relative to the product of Ft and the Borel σ-algebra on

Rd for every fixed t ∈ [0, T ],

(3) continuous in ξ for every fixed t, x, ω.

Suppose further that there exist a constant K > 0 and a nonnegative function

f(t, x, ω) possessing the same measurability properties as A(0, t, x, ω) and such that,

for all t, x, ω, ξ, α1, . . . , αm,

|Aα1,...,αm(ξ, t, x, ω)| ≤ f1/q(t, x, ω) +K∑

β1,...,βm

∣∣ξβ1,...,βm∣∣p−1

, (4.4)



|B(ξ, t, x, ω)|2E ≤ f(t, x, ω) +K∑

β1,...,βm

∣∣ξβ1,...,βm∣∣p +K

∣∣ξ0,...,0∣∣2 . (4.5)

Moreover, it is assumed that u0(x, ω) is measurable relative to the product of F0

and the Borel σ-algebra on Rd, and, for all t, ω,

‖u0‖2 <∞, ‖f(t)‖1 <∞, E ‖u0‖22 <∞,

∫ T

0

‖f(t)‖1 dt <∞.

These assumptions enable us to give a precise meaning to equation (4.1), to

the initial condition (4.2), and to the boundary conditions (4.3) in the following

manner.d We note that by (4.4) and the Holder inequality

(Aα1...αm(Dβ1 . . .Dβmu), Dα1 . . .Dαmv

)0≤ c

(‖f(t)‖1/q

1 + ‖u‖p−1m,p

)‖v‖m,p . (4.6)

Therefore, for t ∈ [0, T ] the left side of (4.6) is a linear functional on Wmp (G)

and, in particular, on V =Wm

p (G). As we know, there exists a unique element

A(u, t, ω) ∈ V ∗ = W−mq (G) for which for all v ∈ V

〈〈v, A(u, t, ω)〉〉 = −(Aα1...αm(Dβ1 . . . Dβmu, t, ω), Dα1 . . .Dαmv

)0.

It is clear that the function A(u, t, ω) satisfies the measurability conditions and

conditions (A1), (A4) of Section 3.2. Further, we note that for g ∈ H by (4.5)( ∣∣B(Dβ1 . . . Dβmu, t)

∣∣E, g)0≤ ‖f(t)‖1/2

1 · ‖g‖H +N(‖u‖p/2V · ‖g‖H + ‖u‖H · ‖g‖

).

This implies that B(u, t, ω) ∈ L(E,H) is defined for t ∈ [0, T ], ω ∈ Ω, u ∈ V

according to the formula

(B(u, t, ω)e, g)0 =((B(Dβ1 . . . Dβmu, t, ·, ω), e)E, g

)0,

where e ∈ E, g ∈ H . Moreover, if ei, hi are orthonormal bases in E, H , respectively,

then

‖B(u, t)‖2=∑

i,j

(B(u, t)ei, hj

)2 ≤∥∥ ∣∣B(Dβ1 . . . Dβmu, t)

∣∣E

∥∥2

2

≤ ‖f(t)‖1 +N(‖u‖pV + ‖u‖2

H

).

(4.7)

A similar inequality obviously holds for∥∥B(u, t)Q1/2

∥∥2. Hence, B(u, t) satisfies

inequality (3.7). Moreover, it is clear that if u = u(t, ω) is a function with values

in V which is measurable in (t, ω) and Ft-consistent, then B(u(t, ω), t, ω) possesses

the same measurability properties as an element of LQ(E,H).

Thus, equation (3.4) can be considered for the operators A, B defined above.

Definition 4.11. A V -solution (H-solution) of problem (4.1)–(4.3) is a V -solution

(H-solution) of equation (3.4). A continuous modification of problem (4.1)–(4.3) in

H is defined similarly.dFor notational convenience we will occasionally omit all or some of the arguments in variousfunctions.



Remark 4.12. A function v(t) ∈ V is a V -solution of problem (4.1)–(4.3) if and

only if it is appropriately measurable, inequality (3.5) is satisfied, and, for every

η ∈ V and almost all (t, ω),

(v(t), η)0 = (u0, η)0 −∫ t

0

(Aα1...αm(Dβ1 . . . Dβmv, s), Dα1 . . . Dαmη

)0

+

∫ t

0

∫

Rd

B(Dβ1 . . . Dβmv, s, x)η(x)dxdw(s) + (z(t), η)0.

(4.8)

In others words, a V -solution is a solution of problem (4.1)–(4.3) in the sense of the

integral identity.

To prove this, it suffices to use Remark 3.3 and the fact that, for η ∈ V , the

norm∣∣B(Dβ1 . . .Dβmv(t, x), t, x)

∣∣E

multiplied by |η(x)| is integrable over Rd, and

hence for any e ∈ E

ηB(v, t)e = (η,B(v, t)e)H =

∫

Rd

(B(Dβ1 . . . Dβmv, t, x), e

)Eη(x)dx

=

(∫

Rd

B(Dβ1 . . .Dβmv, t, x)η(x)dx, e

)

E

.

A similar remark holds for an H-solution of problem (4.1)–(4.3).

Remark 4.13. Relation (4.8) is obtained if we formally multiply (4.1) by η, inte-

grate over t, x, integrate by parts in x, and use the boundary conditions (4.3). In

our interpretation of a solution of (4.1)–(4.3) we start from (4.8) in which there are

no boundary conditions. They are accounted in (4.8) only through the membership

of v in the spaceWm

p (G). In this connection we note that for p > d by one of the

Sobolev embedding theorems each function ofWm

p (G) is equal almost everywhere

to a function which has derivatives of order less than or equal to m− 1, and these

derivatives are continuous in the closure of G and vanish on the boundary of G.

Definition 4.14. We say that equation (4.1) satisfies the condition of strong

parabolicity if the operators A, B satisfy conditions (A2), (A3) of Section 3.2.

Conditions which are sufficient for strong parabolicity in terms of the original

functions Aα1...αm(ξ, t, x, ω), B(ξ, t, x, ω) will be given below. The next theorem

follows automatically from the results of Section 3.2.

Theorem 4.15. If equation (4.1) is strongly parabolic, then the assertions of The-

orems 3.6, 3.11 hold.

Verification of the condition of strong parabolicity in the general case is a prob-

lem of colossal difficulty even for B ≡ 0. However, by generalizing the results of

Ref. [62], in certain cases it is possible to give simple sufficient conditions for strong

parabolicity of equation (4.1).



We say that the algebraic condition of strong parabolicity is satisfied if, for every

fixed t, x, ω, the functions Aα1...αm(ξβ1...βm , t, x), and B(ξβ1...βm , t, x) are differen-

tiable once with respect to ξ everywhere except for a set having a finite number of

points of intersection with each line in the space of coordinates ξ, the derivatives

are locally summable on each line in this space, and there exist constants ε > 0 and

N ∈ R such that, for all ξα1...αm , ηα1...αm , t, x, ω,

− 2Aβ1...βmα1...αm

(ξγ1...γm , t, x)ηα1...αmηβ1...βm +∣∣Bβ1...βm(ξγ1...γm , t, x)ηβ1...βm

∣∣2Q

+ ε∑

|α1|+...+|αm|=m|ξα1...αm |p−2 |ηα1...αm |2 ≤ N

(η0...0

)2,

(4.9)

where

Aβ1...βmα1...αm

=∂Aα1...αm

∂ξβ1...βm, Bβ1...βm =

∂B

∂ξβ1...βm,

and in the second term of the left side of (4.9) summation over all β1..βm is carried

out before computing the norm |·|Q.

The next theorem justifies the name of condition (4.9).

Theorem 4.16. Suppose that the algebraic condition of strong parabolicity is sat-

isfied, together with all other assumptions of this section regarding the functions

A(ξβ1...βm), B(ξβ1...βm). Then equation (4.1) is strongly parabolic.

Proof. We use the formula of Hadamard

B(ξ) −B(η) =

∫ 1

0

Bβ1...βm(ξt+ (1 − t)η)(ξβ1 ...βm − ηβ1...βm)dt,

and also the Cauchy-Schwartz inequality and the definition of the operators A, B.

We then obtain

I(v1, v2) ≡ 2〈〈v1 − v2, A(v1) −A(v2)〉〉 + |B(v1) −B(v2)|2Q= 2(Dα1 . . .Dαm(v1 − v2), Aα1...αm(Dβ1 . . .Dβmv1) −Aα1...αm(Dβ1 . . . Dβmv2)

)0

+∥∥∥∣∣B(Dβ1 . . .Dβmv1) −B(Dβ1 . . . Dβmv2)

∣∣Q

∥∥∥2

2≤ N ‖v1 − v2‖2

H

− ε∑

|α1|+...+|αm|=m

∫

G

(∫ 1

0

∣∣Dα1 . . .Dαm(v2 + t(v1 − v2)

)∣∣p−2dt

)

× |Dα1 . . . Dαm(v1 − v2)|2 dx.(4.10)

We have thus proved that the condition of monotonicity (A2) of Section 3.2 is

satisfied. In fact, equality (4.10) enables us to verify the coercivity condition (A3):



From (4.10) and (4.7) we have, with v1 = v, v2 = 0,

2〈〈v, A(v)〉〉 + |B(v)|2Q ≤ I(v, 0) + 2〈〈v, A(0)〉〉 − |B(0)|2Q + 2 |B(v)|Q |B(0)|Q

≤ N

(1

δ+ δ

)‖f(t)‖1 +Nδ ‖v‖pV + 2δ ‖v‖pV +

2

δ‖A(0)‖qV ∗ +N ‖v‖2

H

− ε

p− 1

∑

|α1|+...+|αm|=m‖Dα1 . . .Dαmv‖pp ,

where δ > 0 and N does not depend on δ.

Using Theorems 4.6 and 4.7, noting that the growth condition (A4) has already

been verified, and choosing δ sufficiently small, we conclude that the coercivity

condition is satisfied. The proof of the theorem is complete.

Example 4.17. Let d = 1, E = R, and suppose that (4.1) has the form

du(t, x) = a(t, ω)(−1)m+1 ∂m

∂xm

(∣∣∣∣∂m

∂xmu(t, x)

∣∣∣∣p−2

∂m

∂xmu(t, x)

)dt

+ b(t, ω)

∣∣∣∣∂m

∂xmu(t, x)

∣∣∣∣p/2

dw(t),

(4.11)

where a, b are appropriately measurable and also bounded processes. Here the

algebraic condition of strong parabolicity becomes

−2(p− 1)a+p2

4b2 ≤ −ε,

where ε > 0 is a constant. If this condition is satisfied, then by Theorem 4.15 we

obtain assertions regarding the existence, uniqueness, stability with respect to the

initial data, and the Markov property of solutions of (4.11) with the “boundary”

condition u ∈Wm

p (G).

Equation (4.11) coincides with (3.2). Equations of the form (3.3) are studied in

the next section.

4.3. Cauchy Problem for Linear Second-Order Equations

In this section we continue the study of equation (4.1), assuming that m = 1,

G = Rd, and A and B are linear functions of ξ which are generally not equal to

zero for ξ = 0. Moreover, all assumptions of Section 4.2 are naturally assumed to

be satisfied.

Problem (4.1)–(4.3) becomes

du(t, x) = Dα(aαβ(t, x)Dβu(t, x) + fα(t, x)

)dt

+(bα(t, x)Dαu(t, x) + g(t, x)

)dw(t) + dz(t, x),

(4.12)

u(t, ·) ∈ L2(Rd), u(0, x) = u0(x), x ∈ Rd, (4.13)



where aαβ , fα are real functions and bα and g are functions with values in E.

Conditions (4.4) and (4.5) are equivalent to the boundedness of aαβ and |bα|E ,

together with the inequality

∑

α

E

∫ T

0

‖fα‖22 dt+ E

∫ T

0

‖ |g|E ‖22 dt <∞.

A solution of problem (4.12), (4.13) is understood in the sense of the integral

identity (4.8); for a V -solution the identity is satisfied for almost all (t, ω), and for

an H-solution, for each t with probability one.

Lemma 4.18. Suppose that, for all x, η ∈ Rd, t ∈ [0, T ], ω ∈ Ω,

2

d∑

i,j=1

aij(t, x)ηiηj −∣∣∣∣∣d∑

i=1

bi(t, x)ηi

∣∣∣∣∣

2

Q

≥ ε |η|2 . (4.14)

where ε > 0 is a constant, aij = aαβ, bi = bα if α is the i-th and β the j-th

coordinate vectors. Then the algebraic condition of strong parabolicity is satisfied.

The lemma is easily proved by means of inequalities of the type

2a0αη0ηi ≤ |a0α ε |ηi|2 + ε−1|a0,α |η0|2.

The next result is a direct corollary of Lemma 4.18, Theorems 4.1 and 4.2, and

also Corollary 3.7.

Theorem 4.19. Suppose that condition (4.14) is satisfied. Then there exists a

function u(t, ω) defined on [0, T ]×Ω with values in L2(Rd), strongly continuous in

t in L2(Rd), Ft-consistent and such that

(a) For almost all (t, ω),

u ∈W 12 (Rd);

(b)

E supt≤T

‖u(t)‖22 + E

∫ T

0

‖u(t)‖21,2 dt <∞;

(c) for each η ∈W 12 (Rd),

(u(t), η)0 = (u0, η)0 +

∫ t

0

(−1)α(Dβu(s), aαβ(s)Dαη

)0

+

∫ t

0

(−1)α(fα(s), Dαη)0ds

+

∫ t

0

(bα(s)Dαu(s) + g(s), η)0dw(s) + (z(t), η)0

(4.15)

for all t ∈ [0, T ] (a.s.)



Theorems 3.8 and 3.11 enable us to establish uniqueness, stability with respect

to the initial data, and the Markov property of the function u considered in Theorem

4.19 [u is an H-solution of problem (4.12), (4.13)]. These properties of u are proved

simply by referring to Theorems 3.8 and 3.11, and we are not going to discuss them.

We now turn to a more important question, the question of raising the smooth-

ness of a solution of problem (4.12), (4.13). One reason for addressing this question

is that filtering of diffusion processes [86] leads to equations analogous to (4.15),

but with the inner product in L2(Rd) replaced by the inner product in Wm2 (Rd),

i.e., the index 0 in the inner product in (4.15) is replaced by m so that

(ϕ, ψ)m = (Dα1 . . . Dαmϕ,Dα1 . . . Dαmψ)0.

Thus, we denote the modified equation (4.15) by (4.15)m. For (4.15)m to be mean-

ingful for sufficiently smooth u0 and coefficients aαβ , fα, bα, g, z, it suffices to

restrict attention to functions u belonging to Wm+12 (Rd) [for a.a. (t, ω)]. However,

the filtering density is equal to the function (1 − ∆)mu multiplied by some func-

tion of time. Thus, assertions are needed regarding the membership of the solution

of equation (4.15)m not in Wm+12 (Rd) but in W 2m

2 (Rd) [for a.a. (t, ω)]. However,

merely an assertion on the existence of a solution of (4.15)m with values in W 2m2 (Rd)

is of little use, since from the filtering theory it is known a priori only that the solu-

tions belongs to Wm+12 (Rd); if we wish to prove its smoothness, we must have not

only a theorem on the existence of a solution of (4.15)m with values in W 2m2 (Rd)

but also a theorem on the uniqueness of a solution with values in Wm+12 (Rd). Thus,

a theorem on raising the smoothness for equation (4.15)m is required. A proof of

the corresponding result is given in Ref. [64]. Not to obscure the exposition with

technical details, we here prove the theorem on raising smoothness only for equation

(4.15).

Throughout the remainder of the paper we fix an integer m > 0 and assume that

z(t) is a square-integrable martingale with values in Wm2 (Rd) which is continuous in

t in Wm2 (Rd) for all t, ω, the functions aαβ (resp. bα) have m derivatives (resp. weak

derivatives) in x, are continuous (weakly continuous) in x, and these derivatives of

aαβ , bα are bounded (for bα in the norm of E) uniformly with respect to t, x, ω.

Suppose that, for all (t, ω), the functions fα ∈Wm2 (Rd), u0 ∈Wm

2 (Rd), and

E ‖u0‖2m,2 + E

∫ T

0

‖fα(t)‖2m,2 dt <∞.

Let g ≡ 0. This condition involves no loss of generality, since the integral of g with

respect to dw(t) can be included in z(t). Finally, we assume that condition (4.14)

is satisfied.

Theorem 4.20. There exists a set Ω′ ⊂ Ω such that P (Ω′) = 1 and for ω ∈ Ω′

the function u(t) of Theorem 4.19 belongs to Wm2 (Rd) and is continuous in t in the



norm of Wm2 (Rd). Moreover, u ∈ Wm+1

2 (Rd) a.e. in (t, ω), and

Esupt≤T

‖u‖2m,2 + E

∫ T

0

‖u(s)‖2m+1 ds <∞.

Proof. We set H = Wm2 (Rd) and identify H with its dual by means of (·, ·)m.

It is then easily seen from the Parseval equality that Wm−12 is identified with V ∗,

the space dual to V = Wm+12 (Rd). Thus, V ⊂ H ⊂ V ∗, and it is obvious that each

embedding is dense and continuous. As in the preceding section, it is easy to verify

that the formulas

〈〈η,A(t)u〉〉 =(aαβ(t)Dβu+ fα(t), (−1)αDαη

)m,

((B(t)u) e, η

)m

=((bα(t), e)ED

αu, η)m

define bounded linear operators A(t) : V → V ∗, B(t)u : E → H . The reader

can also verify without difficulty that the functions A(t)u, B(t)u satisfy conditions

(A1)–(A4) of Section 3.2 and also the measurability conditions of this section (see,

e.g. Ref. [64]).

Hence, by Corollary 3.7 in our case equation (4.1) has an H-solution u(t). Ap-

plication of Remark 4.12 shows that, for each η ∈ Wm2 (Rd),

(u(t), η)m = (u0, η)m +

∫ t

0

(aαβ(s)Dβu(s) + f(s), (−1)|α|Dαη

)m

+

∫ t

0

(bα(s)Dαu(s), η

)mdw(s) + (z(t), η)m

(4.16)

for all t ∈ [0, T ] (a.s.)

To avoid misunderstanding we note that (4.16) does not coincide with (4.15)mif m > 0 and aαβ , bα depend on x.

If in (4.16) in place of η we substitute (1 −∆)−mη and use the fact that by the

Parseval equality (f, g)m = (f, (1 − ∆)mg)0, where g ∈W 2m2 (Rd), then we see that

in (4.16) in place of m it is possible to write 0. After this, by Theorem 3.8 we obtain

supt≤T

‖u(t) − u(t)‖2 = 0 (a.s.),

and u(t) possesses the same properties as u(t).

The proof of the theorem is complete.

In conclusion, we discuss the significance of condition (4.14) for the validity of

Theorem 4.19. For d = 1 and E = R we consider the equation

du(t, x) =1

2

∂2u(t, x)

∂x2dt+ σ

∂u(t, x)

∂xdw(t),

where σ is a constant and the initial condition u0(x) is nonrandom. If Theorem

4.19 is valid for this equation, then from (4.15) and the Parseval equality it follows



that∫

R

u(t, y)η(y)dy =

∫

R

u0(y)η(y)dy − 1

2

∫ t

0

(∫

R

y2u(s, y)η(y)dy

)ds

−√−1σ

∫ t

0

(∫

R

yu(s, y)η(y)dy

)dw(s),

where u, η are the Fourier transforms of u, η. Here it is easy to interchange the

integrals if η(y) is a compactly supported function, and we then find that for almost

all (t, y, ω)

u(t, y) = u0(y) − 1

2y2

∫ t

0

u(s, y)ds−√−1σy

∫ t

0

u(s, y)dw(s). (4.17)

We fix a y for which equation (4.17) holds for almost all (t, ω), and we denote the

right side of (4.17) by ϕ(t, y). Then ϕ(t, y) satisfies (4.17) for all t (a.s.). The

solution of the equation for ϕ is known:

ϕ(t, y) = e−12 (1−σ2)y2t−

√−1σyw(t) u0(y).

Since u(t, y) = ϕ(t, y) a.e. in (t, y, ω), it follows that

E

∫ T

0

‖u(t)‖21,2 dt =

∫

R

(1 + y−2)e−(1−σ2)y2T − 1

1 − σ2u2

0(y) dy. (4.18)

This implies that the left side of (4.18) is finite for all u0 ∈ L2(R) if and only if

|σ| < 1. The last condition in the present case is equivalent to (4.14). This example

demonstrates the necessity of condition (4.14) for the validity of Theorem 4.19, and

also the necessity of the coercivity condition (A3) of Section 3.2 for the validity of

the results in that section.

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Chapter 2

Predictability of the Burgers Dynamics Under Model

Uncertainty

Dirk Blomker and Jinqiao Duan∗

Institut fur Mathematik

RWTH Aachen, 52062 Aachen, Germany

Complex systems may be subject to various uncertainties. A great effort has beenconcentrated on predicting the dynamics under uncertainty in initial conditions.In the present work, we consider the well-known Burgers equation with randomboundary forcing or with random body forcing. Our goal is to attempt to under-stand the stochastic Burgers dynamics by predicting or estimating the solutionprocesses in various diagnostic metrics, such as mean length scale, correlationfunction and mean energy. First, for the linearized model, we observe that theimportant statistical quantities like mean energy or correlation functions are thesame for the two types of random forcing, even though the solutions behave verydifferently. Second, for the full nonlinear model, we estimate the mean energyfor various types of random body forcing, highlighting the different impact onthe overall dynamics of space-time white noises, trace class white-in-time andcolored-in-space noises, point noises, additive noises or multiplicative noises.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2 Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.1 Mean Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.2 Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 Nonlinear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.1 Body forcing - Mean energy bounds . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2 Point forcing - Mean energy bounds . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3 Body forcing - Transient behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.4 Trace class noise: Additive vs. multiplicative body noises . . . . . . . . . . . . 87

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

1. Introduction

The Burgers equation has been used as a simplified prototype model for hydrody-

namics and infinite dimensional systems. It is often regarded as a one-dimensional

Navier-Stokes equation. Our motivation for considering this equation comes from

the modeling of the hydrodynamics and thermodynamics of the coupled atmosphere-∗Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

71


72 D. Blomker and J. Duan

ocean system. At the air-sea interface, the atmosphere and ocean interact through

heat flux and freshwater flux with a fair amount of uncertainty [1–3]. These trans-

late into random Neumann boundary conditions for temperature or salinity. The

Dirichlet boundary condition is also appropriate under other physical situations.

The fluctuating wind stress forcing corresponds to a random body forcing for the

fluid velocity field. The coupled atmosphere-ocean system is quite complicated and

numerical simulation is the usual approach at this time. In this paper, we consider a

simplified model for this system, i.e., we consider the Burgers equation with random

Neumann boundary conditions and random body forcing. Although the stochastic

Burgers equation is widely studied, most work we know are for Dirichlet bound-

ary conditions or periodic boundary conditions [4–7]. The reference [8] studied the

control of deterministic Burgers equation with Neumann boundary conditions.

We consider the stochastic Burgers equation with boundary forcing on the in-

terval [0, L]

∂tu+ u · ∂xu = ν∂2xu (1.1)

∂xu(·, 0) = αη ∂xu(·, L) = 0. (1.2)

Here α > 0 denotes the noise strength and η is white noise, i.e., η is a generalized

Gaussian process with Eη(t) = 0 and Eη(t)η(s) = δ(t− s). The restriction to noise

on the left boundary is only for simplicity. Analogous results will be true, if forces

act on both sides of the domain.

We will see that boundary forcing coincides with point forcing at the boundary.

Thus we also look at point forcing. As a simple example for point forcing, we

consider

∂tv + v · ∂xv = ν∂2xv + αδ0η (1.3)

u(·,−L) = u(·, L) = 0 , (1.4)

where δ0 is the Delta-distribution.

We will compare solutions of (1.1) and (1.3) with solutions of the stochastic

Burgers equation with body forcing.

∂tv + v · ∂xv = ν∂2xv + σξ (1.5)

either subject to Dirichlet or Neumann boundary conditions. Here the noise

strength is denoted by σ > 0 and ξ is space-time white noise. I.e., ξ is a gen-

eralized Gaussian process with Eξ(t, x) = 0 and Eξ(t, x)ξ(s, y) = δ(t − s)δ(x − y).

We will also consider trace class body noise, i.e., noise that is white in time but

colored in space.

For the linearized equations, we will compare statistical quantities of both solu-

tions, which are frequently used. One of them is the mean energy

1

L

∫ L

0

E[u(t, x) − u(t)]2dx, (1.6)


Predictability of the Burgers Dynamics Under Model Uncertainty 73

where

u(t) =1

L

∫ L

0

u(t, x)dx.

Another important quantity, which gives information about the characteristic size

of pattern, is the mean correlation function

C(t, r) :=1

L

∫ L

0

E[u(t, x) − u(t)] · [u(t, x+ r) − u(t)]dx, (1.7)

which is usually averaged over all points r with a given distance from 0. We obtain

the averaged mean correlation function

C(t, r) =1

2[C(t, r) + C(t,−r)] (1.8)

where we employ the canonical odd and 2L-periodic extension of u in order to define

C(t, r) for any r ∈ R.

For the linearized equation the main result states that mean energy and averaged

mean correlation function are the same for solutions of (1.1) and (1.5). Nevertheless

the solutions behave completely different. Furthermore, we give some qualitative

properties like, for instance, the typical pattern size. This should carry over to a

transient regime (i.e., small times) for the corresponding nonlinear equations.

For the full nonlinear Burgers model, we estimate the mean energy for various

types of random body forcing, highlighting the different impact on the overall dy-

namics of space-time white noises, trace class white-in-time and colored-in-space

noises, point noises, additive noises or multiplicative noises.

In the following, we discuss linear dynamics in §2 and nonlinear dynamics in §3.

2. Linear Theory

Define

A = ν∂2x

with

D(A) = w ∈ H2([0, L]) : ∂xw(0) = 0, ∂xw(L) = 0It is well-known (cf. e.g. [9]) that A has an orthonormal base of eigenfunctions

ekk∈N0 in L2([0, L]) with corresponding eigenvalues λkk∈N0 . In our situation

e0(x) = 1/√L, ek(x) =

√2/L · cos(πkx/L) for k ∈ N, and λk = −ν(kπ/L)2.

Moreover A generates an analytic semigroup etAt≥0. (cf. e.g. [10]).

In fact, etAv0 is the solution of the following evolution problem

∂tv = Av, ∂xv(·, 0) = ∂xv(·, L) = 0, v(0, x) = v0(x). (2.1)

The solution is

etAv0(x) := v(t, x) =

∞∑

k=0

< v0, ek > eλktek, t > 0, 0 < x < L, (2.2)



where < ·, · > is the usual scalar product in L2(0, L).

We now consider the following linearized problems. First

∂tu = Au, ∂xu(·, 0) = α∂tβ, ∂xu(·, L) = 0. (2.3)

Here the white noise η is given by the generalized derivative of a standard Brownian

motion (cf. e.g. [11]), and α is the noise intensity.

Secondly,

∂tv = Av + σ∂tW, ∂xv(·, 0) = 0, ∂xv(·, L) = 0, (2.4)

where the space-time white noise is given by the generalized derivative of an Id-

Wiener process. Namely, W (t) =∑

k∈N0βk(t)ek, where βkk∈N is a family of

independent standard Brownian motions, and σ is the noise intensity.

It is known (cf. e.g. [12]) that (2.4) has a unique weak solution given by the

stochastic convolution (taking initial condition to be zero)

WA(t) = σ ·∫ t

0

e(t−τ)AdW (τ) = σ ·∑

k∈N0

∫ t

0

e(t−τ)λkdβk(τ)ek . (2.5)

We define the Neumann map D by

(1 −A)Dγ = 0, ∂xDγ(0) = γ, ∂xDγ(L) = 0

for any γ ∈ R. It is known that D : R 7→ H2([0, L]) is a continuous linear operator.

In fact, we have explicit expression for this linear operator

D(γ) =ex + e2Le−x

1 − e2Lγ. (2.6)

From Ref. [13] or Ref. [14] we immediately obtain, that (2.3) has a unique weak

solution (taking initial condition to be zero)

Z(t) = (1 −A)

∫ t

0

e(t−τ)ADαdβ(τ). (2.7)

In the next section we derive explicit formulas for Z in term of Fourier series.

2.1. Mean Energy

To obtain the Fourier series expansion for Z, consider for e ∈ D(A) and γ ∈ R

< D(γ), (1 −A)e >L2([0,L]) = < D(γ), e > −∫ L

0

D(γ) · exxdx

= < D(γ), e > −∫ L

0

D(γ)xx · edx+ D(γ)x · e|x=Lx=0

= −γe(0), (2.8)



by the definition of D. Hence,

< Z(t), ek > = <

∫ t

0

e(t−τ)ADαdβ(τ), (1 −A)ek >

=

∫ t

0

e(t−τ)λk < Dαdβ(τ), (1 −A)ek >

= αek(0) ·∫ t

0

e(t−τ)λkdβ(τ). (2.9)

We now obtain

Z(t) = α ·∑

k∈N0

ek(0)

∫ t

0

e(t−τ)λkdβ(τ)ek . (2.10)

Finally,

Z(t) = αe1(0) ·∑

k∈N

∫ t

0

e(t−τ)λkdβ(τ)ek + αe20(0)β(t) (2.11)

and

WA(t) = σ ·∑

k∈N

∫ t

0

e(t−τ)λkdβk(τ)ek + σβ0(t) . (2.12)

If we now choose σ = αe1(0), we readily obtain that

E‖Z(t) − Z(t)‖2 = σ2 ·∑

k∈N

∫ t

0

e2τλkdτ = E‖WA(t) −WA(t)‖2 ,

where ‖ · ‖ is the norm in L2([0, T ]). Hence, the mean energy in both cases is given

by σ2L−1∑

k∈N

∫ t0e2τλkdτ .

For the mean energy we can prove the following theorem, which is similar to the

results of [15] and [16].

Theorem 2.1. Fix σ2 = α2/L, then the mean energy CZ(t, 0) = CWA(t, 0) behaves

like C1(α2/L)√t/ν for t L2/ν, and like C2α

2/ν for t L2/ν.

The main difference to body forcing is the scaling in the length-scale L. The long-

time scaling is independent of L, while the transient scaling is.

2.2. Correlation Function

To obtain results for the correlation function, we think of Z and WA to be periodic

on [−L,L], and symmetric w.r.t. 0. I.e., we choose the standard 2L-periodic ex-

tension respecting the Neumann boundary conditions on [0, L]. To be more precise,

we extend Z and WA in a Fourier series in the basis ek, which we then consider to

be defined on the whole R.



-

6

∼L2

νt

CZ(t, 0)

∼α2

ν

∼α2

L

√t

ν

Fig. 2.1. The scaling of the mean energy for boundary forcing.

We consider firstly for k, l 6= 0

∫ L

0

ek(x)el(x+ r)dx =

Lek(0)el(0) l((−1)k+l−1)

π(l2−k2) sin(πlr/L) : k 6= l

Lek(0)2 cos(πkr/L) : k = l

.

Now relying on the independence of the Brownian motions, it is straightforward to

verify

CWA(t, r) =1

LE < WA(t, x) −WA(t),WA(t, x+ r) −WA(t) >

=α2e21(0)

L·∑

k∈N

∫ t

0

e2τλkdτ cos(πkr/L) , (2.13)

as α2e21(0) = σ2. Furthermore,

CZ(t, r) =1

LE < Z(t, x) − Z(t), Z(t, x+ r) − Z(t) > (2.14)

= CWA(t, r) +e21(0)

π

∞∑

k,l=1k 6=l

∫ t

0

eτ(λk+λl)dτl((−1)k+l − 1)

(l2 − k2)sin(πlr/L) .

Obviously, CZ and CWA do not coincide, but let us now look at the averaged

correlation function

C(t, r) =1

2[C(t, r) + C(t,−r)] .

Then it is obvious that

CWA(t, r) = CWA(t, r) = CZ(t, r) 6= CZ(t, r) . (2.15)

Now

Theorem 2.2. For α2e21(0) = σ2 the mean energy and the averaged mean correla-

tion functions CWA and CZ for Z and WA coincide for any t ≥ 0.

This is somewhat surprising, as realizations of Z and WA behave completely

different, when the condition α2e21(0) = σ2 is satisfied. See e.g. Figure 2.2 and

Figure 2.3.



−1

−0.5

0

0.5

1

0

0.5

1

1.5

2−5

0

5

xt

u

Fig. 2.2. Random boundary condition: One realization of the solution of the equation (2.3) forL = 1, ν = 1, α = 1 and initial condition u(x, 0) = 0.

It is even more surprising, as the scaling behavior of quantities like mean energy

and mean correlation functions are an important tool in applied science, which

for example is used to determine the size of characteristic length scales and the

universality class the model belongs to. Here both linear models lie in the same

class, although their behavior differs completely.

The scaling behavior with respect to L and t of the mean energy can be described

using the results of [15], where the mean surface width for very general models was

discussed. Therefore we focus on the scaling properties of the mean correlation

function. Here we also want to investigate the dependence on α and ν.

First we consider the scaling properties of the correlation function CZ(t, r) or

CWA , as given in (2.14). We are especially interested in the smallest zero of the

function, which gives information about characteristic length scales or pattern sizes.

For this, we use the normalized correlation function.

ρZ(t, r) =CZ(t, r)

CZ(t, 0). (2.16)

Note that CZ(t, 0) is the mean energy and the maximum of r 7→ CZ(t, r).

We begin with some technical results. For any continuously differentiable and

integrable function f : R+ → R we obtain using the mean value theorem

∣∣∣∣∣

∫ ∞

0

f(x)dx −∞∑

k=1

f(k)

∣∣∣∣∣ ≤∞∑

k=1

supη∈(k−1,k)

|f ′(η)|. (2.17)



−1

−0.5

0

0.5

1

0

2

4

6

8

10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

xt

v

Fig. 2.3. Random body forcing: One realization of the solution of the equation (2.4) for L = 1,ν = 1, σ = 1 and initial condition v(x, 0) = 0.

For f(k) := e−2τνk2π2/L2

cos(kπr/L) it is easy to verify that

|f ′(k)| ≤ e−τνk2π2/L2 · π

L·[r +

4√τν√2e

],

where we used that xse−x2α ≤ (2αe)−1/2 for any x, α ≥ 0. Hence,

∞∑

k=1

supη∈(k−1,k)

|f ′(η)| ≤∞∑

k=1

e−τν(k−1)2π2/L2 · πL

· [r +4√2e

√τν]

≤ π

L[r +

4√2e

√τν](1 +

∫ ∞

0

e−τνk2π2/L2

dk)

=π

L[r +

4√2e

√τν](1 +

L

π√τν

·∫ ∞

0

e−k2

dk) (2.18)

and∫ t

0

∞∑

k=1

supη∈(k−1,k)

|f ′(η)|dτ ≤ C1

L

√t

ν[r +

√tν][L+

√tν]. (2.19)

Moreover,

1

L

∫ t

0

∫ ∞

0

f(k)dkdτ =1

π

∫ ∞

0

1 − e−2tνk2

2νk2cos(kr)dk

=1

π

√t

ν·G(

r√νt

), (2.20)

with G(x) :=∫∞0

1−e−2k2

2k2 cos(kx)dk.



0

0.4

0.8

1.2

1.6

1 2 3 4 5 6

Fig. 2.4. A sketch of G

Using (2.15) we immediately obtain

CZ(t, r) =α2

L

1

2π

√t

ν·G(

r√νt

) + O(α2

√t

L3√ν

[r +√tν][L+

√tν]

).

Note that the approximation with G is not L-periodic in r, while CZ(t, r) is. The

solution is that the error term is O(1) for r near L.

For the normalized correlation function we deduce

ρZ(t, r) :=CZ(t, r)

CZ(t, 0)=G( r√

νt) + O

(1L2 [r +

√tν][L+

√tν])

G(0) + O(

1L2 [

√tν][L+

√tν]) .

From the properties of G we infer the following:

Theorem 2.3. Given δ ∈ (0, 1) and sufficiently small ε2 > 0, there exists some

ε1 > 0 and three constants 0 < C1 < C2 < C3 depending only on δ and ε2 such that

for t < ε1L2/ν the following holds:

ρZ(t, r) ≥ δ for r ∈ [0, C1

√tν]

and

|ρZ(t, r)| < ε2 for r ∈ [C2

√tν, C3

√tν].

Note that we did not show that the correlation function has a zero, but it is arbitrary

small in a point rε ≈√tν. Therefor the theorem says that the typical length-scale

is√tν, at least for times t L2/ν.

For t→ ∞ we immediately obtain that

CZ(∞, r) =α2

π2ν

∞∑

k=1

1

k2cos(kπr/L) =:

α2

νF (r/L)

and

|CZ(t, r) − CZ(∞, r)| ≤ α2

π2νe−2tνπ2/L2

∞∑

k=1

1

k2.

We can look for the explicit representation of F , which is a 2-periodic function,

and compute explicitly the zero, but all we need from F is, that for a given small



–0.05

0

0.05

0.1

0.15

–3 –2 –1 1 2 3

x

Fig. 2.5. A sketch of F

enough δ > 0 there is a xδ > 0 such that F > δ on [0, xδ]. Moreover, there is some

x0 such that F (x0) = 0.

Consider the normalized correlation function

ρZ(t, r) =CZ(t, r)

CZ(t, 0)=F (r/L) + O(e−2tνπ2/L2

)

F (0) + O(e−2tνπ2/L2).

Assume that tν L2 (i.e., there is some small ε > 0 such that εtν > L2). Now,

ρZ(t, x0L) = O(e−2tνπ2/L2

)

and

ρZ(t, xL) ≥ δ

F (0)+ O(e−2tνπ2/L2

) > 0



for any x < xδ .

So for t L2/ν the first zero of ρZ should be of order L. A more precise

formulation is:

Theorem 2.4. Given δ ∈ (0, 0.8) and δ ε2 > 0, there exists some ε1 > 0, a

constant C > 0, and a point xo > 0 depending only on δ and ε2 such that for

t > L2/(νε1) we obtain the following:

ρZ(t, r) ≥ δ for r ∈ [0, CL]

and

|ρZ(t, x0L)| < ε2.

Thus the theorem tells us that for t L2/ν, the typical length scale is of order L,

which is the size of system. This result is true for both boundary and body forcing.

3. Nonlinear Theory

For the nonlinear results we leave the setting of boundary forcing. Mainly, due to

the lack of a-priori estimates. Usually, for Neumann boundary conditions one relies

on the maximum principle to bound solutions, but the solution for boundary forcing

is quite rough. Therefore, we hardly get useful results. Only, transient bounds for

small times are possible to establish. For the next sections, we focus first on body

forcing and later on point forcing. We will see later that boundary forcing is actually

just a point forcing in a point at the boundary.

The main results of this sections are uniform bounds on the energy and thus

on the correlation function C, as |C(t, r)| ≤ C(t, 0), and C(t, 0) is the energy.

Furthermore, we show that for t→ 0 the linear regime dominates. In Ref. [17] also

Holder-continuity for the mean energy was shown for a quasigeostrophic model. We

conclude this section by a qualitative discussion on upper bounds for the energy

using additive and multiplicative trace-class noise.

3.1. Body forcing - Mean energy bounds

Here we provide bounds on the mean energy for the body forcing case. We consider

additive space-time white noise case first, and show that the mean energy and thus

the correlation function is uniformly bounded in time. This result is known (cf.

Ref. [18]) for Burgers equation using the celebrated Cole-Hopf transformation, but

we provide here a simple proof for completeness. Furthermore, our proof is based on

energy estimates and it is easily adapted to other types of equations and additional

terms in the equation. In contrast to that Cole-Hopf transformation is strictly

limited to the standard Burgers equation.

For a long time for space-time white noise only uniform bounds for logarithmic

moments were known. See Ref. [13, Lemma 14.4.1] or Ref. [19]. In Ref. [18] the



transformation to a stochastic heat equation via the celebrated Cole-Hopf transfor-

mation was used to study finiteness of moments. Here we rely on a much simpler

tool, which can also be applied to other equations. See for instance [17] for a

quasigeostrophic model, where our analysis would apply, too.

Consider

∂tu+ u · ∂xu = ν∂2xu+ σ∂tW , (3.1)

u(·,−L) = u(·, L) = 0, u(x, 0) = u0(x). (3.2)

Here W is a Q-Wiener process with a continuous operator Q ∈ L(L2). Thus W

might be cylindrical, and we include the case of space-time white noise.

Using the semigroup etA the solution for this system is (see Ref. [13] or [4]):

u(t) = etAu0 −∫ t

0

e(t−τ)A(λΦλ(τ) +1

2∂xu(τ, x)2)dτ + Φλ(t) , (3.3)

where for some λ ≥ 0 fixed later

Φλ(t) = σα

∫ t

0

e(t−τ)(A−λ)dW (τ)

solves

∂tΦ = ν∂2xΦ − λΦ + σ∂tW

subject to

Φ(·,−L) = Φ(·, L) = 0, Φ(x, 0) = 0 .

Our main result is now:

Theorem 3.1. Consider initial conditions u0 with E‖u0‖2 < ∞, which are inde-

pendent of the Wiener process W (e.g. deterministic). Then the mean energy of

the solution of (3.3) is uniformly bounded in time. I.e.,

supt≥0

E‖u(t) − u(t)‖2 <∞ .

Remark 3.2. Actually, we prove that supt≥0 E‖u(t)‖2 < ∞. The main problem

in the proof is that after applying Gronwall-type estimates we end up with terms

E exp∫ t0 ‖Φλ(s)‖2

L∞. This might blow up in finite time, as second order exponential

moments of the Gaussian may fail to exist, if t is too large. This is why we introduced

artificially additional dissipation in the equation for Φλ, in order to get exponential

moments small.

For the proof of Theorem 3.1 define

v(t) = u(t) − Φλ(t) for t ≥ 0, λ ≥ 0 . (3.4)

We see that v is a weak solution of

∂tv +1

2∂x(v + Φλ)2 = ν∂2

xv + λΦλ (3.5)



v(·,−L) = v(·, L) = 0, v(x, 0) = u0(x). (3.6)

The following calculation is now only formal, but it can easily be made rigorous

using for instance spectral Galerkin approximations. Taking the scalar product in

(3.5) yields

1

2∂t‖v‖2 = −‖vx‖2 +

∫ L

−L(v + Φλ)2vx dx +

∫ L

−LΦλv dx

≤ −‖vx‖2 + ‖Φλ‖H−1‖vx‖ + |vx‖‖Φλ‖2L4 + 2‖vx‖‖Φλ‖L∞‖v‖

≤ −1

2c2p‖v‖2 + 4‖v‖2‖Φλ‖2

∞ + 2λ2‖Φλ‖4L4 + 2‖Φλ‖2

H−1 ,

where we used Young inequality (ab ≤ 18a

2 + 2b2), and Poincare inequality ‖v‖ ≤cp‖vx‖. Now, from Gronwall-type inequalities

‖v(t)‖2 ≤ e−c2pt+8

∫t0‖Φλ‖2

∞dτ‖u(0)‖2 (3.7)

+

∫ t

0

e−c2p(t−s)+8

∫ts‖Φλ‖2

∞dτ4(λ2‖Φλ‖4L4 + ‖Φλ‖2

H−1)ds.

Now we use the following lemma, which is easily proved by Fernique’s theorem, if

we consider Φλ as a Gaussian in L2([0, t0], L∞).

Lemma 3.3. Fix K > 0 and t0 > 0, then there is a λ0 such that

supt∈[0,t0]

E exp16

∫ t

0

‖Φλ(s)‖2L∞ds ≤ K2

for all λ ≥ λ0.

Furthermore, we use that all moments of ‖Φλ‖L∞ and ‖Φλ‖H−1 are uniformly

bounded in time. This is easily proven, using for instance the celebrated factoriza-

tion method.

Now we first fix K > 0, and then t0 such that e−c2ptK < 1

4 . This yields for

t ∈ [0, t0] and λ sufficiently large

E‖v(t)‖2 ≤ e−c2ptKE‖u(0)‖2 + 4K

∫ t

0

e−c2p(t−s)

(E(λ2‖Φλ‖4

L4 + ‖Φλ‖2H−1)2

)1/2

ds ,

using Holder, Lemma 3.3, and the independence of u(0) from Φλ. We now find a

constant C depending on t0 and K such that

supt∈[0,t0]

E‖v(t)‖2 ≤ KE‖u(0)‖2 + C and E‖v(t0)‖2 ≤ 1

4E‖u(0)‖2 + C.

Using E‖u(t)‖2 ≤ 2E‖v(t)‖2 + 2E‖Φλ(t)‖2 yields for a different constant C

supt∈[0,t0]

E‖u(t)‖2 ≤ KE‖u(0)‖2 + C and E‖u(t0)‖2 ≤ 1

2E‖u(0)‖2 + C.



Now we repeat the argument for k ∈ N by defining v(t) = u(kt0 + t) − Φλ(t),

where Φλ(t) has the same distribution than Φλ(t) due to a time shift of the Brow-

nian motion. Now v solves again (3.5) with initial condition u(kt0). Note that by

construction u(kt0) is independent of Φλ.

Repeating the arguments as before yields for k ∈ N0

supt∈[0,t0]

E‖u(t+ kt0)‖2 ≤ KE‖u(kt0)‖2 + C

and

E‖u((k + 1)t0)‖2 ≤ 1

2E‖u(kt0)‖2 + C .

Now the following lemma, which is a trivial statement on discrete dynamical sys-

tems, finishes the proof.

Lemma 3.4. Suppose for q < 1 and some C > 0 we have an+1 < qan +C, then anis bounded by

an <C

1 − q+ a0 .

3.2. Point forcing - Mean energy bounds

Consider hyperviscous Burgers equation with point-forcing. We would like to pro-

ceed exactly the way, we did in the previous section, But we can not, as for point

forcing, the solution of the linear equation might fail to be in L∞. This is why

we add additional damping. Hyperviscous Burgers equation has been studied in

several occasions. See for example [20–22].

Consider for some ε > 0 the operatorAε = −ν(−∆)1+ε, where ∆ is the Laplacian

subject to Dirichlet boundary conditions. Then the hyperviscous Burgers equation

is given by

∂tu+ u · ∂xu = Aεu+ αδ0β (3.8)

u(·,−L) = u(·, L) = 0, u(x, 0) = u0(x). (3.9)

Here, β is a standard Brownian motion and δ0 the Delta-distribution.

Using the semigroup etAε the solution for this system is (see Ref. [13] or [4]):

u(t) = etAεu0 −∫ t

0

e(t−τ)Aε(λΦλ(τ) +1

2∂xu(τ, x)2)dτ + Φλ(t) (3.10)

where for some λ ≥ 0 fixed later

Φλ(t) = α

∫ t

0

e(t−τ)(Aε−λ)δ0(x)dβ(τ)

solves

∂tΦ = AεΦ − λΦ + αδ0β



subject to

Φ(·,−L) = Φ(·, L) = 0, Φ(x, 0) = 0 .

Using the standard orthonormal basis ekk∈N of eigenfunctions of Aε given by

ek(x) =√

1/L · sin(−L+ πkx2L ) with corresponding eigenvalues λk = −(πk/2L)2+2ε,

we see

Φλ(t) = α

∞∑

k=1

∫ t

0

e(t−τ)(λk−λ)dβ(τ)ek(0)ek . (3.11)

Note that the Fourier-coefficients of that series are not at all independent. Thus

we cannot rely on the better regularity results available for the stochastic con-

volution of the previous chapter. Especially, for ε = 0 we cannot show that

Φλ(t) ∈ L∞([−L,L]).

Note that the series expansion of boundary and point forcing is very similar.

Thus we can regard boundary forcing at a point forcing at the boundary, when the

equation is subject to Neumann boundary conditions.

Our main result is now:

Theorem 3.5. For all ε > 0 and all initial conditions u0 independent of β with

E‖u0‖2 < ∞ the solution of (3.10) satisfies that the mean energy is uniformly

bounded in time. I.e.,

supt≥0

E‖u(t) − u(t)‖2 <∞ .

We will proceed exactly as in the previous section. Now v = u − Φλ is a weak

solution of

∂tv +1

2∂x(v + Φλ)2 = ν∂2

xv + λΦλ , (3.12)

again subject to Dirichlet boundary conditions and initial condition v(0) = u0.

Now consider first the nonlinear term for some small δ > 0. Using Holder,

Sobolev embedding of H12− 1

p into Lp and the bound

‖u‖H2γ ≤ C‖Aγ/(1+ε)ε u‖

yields∫ L

−LvΦλvxdx ≤ ‖v‖L2+δ‖Φλ‖L(4+2δ)/δ‖vx‖

≤ C‖A14

11+ε

δ2+δ

ε v‖‖Φλ‖L(4+2δ)/δ‖A12

11+ε

ε v‖ ,

Now we can easily find an δ > 0 sufficiently small such that there is a p = p(ε) ∈(2,∞) such that (using interpolation inequality)

∫ L

−LvΦλvxdx ≤ C‖v‖L2‖Φλ‖Lp‖A

12ε v‖ .



Now we can use the same proof as in the section before. We only need that Φλ(t) ∈L∞(0, L). To be more precise, an easy calculation using the series expansion of

(3.11) shows that for any ε > 0

supt≥0

E‖Φλ(t)‖2L∞ ≤ Cε sup

t≥0E‖Φλ(t)‖2

H1+ε2

→ 0 as λ → ∞ .

It is now straightforward to prove an analog to Lemma 3.3. The remainder of the

proof is analogous to the section before.

Let us remark that we could even simplify that proof a little bit, by avoiding

second order exponentials of Φλ. In that case we could work with λ = 0

3.3. Body forcing - Transient behavior

Let us focus on Burgers equation with body forcing. The results for hyperviscous

Burgers with point-forcing are completely analogous. We will prove:

Theorem 3.6. Let u be a solution of (3.1) and consider for simplicity u(0) = 0.

Denote by

Eu(t) = E‖u(t) − u(t)‖2

the mean energy of u(t), then there is some δ0 such that

Eu(t) = EΦ0 (t) + O(t12+δ0) for t→ 0 .

To be more precise, for some t0 > 0 sufficiently small there is a constant C > 0

such that |Eu(t) −EΦ0 (t)| ≤ Ct12+δ0 for all t ∈ [0, t0].

As we know from results like Theorem 2.1 that EΦ0 (t) behaves like√t for small t,

we can conclude that the linear regime dominates for small t.

We could explicitly calculate δ0, but omit this for simplicity of presentation.

For the proof of Theorem 3.6 use

|Eu(t) −EΦ0(t)| ≤ CE‖v(t)‖2 ,

where we used Cauchy-Schwarz inequality and uniform bounds on E‖u(t)‖2 and

E‖Φ0(t)‖2. Using (3.7) with λ = 0 and u(0) = 0 yields together with Lemma 3.3

E‖v(t)‖2 ≤ C

∫ t

0

(E‖Φ0(t)‖4H−1)1/2dt .

It is now easy to show that E‖Φ0(t)‖4H−1 behaves like t2δ0 for some δ0 > 0, which

can be explicitly calculated using the methods of Theorem 2.1. Theorem 3.6 is now

proved.

A simple corollary using Holders inequality is:

Corollary 3.7. Under the assumptions of Theorem 3.6, we know for the mean

correlation function

Cu(t, r) = CΦ0(t, r) + O(t12+δ0) for t→ 0 and all r .



Notice that this result is only useful for small times and small r, as seen from the

qualitative behavior of CΦ0 , which is similar to the results shown in section 2.2.

3.4. Trace class noise: Additive vs. multiplicative body noises

Consider again a solution of the following Burgers equation:

∂tu+ u · ∂xu = ν∂2xu+ σW (3.13)

u(·, 0) = 0, u(·, L) = 0, u(x, 0) = u0(x), (3.14)

where W (t)t≥0 is a Brownian motion, with covariance Q, taking values in the

Hilbert space L2(0, L) with the usual scalar product 〈·, ·〉. We assume that the

trace Tr(Q) is finite. So W is noise colored in space but white in time.

Applying the Ito’s formula, we obtain

1

2d‖u‖2 = 〈u, dW 〉 + [〈u, uxx − uux〉 +

1

2σ2Tr(Q)]dt. (3.15)

as before 〈u, uux〉 = 0. Thus

d

dtE‖u‖2 = −2‖ux‖2 + σ2Tr(Q). (3.16)

By the Poincare inequality ‖u‖2 ≤ c‖ux‖2 for some positive constant depending

only on the length L, we have

d

dtE‖u‖2 ≤ −2

c‖u‖2 + σ2Tr(Q). (3.17)

Then using the Gronwall inequality, we finally get

E‖u‖2 ≤ E‖u0‖2e−2c t +

1

2cσ2Tr(Q)[1 − e−

2c t]. (3.18)

Note that the first term in this estimate involves with initial data, and the second

term involves with the noise intensity σ as well as the trace of the noise covariance.

We now consider multiplicative body noise forcing.

∂tu+ u · ∂xu = ν∂2xu+ σuw, (3.19)

with the same boundary condition and initial condition as above, where wt is a

scalar Brownian motion. So w is noise homogeneous in space but white in time.

By the Ito’s formula, we obtain

1

2d‖u‖2 = 〈u, σudw〉 + [〈u, νuxx − uux〉 +

1

2σ2‖u‖2]dt. (3.20)

Thus

d

dtE‖u‖2 = −2ν‖ux‖2 + σ2‖u‖2

≤ (σ2 − 2ν

c)‖u‖2. (3.21)



Therefore,

E‖u‖2 ≤ E‖u0‖2e(σ2− 2ν

c )t. (3.22)

Note here that the multiplicative noise affects the mean energy growth or decay

rate, while the additive noise affects the mean energy upper bound.

Acknowledgments

Part of this work was done at the Oberwolfach Mathematical Research Institute,

Germany and the Institute of Applied Mathematics, the Chinese Academy of Sci-

ences, Beijing, China. This work was partly supported by the NSF Grants DMS-

0209326 & DMS-0542450 and DFG Grant KON 613/2006.

References

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[2] H. A. Dijkstra, Nonlinear Physical Oceanography. (Kluwer Academic Publishers,Boston, 2000).

[3] J.-L. Lions, R. Temam, and S. H. Wang, On the equations of the large-scale ocean,Nonlinearity. 5(5), 1007–1053, (1992). ISSN 0951-7715.

[4] G. Da Prato, A. Debussche, and R. Temam, Stochastic Burgers’ equation, NoDEANonlinear Differential Equations Appl. 1(4), 389–402, (1994). ISSN 1021-9722.

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[7] I. D. Chueshov and P. A. Vuillermot, Long-time behavior of solutions to a class ofstochastic parabolic equations with homogeneous white noise: Ito’s case, StochasticAnal. Appl. 18(4), 581–615, (2000). ISSN 0736-2994.

[8] H. V. Ly, K. D. Mease, and E. S. Titi, Distributed and boundary control of theviscous Burgers’ equation, Numer. Funct. Anal. Optim. 18(1-2), 143–188, (1997).ISSN 0163-0563.

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[13] G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems. vol. 229,London Mathematical Society Lecture Note Series, (Cambridge University Press,Cambridge, 1996). ISBN 0-521-57900-7.



[14] G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary condi-tions, Stochastics Stochastics Rep. 42(3-4), 167–182, (1993). ISSN 1045-1129.

[15] D. Blomker, S. Maier-Paape, and T. Wanner, Roughness in surface growth equations,Interfaces Free Bound. 3(4), 465–484, (2001). ISSN 1463-9963.

[16] D. Blomker, S. Maier-Paape, and T. Wanner, Surface roughness in molecular beamepitaxy, Stoch. Dyn. 1(2), 239–260, (2001). ISSN 0219-4937.

[17] D. Blomker, J. Duan, and T. Wanner, Enstrophy dynamics of stochastically forcedlarge-scale geophysical flows, J. Math. Phys. 43(5), 2616–2626, (2002). ISSN 0022-2488.

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Chapter 3

Asymptotics for the Space-Time Wigner Transform with

Applications to Imaging

Liliana Borcea, George Papanicolaou∗ and Chrysoula Tsogka†

Computational and Applied Mathematics, Rice University6100 Main Street, Houston, TX 77005-1892

[email protected]

We consider the space-time Wigner transform of the solution of the randomSchrodinger equation in the white noise limit and for high frequencies. We ana-lyze in particular the strong lateral diversity limit in which the space-time Wignertransform becomes weakly deterministic. We also show how to use these asymp-totic results in broadband array imaging in random media.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2 The parabolic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923 Scaling and the asymptotic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4 The Ito-Liouville equation for the Wigner transform . . . . . . . . . . . . . . . . . . . 95

4.1 The white noise limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 The high frequency limit and the space-time Wigner transform . . . . . . . . . 96

4.3 Statement of the strong lateral diversity limit . . . . . . . . . . . . . . . . . . . 98

4.4 The mean space-time Wigner transform . . . . . . . . . . . . . . . . . . . . . . 995 Self-averaging of the smoothed space-time Wigner transform, in the strong lateral di-

versity limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6 Application to imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1 Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2 Coherent interferometric imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 104

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

1. Introduction

In this paper we analyze the self-averaging property of the space-time Wigner trans-

form for solutions of the random Schrodinger equation, in a particular asymptotic

regime. We start with the wave equation in a random medium and then use the

parabolic or paraxial approximation, which is valid when waves propagate primar-

ily in a preferred direction and backscattering is negligible. This approximation is

∗Department of Mathematics, Stanford University, Stanford, CA 94305.([email protected])†Department of Mathematics, University of Chicago, Chicago, IL 60637.([email protected])

91


92 L. Borcea, G. Papanicolaou, and C. Tsogka

widely used in random wave propagation [1–5] and it is justified in some special

cases in Ref. [6], in the regime that we consider here. The parabolic wave field

satisfies a random Schrodinger equation, which we consider in the white noise limit.

White noise limits for random ordinary differential equations have been analyzed

extensively [7–9]. For random partial differential equations, white noise limits are

considered in [10] for diffusion equations and, more recently, in Refs. [11, 12], for

the random Schrodinger equation.

The resulting Ito-Schrodinger equation for the limit wave field is a stochastic

partial differential equation of independent interest that is analyzed in Ref. [5] and

in a wider context in Ref. [13, 14]. We consider here the high frequency limit of this

equation, using the space-time Wigner transform. This is a slight extension of the

high frequency limits analyzed in Refs. [15, 16] and in Ref. [11], using the spatial

Wigner transform. The limit process satisfies an Ito-Liouville partial differential

equation that arises from a stochastic flow [14, 17].

We analyze this Ito-Liouville equation in the strong lateral diversity limit, where

the propagating wave beam is wide with respect to the correlation length of the

random inhomogeneities in the transverse direction, orthogonal to the axis of the

beam. The importance of this limit in time reversal was pointed out in Ref. [18] and

it was analyzed later in Refs. [15, 16, 19, 20], using the spatial Wigner transform.

Applications to imaging are considered in Refs. [21–23], especially applications of the

space-time Wigner transform, but the strong lateral diversity limit is not analyzed

there.

We dedicate this work to Boris Rozovskii on the occasion of his 60th birthday.

2. The parabolic approximation

Let P (~x, t) be the solution of the acoustic wave equation

1

c2(~x)

∂2P

∂t2− ∆P = 0, t > 0, ~x ∈ R3, (2.1)

with a given excitation source at time t = 0 and in a medium with sound speed

c(~x) that is fluctuating about the mean value co, taken as constant for simplicity.

We model the fluctuations of c(~x) as a random process

c(~x) = co

[1 + σoµ

(~x

`

)]−1/2

(2.2)

where µ is a normalized, bounded and statistically homogeneous random field, with

mean zero and with smooth and rapidly decaying correlation function

Eµ(~x + ~x′)µ(~x′) = R(~x). (2.3)

Here the normalization means that

R(~0) = 1,

∫d~xR(~x) = 1, (2.4)


Asymptotics for a Space-Time Wigner Transform 93

so that ` in (2.2) is the correlation length of the fluctuations.

We consider a regime with weak fluctuations (σo 1) where backscattering of

the waves by the medium can be neglected and where we can study P (~x, t) with

the parabolic approximation [1]. For this, we take the z coordinate in the direction

of propagation of the waves and we let ~x = (z,x), with x the two dimensional

vector of coordinates transverse to the direction of propagation. In the parabolic

approximation the wave field is given by

P (z,x, t) =1

2π

∫P (z,x, ω)e−iωtdω, P (z,x, ω) ≈ eikzψ(z,x, k), (2.5)

where k = ω/co is the wavenumber and ψ is a complex valued amplitude satisfying

the Schrodinger equation

2ik∂ψ

∂z+ ∆xψ + k2σoµ

(z`,x

`

)ψ = 0, z > 0, (2.6)

with ∆x denoting the Laplacian in x. This equation is obtained by substituting

eikzψ in the reduced wave equation for P

∆P + k2n2(~x)P = 0,

with index of refraction n(~x) = co/c(~x) given by

n2(~x) = 1 + σoµ

(~x

`

), (2.7)

and by neglecting the term ∂2ψ∂z2 under the hypothesis that ψ is slowly varying in z

(i.e., k∣∣∣∂ψ∂z

∣∣∣∣∣∣∂

2ψ∂z2

∣∣∣).We now have an initial value problem for the wave amplitude ψ, governed by

equation (2.6) with initial condition

ψ(0,x, k) = ψo(x, k). (2.8)

We assume that ψo is a compactly supported function with frequency dependence

in the positive interval

ω ∈[ωo −

B

2, ωo +

B

2

], (2.9)

centered at ωo and with bandwidth B. The negative image of this interval is also

included if the initial data is real.

3. Scaling and the asymptotic regime

To carry out an asymptotic analysis of the wave field (2.5) we write the Schrodinger

equation (2.6) in dimensionless form

2ik′∂ψ

∂z′+

LzkoL2

x

∆x′ψ + koLzσo(k′)2µ

(Lzz

′

`,Lxx

′

`

)ψ = 0, (3.1)



with scaled variables

x = Lxx′, z = Lzz

′, ω = ωoω′, k = kok

′, c = coc′. (3.2)

Here ko = ωo/co is the central wavenumber, Lz quantifies the distance of propa-

gation and Lx is a transversal length scale which we take to be the width of the

propagating beam. Note that the scaled sound speed has constant mean < c′ >= 1.

Therefore, since the scaled wavenumber k′ is the same as the scaled frequency ω′,we shall replace ω′ by k′ from now on.

To simplify notation we drop the primes on the scaled variables and we introduce

three dimensionless parameters depending on the random medium

ε =`

Lz, δ =

`

Lx, σ = σoδε

− 32 , (3.3)

and the reciprocal of the Fresnel number

θ =LzkoL2

x

=1

2π

(λ0Lz

Lx

)

Lx. (3.4)

Here λo is the central wavelength and the reciprocal of the Fresnel number is written

as the ratio of the focusing spot size in time reversal imaging, λoLz/Lx, and the

transversal length scale Lx.

The scaled form of equation (3.1) is

2ik∂ψ

∂z+ θ∆xψ +

1

ε1/2µ(zε,x

δ

) σk2δ

θψ = 0, z > 0 (3.5)

and we study it in the asymptotic regime

ε δ 1, θ 1, σ = O(1). (3.6)

Thus, we suppose that the waves travel many correlation lengths in the random

medium (ε 1) and, to be consistent with the parabolic approximation, we take

Lx Lz (i.e., ε δ). We also take θ 1, which means that the time reversal

imaging spot size is much smaller than Lx

λoLzLx

Lx. (3.7)

Finally, we scale the strength of the fluctuations in (3.3) and (3.6) so that we can

take the white noise limit ε→ 0 in (3.5).

The asymptotic regime (3.6) can be realized with several scale orderings. In this

paper we assume that

ε θ δ 1, (3.8)

which amounts to taking ε→ 0 as the first in a sequence of three limits. This leads

to an Ito-Schrodinger equation for the limit ψ. The second limit θ/δ → 0 implies

that we are in a high frequency regime

λo`

ε

δ 1. (3.9)



In imaging, the spot size is small in this limit, when compared with the correlation

length

λoLzLx

` Lx. (3.10)

This is a regime in which we can derive an Ito-Liouville equation for the Wigner

transform of ψ, under the additional assumption of isotropy of the fluctuations of the

sound speed. Finally, we take the strong lateral diversity limit δ 1, which allows

us to show that the appropriately smoothed Wigner transform is self-averaging.

Other scale orderings consistent with (3.6) are

θ ε δ 1 (3.11)

and

ε δ ≤ θ 1. (3.12)

The ordering (3.11) is considered in Ref. [15], in a study of statistical stability of time

reversal in random media. It is a high frequency regime and it gives similar results to

those obtained here. The scale ordering (3.12) is consistent with λo ∼ ` and it is used

in numerical simulations in Refs. [21–24], in the context of array imaging of sources

and reflectors. In the parabolic approximation this scaling is analyzed in Ref. [16].

The theory is not so well developed when the parabolic approximation does not

apply. Nevertheless, it appears from the numerical simulations in Refs. [21–24] that

the statistical stability that we have in regimes (3.8) or (3.11) is valid in the case

(3.12).

4. The Ito-Liouville equation for the Wigner transform

In this section we give, without details, the Ito-Liouville equation for the Wigner

transform of ψ in the limits ε → 0 and θ → 0. We then state the main result of

this paper, which is that in the limit δ → 0 we have self-averaging for smooth linear

functionals of the space-time Wigner transform. The proof is given in section 5. We

consider an application of this self-averaging property in section 6, where we look at

coherent interferometric imaging in random media, as introduced in Refs. [22, 23].

4.1. The white noise limit

Let us emphasize with the notation ψε(z,x, k) the dependence on ε of the wave

amplitude satisfying (3.5). This amplitude depends on θ and δ as well, but since

these are kept fixed in the first limit we suppress them from the notation. The

initial wave field ψo is assumed independent of ε.



It follows from [11, 12] that as ε → 0, ψε(z,x, k) converges weakly, in law, to

the solution ψ(z,x, k) of Ito-Schrodinger equation

dψ =

[iθ

2k∆x − k2σ2δ2

8θ2Ro(0)

]ψdz +

ikσδ

2θψdB

(z,

x

δ

), z > 0,

ψ(0,x, k) = ψo(x, k), at z = 0.

(4.1)

Here B(z,x) is a Brownian random field that is smooth in the transverse variable

x. The mean of B is zero and its correlation is given by

EB(z1,x1)B(z2,x2) = z1 ∧ z2Ro(x1 − x2), (4.2)

where z1 ∧ z2 = min z1, z2 and

Ro(x) =

∫ ∞

−∞R(z,x)dz. (4.3)

Because of our assumptions on R in section 2 we have that Ro is smooth and

rapidly decaying. This is used in section 5 to deduce the statistical stability of the

smoothed Wigner transform of ψ, in the limit θ/δ → 0 and δ → 0.

4.2. The high frequency limit and the space-time Wigner transform

As in section 4.1, we now use the notation ψθ(z,x, k) to emphasize the dependence

of the solution of Ito-Schrodinger equation (4.1) on the parameter θ. We study the

high frequency limit θ → 0 with the space-time Wigner transform

W θ(z,x, k,q, r)

=

∫dx

(2π)2

∫dk

2πeiq·x−ikrψθ

(z,x +

θx

2, k +

θk

2

)ψθ

(z,x− θx

2, k − θk

2

),

(4.4)

where the bar on ψθ denotes complex conjugate. The r variable in W θ is dual to

k and it represents the distance traveled by the waves in a medium with constant

speed < c >= 1, during a travel time t = r/ < c >. The argument q in W θ is a

two dimensional vector that is dual to x.

For an arbitrary but fixed z, the L2 norm of the Wigner transform W θ is deter-

mined by the space and frequency L2 norm of the initial wave function ψθo

∥∥W θ(z, ·)∥∥L2 =

[∫dx

∫dk

∫dq

∫dr∣∣W θ(z,x, k,q, r)

∣∣2]1/2

=

∫dx

∫dk

∫dx

(2π)2

∫dk

2π

∣∣∣∣∣ψθ

(z,x +

θx

2, k +

θk

2

)∣∣∣∣∣

2 ∣∣∣∣∣ψθ

(z,x− θx

2, k − θk

2

)∣∣∣∣∣

2

1/2

=

∥∥ψθ(z, ·)∥∥2

L2

(2πθ)3/2=

∥∥ψθo∥∥2

L2

(2πθ)3/2,

(4.5)



because the Ito-Schrodinger equation (4.1) preserves the space and frequency L2

norm of its solution [5]. This means that with a proper definition and scaling of

the initial wave function ψθo [25], we can bound the L2 norm of W θ(z, ·) uniformly

with respect to θ.

We formally obtain an Ito-Liouville equation for the high frequency limit W

as follows [12]. We use Ito’s formula to get from (4.1) an equation for ψθ1ψθ2 =

ψθ(z,x1, k1)ψθ(z,x2, k2), with

x1 = x +θx

2, x2 = x − θx

2, (4.6)

k1 = k +θk

2, k2 = k − θk

2(4.7)

and then we Fourier transform in x and k and take the limit θ/δ → 0. The variables

x, x, k and k are independent of the small parameters. We have

d(ψθ1ψ

θ2

)=[

iθ

2k+θk

(14∆x + 1

θ∇x · ∇x + 1θ2 ∆x

)− iθ

2k−θk

(14∆x − 1

θ∇x · ∇x + 1θ2 ∆x

)

+

(k2− θ2

4 k2)σ2δ2

4θ2 R0

(θ|x|δ

)−

(k2+ θ2

4 k2)σ2δ2

4θ2 R0(0)

]ψθ1ψ

θ2dz

+ iσδ2θ ψ

θ1ψ

θ2

[(k + θ

2 k)dB(z, x

δ + θx2δ

)−(k − θ

2 k)dB(z, x

δ − θx2δ

)]

(4.8)

and, using the smoothness of B and Ro in the transverse variables, we have further

dB(z,

x

δ+θx

2δ

)− dB

(z,

x

δ− θx

2δ

)=θx

δ· ∇xdB(z,x) +O

(θ

δ

)2

(4.9)

and

Ro

(θx

δ

)= Ro(0) +

θ

δx · ∇Ro(0) +

θ2

2δ2

2∑

i,j=1

∂2ijRo(0)xixj +O

(θ

δ

)3

. (4.10)

The equation for W follows by Fourier transforming (4.8) in x and k, using the

expansions (4.9) and (4.10) and letting θ/δ → 0. We simplify the result by assuming

that the fluctuations are isotropic so that Ro(x) = Ro(|x|). This gives

∇Ro(0) = 0 and ∂2ijRo(0) = R

′′(0)δij (4.11)

and we obtain for W the Ito-Liouville equation

dW =[q

k· ∇x − |q|2

2k2∂∂r + k

2Dκ

2 ∆q + δ2Dr

2∂2

∂r2

]Wdz

+σk2 ∇qW · ∇xdB

(z, x

δ

)− σδ

2∂W∂r dB

(z, x

δ

), z > 0

(4.12)

with initial condition

W (z = 0,x, k,q, r) = Wo(x, k,q, r) (4.13)



and with the positive diffusion coefficients

Dκ = −σ2R′′

o (0)

4and Dr =

σ2Ro(0)

4. (4.14)

Equation (4.12) was also derived in Ref. [22] and we note that it is a stochastic flow

equation [14, 17] that is the starting point of the analysis in this paper. We want

to study the limit of the process W as δ → 0.

4.3. Statement of the strong lateral diversity limit

Now that we have the Ito-Liouville equation (4.12), we emphasize the dependence

of the process on δ by writing W δ(z,x, k,q, r). We assume that the initial condition

Wo is independent of δ.

The mean Wδ = EW δ is considered in section 4.4 and it follows, as is easily

seen from (4.12), that as δ → 0, Wδ converges weakly to the solution W of the

phase space advection-diffusion equation

∂W∂z

= LW (4.15)

with initial conditions

W(0,x, k,q, r) = Wo(x, k,q, r), (4.16)

where

L =q

k· ∇x − |q|2

2k2

∂

∂r+k

2Dκ

2∆q. (4.17)

This deterministic equation is solved explicitly in Ref. [22].

However, the point-wise variance of W δ is not zero for any δ and it does not van-

ish as δ → 0. This means that W δ is randomly fluctuating and it does not converge

to a deterministic process as δ → 0 in the strong, point-wise sense. Nevertheless,

we do have convergence in a weak sense as follows.

Theorem 4.1. Suppose that Wo is in L2 and it does not depend on δ. Then, given

any smooth and rapidly decaying test function ϕ(x, k,q, r), we have that

< W δ , ϕ > (z) =

∫dx

∫dk

∫dq

∫dr ϕ(x, k,q, r)W δ(z,x, k,q, r) (4.18)

converges in probability as δ → 0 to <W , ϕ > (z), for any z > 0.

Theorem 4.1 is proved in section 5. It states that even though W δ does not have

a deterministic point-wise limit, it is weakly self-averaging. That is, smooth linear

functionals of W δ become deterministic in the limit δ → 0. This is the property that

can be exploited in applications such as imaging in random media, as we explain in

section 6.



4.4. The mean space-time Wigner transform

Taking expectations in (4.12) we get that Wδ(z,x,q, r) = EW δ(z,x,q, r) satisfies

the phase space advection-diffusion equation

∂Wδ

∂z= LδW , (4.19)

with initial condition Wo(x, k,q, r), where

Lδ = L +δ2Dr

2

∂2

∂r2. (4.20)

Equivalently, Wδ is given as an expectation

Wδ(z,x,κ, r) = EWo

(Xδ(z), k,Qδ(z),Rδ(z)

), (4.21)

whereXδ(z),Qδ(z),Rδ(z)

is the Ito diffusion process with generator Lδ and with

initial condition

Xδ(0) = x, Qδ(0) = q, Rδ(0) = r. (4.22)

For z > 0, the Ito stochastic differential equations are

dXδ(z) =1

kQδ(z)dz,

dQjδ(z) = k

√Dκ dBj(z), j = 1, 2, Qδ =

(Qδ1, Q

δ2

), (4.23)

dRδ(z) = −|Qδ(z)|2

2k2 dz − δ

√DrdB(z),

where the driving is with three independent standard Brownian motions B(z) and

Bj(z), for j = 1, 2. The same processXδ(z),Qδ(z),Rδ(z)

also determines expec-

tations of higher powers of W δ

E∣∣W δ(z,x,q, r)

∣∣n

= E∣∣Wo

(Xδ(z), k,Qδ(z),Rδ(z)

)∣∣n, n ≥ 1. (4.24)

As δ → 0, we see that Wδ converges to the solution of (4.15), computed explicitly

in Ref. [22]. Actually, all one point moments converge as δ → 0,

E∣∣W δ(z,x,q, r)

∣∣n→ E

∣∣Wo

(X(z), k,Q(z),R(z)

)∣∣n, n ≥ 1, (4.25)

where X(z),Q(z),R(z) is the δ independent Ito diffusion

dX(z) =1

kQ(z)dz,

dQj(z) = k√Dκ dBj(z), j = 1, 2, Q = (Q1, Q2) , (4.26)

dR(z) = −|Q(z)|2

2k2 dz, z > 0,

with initial conditions

X(0) = x, Q(0) = q and R(0) = r. (4.27)



Clearly, W δ does not have a point-wise deterministic limit because the limit variance

is not zero

E∣∣Wo

(X(z), k,Q(z),R(z)

)∣∣2−∣∣EWo

(X(z), k,Q(z),R(z)

)∣∣2 6= 0. (4.28)

5. Self-averaging of the smoothed space-time Wigner transform, in

the strong lateral diversity limit

In this section we prove Theorem 4.1. We begin by calculating the form of the

infinitesimal generator Aδ of Ito-Liouville process W δ(z,x, k,κ, r), considered as

a process in the space of continuous functions in z with values in the space S ′ of

Schwartz distributions w over R2 × R × R2 × R.

Let F be a real valued test function on R and define for each test function ϕ in

S over R2 × R × R2 × R the function f(w) by

f(w) = F (< w,ϕ >). (5.1)

We have that

Aδf(w) =d

dzEF (< W δ(z), ϕ >)|W δ(0) = w|z=0

= Dδ(w)F′(< w,ϕ >) + Mδ(w)F

′′(< w,ϕ >) (5.2)

where

Dδ(w) =< w,L?δϕ >, (5.3)

with L?δ the adjoint of Lδ in (4.20). The Mδ can be written as the sum of three

terms

Mδ1(w) =

σ2δ2

8

∫dx

∫dk

∫dq

∫dr

∫dx′∫dk

′∫dq′∫dr′ w(x, k,q, r)w(x′, k

′,q′, r′)

×Ro(

x − x′

δ

)∂ϕ(x, k,q, r)

∂r

∂ϕ(x′, k′,q′, r′)

∂r′, (5.4)

Mδ2(w)

= −σ2δ

4

∫dx

∫dk

∫dq

∫dr

∫dx′∫dk


′,q′, r′)k

′ ×

∇x′Ro

(x− x′

δ

)· ∇q′ϕ(x′, k

′,q′, r′)

∂ϕ(x, k,q, r)

∂r, (5.5)

Mδ3(w)

= −σ2

8

∫dx

∫dk

∫dq

∫dr

∫dx′∫dk


′,q′, r′)kk

′ ×2∑

j,l=1

∂2ljRo

(x − x′

δ

)∂ϕ(x, k,q, r)

∂qj

∂ϕ(x′, k′,q′, r′)

∂q′l. (5.6)



Now we get from (5.3) that as δ → 0,

limδ→0

Dδ(w) = D(w) =< w,L?ϕ >, (5.7)

uniformly for w bounded in L2. Here L? is the adjoint of L defined by (4.17).

Furthermore,

limδ→0

Mδj (w) = 0, for j = 1, 2, 3,

uniformly for w bounded in L2, as we show next.

From (5.4)-(5.6) we see that it is enough to show that Mδ3(w) → 0. By the

Cauchy-Schwartz inequality, we have

|Mδ3(w)| ≤ ‖w‖2

L2Jδ(ϕ), (5.8)

where

[Jδ(ϕ)]2 =

(σ2

8

)2 ∫dx

∫dk

∫dq

∫dr

∫dx′∫dk

′∫dq′∫dr′

kk′

2∑

j,l=1

∂2ljRo

(x − x′

δ

)∂ϕ(x, k,q, r)

∂qj

∂ϕ(x′, k′,q′, r′)

∂q′l

2

.

Since Ro is rapidly decaying at infinity, we see that for any fixed test function ϕ,

Jδ(ϕ) tends to zero as δ → 0.

We have shown therefore that for functions f(w) of the form (5.1), with ϕ in Sfixed and uniformly for w bounded in L2,

Aδf(w) → Af(w) = D(w)F′(< w,ϕ >), (5.9)

where

D(w) =< w,L?ϕ > . (5.10)

The operator A is the generator of the deterministic process W(z,x, k,q, r), that

is the solution of (4.15). Since the limit process is deterministic, it follows that

convergence in law implies convergence in probability, weakly in S ′. It also follows

that functions of the form (5.1) are sufficient again since the limit is deterministic

[14]. The moment condition needed for tightness for processes in C([0, Z],S ′) or in

D([0, Z],S ′) [26] is easily obtained as in Ref. [20], and we omit it here.

6. Application to imaging

In this section we consider applications to imaging a source in a random medium

from measurements of the wave field P at an array of transducers. The setup is

shown in Figure 3.1, where we introduce a new coordinate system, with scaled range

ζ = L− z measured from the array and with cross range (transverse) coordinates x

defined with respect to the center ~y? = (L,0) of the source, which can be small or



ζ = Lζ = 0

ζ

source~xr = (0,xr)

~y? = (L,0)

~yS = (L+ ηS , ξS)

Fig. 3.1. Setup for imaging a distributed source with a planar array of transducers

spatially distributed. The array is in the plane ζ = 0 and it consists of receivers at

N discrete locations ~xr = (0,xr), where we record the traces P (~xr, t) over a time

window t ∈ [0, T ] that we suppose is long enough for

P (~xr, t) ≈ 0 for t > T

to hold. This allows us to simplify the analysis by neglecting the effect of a finite

measurement time window.

The goal in imaging is to estimate the support of the source from the traces

at the array and this is done very efficiently with Kirchhoff migration [27, 28], if

there are no fluctuations of the sound speed. However, in random media Kirchhoff

migration gives noisy and unpredictable results, in the sense that they lack statistical

stability. As an alternative to Kirchhoff migration we introduced in Refs. [22, 23]

a new, coherent interferometric (CINT) imaging functional, which is a statistically

smoothed migration. The resolution analysis of CINT is given in Ref. [22]. Here we

use the result of section 4.3 to show that it is statistically stable in the asymptotic

regime (3.8).

6.1. Migration

In this and the following sections all variables are scaled as in section 3. The wave

field P at receiver location ~xr = (0,xr) is

P (xr , ω) ≈ ei(δ/ε)2k/θLψ(L,xr, k), (6.1)

where we used that koLz = δ2/(θε2) and we dropped the range coordinate in the

argument of P , as it is always ζ = 0 at the array. We also kept the definition

ψ = ψ(z,x, k), with z = L− ζ, which means that at the array the range coordinate

in ψ is z = L. The parabolic amplitude ψ solves equation (3.5) with initial data

ψo(x, k) and it depends on the three small parameters ε, θ and δ. In the previous



sections we emphasized this dependence using superscripts, before taking limits.

Here we don’t use the superscripts to simplify notation and we keep ε, θ and δ fixed

until the very end where we apply the theoretical results of section 4.3.

Classic Kirchhoff migration imaging [27, 28] estimates the support of the source

by migrating (back propagating) the traces P (xr, t) to search points ~yS = (L +

ηS , ξS), in a fictitious, homogeneous medium, with scaled sound speed < c >= 1

and by summing over the receivers. The scaled distance from ~xr to ~yS is

[(L+ ηS)2 +

(LxLz

)2 ∣∣∣ξS − xr

∣∣∣2] 1

2

≈ L+ ηS +ε2

2δ2

∣∣∣ξS − xr

∣∣∣2

(L+ ηS)(6.2)

and it equals the scaled travel time τ(xr , ~yS), since the scaled mean speed is

< c >= 1. This gives the migration phase

(koLz)ωτ(~xr , ~yS) ≈ δ2

ε2θk(L+ ηS) +

k

2θ

∣∣∣ξS − xr

∣∣∣2

(L+ ηS)(6.3)

and the migrated wave field to ~yS

P (xr, ω)e−i(koLz)ωτ(~xr,~yS) ≈ ψ(L,xr, k) exp

−i δ

2

ε2k

θηS − i

k

2θ

∣∣∣ξS − xr

∣∣∣2

(L+ ηS)

. (6.4)

The Kirchhoff migration image is given by

I KM(~yS) =

N∑

r=1

∫dω P (xr, ω)e−i(koLz)ωτ(~xr,~y

S) (6.5)

and, as shown in Refs. [21–23], it lacks statistical stability with respect to the

realizations of the random medium and it gives noisy results that are difficult to

interpret.

We consider next coherent interferometric imaging, which is a statistically

smoothed version of migration [22, 23]. Before describing this method, let us make

the assumption that the array receivers are placed on a square mesh, in a square

aperture of area a2. The scaled mesh size is h and we suppose that it is small, so

we can write

N∑

r=1

≈ 1

h2

∫dx ∼

∫dx, (6.6)

with x varying continuously in the array aperture and with symbol ∼ denoting

approximate, up to a multiplicative constant.



6.2. Coherent interferometric imaging

The coherent interferometric imaging technique was introduced in Refs. [22, 23] and

it uses the coherence in the data traces P (x, t) to obtain reliable images in random

media. There are two characteristic coherent parameters in the data:

• The decoherence frequency Ωd, which is the difference in the frequencies ω1 and

ω2 over which ψ(z,x, k1) and ψ(z,x, k2) become uncorrelated.

• The decoherence lengthXd, which is the distance |x1−x2| over which ψ(z,x1, k)

and ψ(z,x2, k) become uncorrelated.

These decoherence parameters depend on the statistics of the random medium and

the range z and they are described in detail in the next section, for the asymptotic

regime (3.8).

Coherent interferometry (CINT) is a migration technique that works with cross

correlations of the traces, instead of the traces themselves. These cross correlations

are computed locally over space-time windows of size Xd × Ωd and they are called

coherent interferograms. We give in section 6.2.2 the mathematical expression of

the CINT imaging function and then we study its statistical stability. The CINT

functional and its resolution properties are motivated and analyzed in Refs. [22, 23].

6.2.1. The decoherence length and frequency

The decoherence length and frequency can be determined from the decay over x =

(x1 − x2)/θ and k = (k1 − k2)/θ of the expectation⟨ψ

(z,x +

θx

2, k +

θk

2

)ψ

(z,x− θx

2, k − θk

2

)⟩,

which we calculated explicitly in Ref. [22], by solving equation (4.19). The moment

formula is given by⟨ψ

(z,x +

θx

2, k +

θk

2

)ψ

(z,x− θx

2, k − θk

2

)⟩

≈ −k2ϕ1(z, k)

4π2z2exp

− k

2δ2Drz

2− k

2Dκϕ2(z, k)z|x|2

6

∫dξ

∫dξ exp

ik|x − ξ|2

2z+ik

z(x − ξ) · (x − ξ) + k

2ϕ3(z, k)x · ξ − k

2ϕ4(z, k)|ξ|2

−k2Dκϕ2(z, k)ϕ5(z, k)z

6

[x · ξ + ϕ5(z, k)|ξ|2

]

ψo

(ξ +

θξ

2, k +

θk

2

)ψo

(ξ − θξ

2, k − θk

2

).

(6.7)



This result is obtained in the white noise limit and the approximation involves a

simplification of the exact formula for small θ. The coefficients in this moment

formula (6.7) are given by

ϕ1(z, k) =z

√−ikDκ

sinh1/2

(z

√−ikDκ

) coth1/2

(z

√−ikDκ

), (6.8)

ϕ2(z, k) =3i

kzDκ

√−ikDκ

tanh(z

√−ikDκ)

− 1

z

, (6.9)

ϕ3(z, k) =i

2kz

3z

√−ikDκ

sinh(z

√−ikDκ)

− 1

cosh(z

√−ikDκ)

− 2

, (6.10)

ϕ4(z, k) =Dκ tanh(z

√−ikDκ)

2

√−ikDκ

1 −

tanh(z

√−ikDκ)

z

√−ikDκ

(6.11)

ϕ5(z, k) =1

cosh(z

√−ikDκ)

. (6.12)

To determine the decoherence length we let k → 0 and then study the decay over

|x| of the right hand side in (6.7).

In the limit k → 0, we get from (6.8)-(6.12) that

ϕj(z, k) =

1 +O(k), j = 1, 2, 5

O(k1/2), j = 3, 4(6.13)

and equation (6.7) simplifies to

⟨ψ

(z,x +

θx

2, k

)ψ

(z,x− θx

2, k

)⟩≈ −k2

4π2z2exp

−k

2Dκz|x|2

6

∫dξ

∫dξψo

(ξ +

θξ

2, k

)ψo

(ξ − θξ

2, k

)

exp

ik

z(x − ξ) · (x − ξ) − k

2Dκz

6

[x · ξ + |ξ|2

],

(6.14)

with the explicit integration depending on the spatial support of the wave source

function ψo. For example, in the case of a spatially distributed source, where

ψo(ξ ± θξ/2, k) ≈ ψo(ξ, k),



the integration over ξ gives⟨ψ(z,x + θx

2 , k)ψ(z,x− θx

2 , k)⟩

≈ −32πz3Dκ

exp−k

2Dκz|x|2

2

∫dξ |ψo(ξ, k)|2

exp

− 3

2z3Dκ

∣∣∣x − ξ − ikz2Dk

2 x∣∣∣2.

In the case of a small source with support of O(θ), where

ψo θ−2ψo

(ξ

θ, k

),

we can let ξ θξ in (6.14) and obtain⟨ψ

(z,x +

θx

2, k

)ψ

(z,x− θx

2, k

)⟩∼ −k2

4π2z2exp

−k

2Dκz|x|2

8+ik

zx · x

∫dξ

∫dξ ψo

(ξ +

ξ

2, k

)ψo

(ξ − ξ

2, k

)exp

−k

2Dκz|x|2

6

∣∣∣∣ξ +x

2

∣∣∣∣2

− ik

zξ · x

.

In either case, the decay in x occurs as a Gaussian function, with standard deviation

of O(

1k√zDκ

). This means that the scaled decoherence length is

Xd(k) ∼ θ

k√zDκ

(6.15)

and it corresponds to the scaled expected time reversal spot size derived in Refs.

[15, 22]. From the analysis in Refs. [15, 18, 22] we know that the effective aperture

is given, in scaled variables, by

ae =√Dκz3 (6.16)

We can now write

Xd(k) ∼ θ

kκd(6.17)

where

κd =aez

=√Dκz. (6.18)

The uncertainty in the direction of arrival of the waves in the random medium [23]

is κd/θ.

Next, we estimate the decoherence frequency by setting x → 0 in (6.7)⟨ψ(z,x, k + θk

2

)ψ(z,x, k − θk

2

)⟩= −k2

ϕ1(z,k)4π2z2 exp

− k2δ2Drz

2

∫dξ

∫dξ

exp

ik|x−ξ|2

2z − ikz (x − ξ) · ξ − k

2ϕ4(z, k)|ξ|2 − k

2Dκϕ2(z,k)ϕ

25(z,k)z|

˜ξ|26

ψo

(ξ +

θ˜ξ2 , k + θk

2

)ψo

(ξ +

θ˜ξ2 , k − θk

2

)

(6.19)



and by taking the large k approximation in (6.19). We obtain from (6.8)-(6.12) that

ϕ1(z, k) ≈ z

√−2ikDκe

− z2

√kDκ

2 (1−i), (6.20)

ϕj(z, k) = O(k−1/2), j = 2, 4 (6.21)

ϕ5(z, k) ≈ e−z√

kDκ2 (1−i), (6.22)

which means that as k increases, the decay in (6.19) is determined by the factor

exp

−z

2

√kDκ

2− k2δ2Drz

2

and therefore, that the scaled decoherence frequency is

Ωd ∼ min

θ

z2Dκ,

θ

δ√Drz

≈ θ

z2Dκas δ → 0. (6.23)

In conclusion, both Xd and Ωd are small, of order θ in our scaling, which means

that we can cover many decoherence lengths with an array aperture of O(1) and we

can fit many frequency intervals of width Ωd in a broad bandwidth B/ωo = O(1).

This is a key point for achieving the self-averaging property of the CINT imaging

function discussed below.

6.2.2. The coherent interferometric imaging function as a smoothed space-

time Wigner transform

Consider a smooth window χ(r; ρ) of length O(ρ), with Fourier transform

χ(k; ρ−1) =

∫χ(r; ρ)eikrdr, (6.24)

supported in the wavenumber interval |k| ≤ ρ−1, where

ρ ∼ θ

Ωd= O(1). (6.25)

Let also Φ(κ;κd) be a smooth function of two dimensional vectors κ, with support

in a disk of radius O(κd), with κd quantifying the uncertainty in the direction of

arrival of the waves in the random medium, as explained in section 6.2.1. The

Fourier transform of Φ is denoted by

Φ(kx;κ−1d ) =

∫Φ(κ;κd)e

−ikκ·xdκ (6.26)

and it is supported in the disk k|x| ≤ κ−1d .



Using the windows (6.24), (6.26) and the migrated wave field (6.4), we define

the coherent interferometric imaging function [23]

I CINT(~yS ; ρ, κd) ∼∫dk

∫dx

∫dk χ(k; ρ−1) e−i(δ/ε)

2kηS

∫dx Φ(kx;κ−1

d )

ψ

(L,x +

θx

2, k +

θk

2

)ψ

(L,x − θx

2, k − θk

2

)(6.27)

exp

−i (k + θk

2 )

2θ

∣∣∣ξS − x − θx2

∣∣∣2

(L+ ηS)+ i

(k − θk2 )

2θ

∣∣∣ξS − x + θx2

∣∣∣2

(L+ ηS)

that becomes after simplifying the exponent,


∫dx

∫dk χ(k; ρ−1) e−i(δ/ε)

2kηS

∫dx Φ(kx;κ−1

d )

ψ

(L,x +

θx

2, k +

θk

2

)ψ

(L,x− θx

2, k − θk

2

)(6.28)

exp

−ikx ·

(x − ξS

)

(L+ ηS)− ik

∣∣∣ξS − x∣∣∣2

2(L+ ηS)

.

Now note that because the support of χ(r; ρ) isO(1), the imaging function is nonzero

if the range offset satisfies

ηS ≤ O(ε2/δ2

) 1.

We set then ηS ε2

δ2 ηS and write approximately for ~yS =

(L+ ε2

δ2 ηS , ξS

),


∫dx

∫dk χ(k; ρ−1)

∫dx Φ(kx;κ−1

d )

ψ

(L,x +

θx

2, k +

θk

2

)ψ

(L,x− θx

2, k − θk

2

)(6.29)

exp

ikx ·

(ξS − x

)

L− ik

ηS +

∣∣∣ξS − x∣∣∣2

2L

.



Next, we use the definition (4.4) of the Wigner transform in (6.29) and obtain


∫dx

∫dq

∫drW (L,x, k,q, r)

∫dk χ(k; ρ−1) exp

−ik

ηS +

∣∣∣ξS − x∣∣∣2

2L− r

(6.30)

∫dx Φ(kx;κ−1

d ) exp

[ikx · ξS − x

L− iq · x

].

Finally, changing variables q = kκ, we get

I CINT(~yS ; ρ, κd) ∼∫dx

∫dκ

∫dr χ

ηS +

∣∣∣ξS − x∣∣∣2

2L− r; ρ

Φ

(ξS − x

L− κ;κd

)∫dkW (L,x, k, kκ, r). (6.31)

In conclusion, the coherent imaging function is given by the Wigner transform,

smoothed by convolution over directions κ and range r and by integration over the

array locations x and wavenumbers k. The self-averaging of I CINT follows from

Theorem 4.1.

Acknowledgments

The work of L. Borcea was partially supported by the Office of Naval Research,

under grant N00014-02-1-0088 and by the National Science Foundation, grants

DMS-0604008, DMS-0305056, DMS-0354658. It was also supported by INRIA in

the group POEMS of P. Joly. The work of G. Papanicolaou was supported by

grants ONR N00014-02-1-0088, 02-SC-ARO-1067-MOD 1 and NSF DMS-0354674-

001. The work of C. Tsogka was partially supported by the Office of Naval Research,

under grant N00014-02-1-0088 and by 02-SC-ARO-1067-MOD 1.

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Chapter 4

The Korteweg-de Vries Equation with Multiplicative

Homogeneous Noise

Anne de Bouard and Arnaud Debussche∗

CNRS et Universite Paris-Sud,UMR 8628, Bat. 425, 91405 ORSAY CEDEX, FRANCE

[email protected].

We prove the global existence and uniqueness of solutions both in the energy spaceand in the space of square integrable functions for a Korteweg-de Vries equationwith noise. The noise is multiplicative, white in time, and is the multiplicationby the solution of a homogeneous noise in the space variable.

Contents

1 Introduction and statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . 113

2 Preliminaries and existence for a truncated equation . . . . . . . . . . . . . . . . . . . 116

3 Global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

1. Introduction and statement of the results

The aim of the paper is to prove the global existence and uniqueness of strong

solutions for a Korteweg-de Vries equation with noise, which may be written in Ito

form as

du+ (∂3xu+

1

2∂x(u2))dt = uϕdW (1.1)

where u is a random process defined on (t, x) ∈ R+ × R, W is a cylindrical Wiener

process on L2(R) and ϕ is a convolution operator on L2(R) defined by

ϕf(x) =

∫

R

k(x− y)f(y)dy, for f ∈ L2(R)

where the convolution kernel k is an H1(R)∩L1(R) function of x ∈ R. Here H1(R)

is the usual Sobolev space of square integrable functions of the space variable x,

having their first order derivative in L2(R). Considering a complete orthonormal

∗ENS de Cachan, Antenne de Bretagne, Campus de Ker Lann, Av. R. Schuman, 35170 BRUZ,FRANCE, [email protected]

113


114 A. de Bouard and A. Debussche

system (ei)i∈N in L2(R), we may alternatively write W as

W (t, x) =∑

i∈N

βi(t)ϕei(x), (1.2)

(βi)i∈N being an independent family of real valued Brownian motions. Hence, the

correlation function of the process ϕW is

E(ϕW (t, x)ϕW (s, y)) = c(x − y)(s ∧ t), x, y ∈ R, s, t > 0,

where

c(z) =

∫

R

k(z + u)k(u)du.

The existence and uniqueness of solutions for stochastic KdV equations of the type

(1.1) but with an additive noise have been studied in Refs. [1], [2], [3]. Here we

extend those results to equation (1.1), that is the multiplicative case with homoge-

neous noise.

Note that an equation of this form, but with an additional weak dissipation has

been considered in Ref. [4]. Indeed, in this latter case where a dissipative term is

added, such a noise may be viewed as a perturbation of the dissipation. Although

our existence and uniqueness results would easily extend to the case where weak

dissipation is added, the dissipative term is of no help in the existence proof, so we

prefer stating the result for equation (1.1).

Assuming k ∈ H1(R) ∩ L1(R) will allow us to prove the global existence and

uniqueness of solutions to equation (1.1) in the energy space H1(R), that is in the

space where both quantities

m(u) =1

2

∫

R

u2(x)dx (1.3)

and

H(u) =1

2

∫

R

(∂xu)2dx− 1

6

∫

R

u3dx (1.4)

are well defined. Note that m and H are conserved quantities for the equation

without noise, that is

∂tu+ ∂3xu+

1

2∂x(u2) = 0. (1.5)

It is important to solve equation (1.1) in the energy space, indeed most of the studies

on the qualitative behavior of the solutions are done in this space. One of our aim

in the future is to analyse the qualitative influence of a noise on a soliton solution

of the deterministic equation, as we did in the additive case in Ref. [5], and this

requires the use of the hamiltonian (1.4). However, our method of construction of

solutions easily extends to treat the case of a kernel k ∈ L2(R) ∩ L1(R), obtaining

global existence and uniqueness in L2(R). It seems difficult to get a result with less

regularity.


KdV Equation with Homogeneous Multiplicative Noise 115

It may be noted also that the use of a noise of the form given in (1.1) naturally

brings some localization in the noisy part of the equation, at least in the limit where

the amplitude of the noise goes to zero, and when the initial state is a solitary wave

– or soliton – solution of the deterministic equation, that is a well localized solution

which propagates with a constant shape and velocity. This localization in the noise

was a missing ingredient in the study of the influence of an additive noise on the

propagation of a soliton (see Ref. [5]).

The precise existence result is the following, and the method we use to prove it

closely follows the method in Ref. [2].

Theorem 1.1. Assume that the kernel k of the noise satisfies k ∈ Hs(R) ∩L1(R),

s = 0 or 1. Then for any u0 in Hs(R), there is a unique adapted solution

u with paths almost surely in C(R+;Hs(R)) of equation (1.1). Moreover, u ∈L2(Ω;C(R+;L2(R))).

As in Refs. [2, 3], we use the functional framework introduced by Bourgain to

study dispersive equations. Following [6], [7], [8], for s, b ∈ R, Xb,s denotes the

space of tempered distributions f ∈ S ′(R2) for which the norm

‖f‖Xb,s=

(∫ ∫

R2

(1 + |ξ|)2s(1 + |τ − ξ3|)2b|f(τ, ξ)|2dτdξ)1/2

is finite, where f(τ, ξ) stands for the space-time Fourier transform of f(t, x). In the

same way, we set for b, s1, s2 ∈ R,

‖f‖Xb,s1,s2=

(∫ ∫

R2

|ξ|2s2(1 + |ξ|)2s1 (1 + |τ − ξ3|)2b|f(τ, ξ)|2dτdξ)1/2

and

Xb,s1,s2 =f ∈ S ′(R2), ‖f‖Xb,s1,s2

< +∞.

Note that the use of the space Xb,s1,s2 is necessary here. Indeed, since we work

with stochastic equations driven by white in time noises, we cannot require too

much time regularity, and we have to choose 0 < b < 1/2. But then the bilinear

estimate which allows to treat the KdV equation is not true in the space Xb,s as

was already mentioned for the additive case in Ref. [2].

For T ≥ 0, we also introduce the spaces XTb,s and XT

b,s1,s2of restrictions to [0, T ]

of functions in Xb,s and Xb,s1,s2 . They are endowed with

‖f‖XTb,s

= inf‖f‖Xb,s

, f ∈ Xb,s and f |[0,T ] = f |[0,T ]

‖f‖XTb,s1,s2

= inf‖f‖Xb,s1,s2

, f ∈ Xb,s1,s2 and f |[0,T ] = f |[0,T ]

.

Because equation (1.1) is a multiplicative equation with a nonlinear deterministic

part, we have to consider first a cut-off version of this equation (see Section 2). As

we make use of the functional framework defined above, the cut-off will arise as a



function of the norm of the solution of the type ‖ · ‖Xtb,s

. Moreover, this function

of the norm must be a regular function, in order to allow us to use a fixed point

argument (i.e. in order that our mapping is a contraction mapping, see Section

2). The fact that the functional spaces we consider are nonlocal spaces in the time

variable then brings a lot of technical difficulties, concerning points that would be

obvious if we were dealing with more classical function spaces (see e.g. the proof of

Lemma 2.1).

The paper is organized as follows: Section 2 is devoted to the proof of several

preliminary lemmas and propositions, which once brought together lead quite easily

to the proof of global existence and uniqueness for the cut-off version of the equation

– or to the local existence and uniqueness for equation (1.1). In Section 3 we prove

that the solutions of equation (1.1) are global in time, by using estimates on the

moments of the L2-norm of the solution. Again, due to the spaces we consider for

the local existence, the globalization argument is not obvious.

2. Preliminaries and existence for a truncated equation

As is usual, we introduce the mild form of the stochastic Korteweg-de Vries equation

(1.1). We denote by U(t) = e−t∂3x the unitary group on L2(R) generated by the

linear equation

∂u

∂t+∂3u

∂x3= 0.

Using Fourier transform, we have F(U(t)v)(ξ) = eitξ3F(v)(ξ). We then rewrite

(1.1) as follows

u(t) = U(t)u0−1

2

∫ t

0

U(t−r)∂x(u2(r))dr+

∫ t

0

U(t−r)(u(r)ϕdW (r)), t ≥ 0. (2.1)

The Xb,s and Xb,s1,s2 norms defined in the introduction have the nice property

that they are increasing with T . However, it is more convenient to work with other

norms, given by the multiplication by the function 1l[0,T ]. In the case we consider

here, that is 0 ≤ b < 1/2, we can prove the following result, stating that the two

norms are equivalent.

Lemma 2.1. Let s ≥ 0 and 0 ≤ b < 1/2, then there exist two constants C1, C2

depending on b but not on T such that for any f ∈ Xb,s

C1‖f‖XTb,s

≤ ‖1l[0,T ](t)f‖Xb,s≤ C2‖f‖XT

b,s.

Proof. The first inequality is clear and in fact we may choose C1 = 1. For the



other inequality, let us set g(t) = 1l[0,T ](t)U(−t)f(t) so that

‖1l[0,T ](t)f‖2Xb,s

=

∫ ∫

R2

(1 + |ξ|)2s(1 + |τ |)2b|g(τ, ξ)|2dτdξ

=

∫ ∫

R2

(1 + |ξ|)2s‖(Fxg)(·, ξ)‖2Hb

tdξ.

The result follows from the following inequality

‖1l[0,T ]h‖Hb(R) ≤ C‖h‖Hb(R), h ∈ Hb(R),

which holds for a constant C ≥ 0 depending on 0 < b < 1/2. To prove this, we use

the following equivalent norm on Hb(R) (see for instance [9]):

‖h‖2Hb(R) =

∫ ∫

R2

|h(t) − h(r)|2|t− r|1+2b

dtdr + ‖h‖2L2(R).

Clearly, ‖1l[0,T ]h‖2L2(R) ≤ ‖h‖2

L2(R). Moreover

∫ ∫

R2

∣∣1l[0,T ](t)h(t) − 1l[0,T ](r)h(r)∣∣2

|t− r|1+2bdrdt

= 2

∫ ∫

r<t

∣∣1l[0,T ](t)h(t) − 1l[0,T ](r)h(r)∣∣2

|t− r|1+2bdrdt

= 2

∫ T

0

∫ t

0

|h(t) − h(r)|2|t− r|1+2b

drdt + 2

∫ T

0

∫ 0

−∞

|h(t)|2|t− r|1+2b

drdt

+ 2

∫ ∞

T

∫ T

0

|h(r)|2|t− r|1+2b

drdt = I + II + III.

The first term I is less than ‖h‖Hb(R). The second and third terms are equal to

II =1

b

∫ T

0

|t|−2b|h(t)|2dt, III =1

b

∫ T

0

|T − r|−2b|h(r)|2dr.

Both are bounded by C‖h‖Hb(0,T ). To see this, we note that it is obvious for b = 0

and results from Hardy inequality for b = 1 when Hb(0, T ) is replaced by H10 (0, T ).

The result follows by interpolation since, for 0 ≤ b < 1/2, H b(0, T ) = Hb0(0, T ) .

We now define Y Tb,0 = XTb,0 ∩ XT

b,0,−3/8 endowed with the norm

‖f‖Y Tb,0

= max‖f‖XTb,0, ‖f‖XT

b,0,−3/8.

We also use the space Y Tb,1 = XTb,1 ∩ XT

b,1,−3/8 with a similar definition of its

norm. From now on and thanks to Lemma 2.1, we will use the definition

‖v‖TYb,s= ‖1l[0,T ]v‖Yb,s

each time we take a norm in Y Tb,s with 0 ≤ b < 1/2.

For u0 ∈ L2(Ω;H1(R)), we set

z(t) = U(t)u0, and v(t) = u(t) − z(t).



Then (2.1) is rewritten as

v(t) = − 12

∫ t

0

U(t− r)[∂x(v2(r)) + ∂x(z2(r)) + 2∂x(z(r)v(r))

]dr

+

∫ t

0

U(t− r)((z(r) + v(r))ϕdW (r)), t ≥ 0.

(2.2)

Let θ be a cut-off function – θ(x) = 0 for x ≥ 2, θ(x) = 1 for 0 ≤ x ≤ 1, with

θ ∈ C∞0 (R+) – and let θR = θ( .R ); we consider the cut-off version of (2.2) written

for R > 0 as:

vR(t) = −1

2

∫ t

0

U(t− r)[θ2R

(‖vR‖Y r

b,0

)∂x(v2

R(r))]dr

−∫ t

0

U(t− r)[θR

(‖vR‖Y r

b,0

)∂x(z(r)vR(r))

]dr

−1

2

∫ t

0

U(t− r)[∂x(z2(r))

]dr

+

∫ t

0

U(t− r)((z(r) + vR(r))ϕdW (r)), t ≥ 0.

(2.3)

We find vR as a fixed point of the mapping TR, TRvR being defined by the right

hand side above. Note that the cut-off is made in the L2 in space norm, even for

the H1 result. We will choose 0 < b < 1/2 and 1/2 < c.

We use the following Lemma.

Lemma 2.2. For any 0 ≤ b < 1/2, R > 0, v ∈ Y Tb,1, there exists C(R) such that

∥∥∥θR(‖v‖Y t

b,0

)v(t)

∥∥∥Y T

b,0

≤ C(R)

and, for s = 0 or 1, there is a positive constant C, independent of R, such that


b,0

)v(t)

∥∥∥Y T

b,s

≤ C ‖v(t)‖Y Tb,s.

Proof. We use arguments similar to the proof of Lemma 2.1. Let w(t) = U(−t)v(t)

then, using the same norm on Hb(R) as in Lemma 2.1,


b,0

)v(t)

∥∥∥2

XTb,0

≤ C

∫

R


b,0

)(Fxw)(t, ξ)

∥∥∥2

Hbt ([0,T ])

dξ.

The L2 part of the Hb norm above is easily estimated, while the other part is



bounded above by

C

∫

R

∫ T

0

∫ t

0

θ2R

(‖v‖Y t

b,0

) |(Fxw)(t, ξ) − (Fxw)(r, ξ)|2|t− r|1+2b

drdtdξ

+C

∫

R

∫ T

0

∫ t

0

(θR

(‖v‖Y t

b,0

)− θR

(‖v‖Y r

b,0

))2 |(Fxw)(r, ξ)|2|t− r|1+2b

drdtdξ

= I + II.

Next, we define τR = inft ≥ 0, ‖v‖Y tb,0

≥ 2R; then θR

(‖v‖Y t

b,0

)= 0 for t ≥ τR

and

I ≤ C

∫

R

∫ τR

0

∫ t

0

|(Fxw)(t, ξ) − (Fxw)(r, ξ)|2|t− r|1+2b

drdtdξ

≤ C

∫

R

‖(Fxw)(·, ξ)‖2Hb(0,τR) dξ ≤ C‖v‖XτR

b,0≤ 2CR.

In order to estimate II , we use the fact that for r < t,

(θR

(‖v‖Y t

b,0

)− θR

(‖v‖Y r

b,0

))2

≤ C

R2|θ′|2L∞ |‖v‖Y t

b,0− ‖v‖Y r

b,0|2

≤ C

R2|‖1l[0,t]v‖Yb,0

− ‖1l[0,r]v‖Yb,0|2 ≤ C

R2‖1l[r,t]v‖2

Yb,0

≤ C

R2

∫

R

(1 + |η|−3/4)‖1l[r,t](Fxw)(., η)‖2Hbdη.

We leave to the reader the estimate of the contribution to II of the L2 part of the

Hb norm above; indeed, it follows the same line as the estimate of the remaining

contribution, which is bounded above by

C

R2

∫

R

∫

R

(1 + |η|−3/4)

∫ τR

0

∫ t

0

∫ t

r

∫ σ2

r

|(Fxw)(σ2, η) − (Fxw)(σ1, η)|2|σ2 − σ1|1+2b

dσ1dσ2

×|(Fxw)(r, ξ)|2|t− r|1+2b

drdtdηdξ

≤ C

R2

∫

R

∫

R

(1 + |η|−3/4)

∫ τR

0

∫ σ2

0

∫ σ1

0

(∫ τR

σ2

dt

|t− r|1+2b

)|(Fxw)(r, ξ)|2dr

×|(Fxw)(σ2, η) − (Fxw)(σ1, η)|2|σ2 − σ1|1+2b

dσ1dσ2dηdξ

where we have inverted the integrals in the time variables; this last term is in turn



bounded above by

C

R2

∫

R

∫

R

(1 + |η|−3/4)

∫ τR

0

∫ σ2

0

(∫ σ1

0

|σ1 − r|−2b|(Fxw)(r, ξ)|2dr)

×|(Fxw)(σ2, η) − (Fxw)(σ1, η)|2|σ1 − σ2|1+2b

dσ1dσ2dηdξ ,

by the fact that |τR−r|−2b ≤ |σ2−r|−2b ≤ |σ1−r|−2b for 0 ≤ r ≤ σ1 ≤ σ2 ≤ t ≤ τR.

Using then the same arguments as in the proof of Lemma 2.1, we finally get

II ≤ C

R2

∫

R

‖(Fxw)(., ξ)‖2Hb(0,τR)dξ

∫

R

(1 + |η|−3/4)‖(Fxw)(., η)‖2Hb(0,τR)dη

≤ C

R2‖v‖2

YτR

b,0‖v‖2

XτRb,0.

This, together with the estimate of I implies the first inequality of the Lemma for

the Xb,0 part of the Yb,0 norm; the Xb,0,−3/8 part, and the second inequality of the

Lemma are proved in the same way.

Next results state the estimates on the bilinear term appearing in (2.3).

Proposition 2.3. Let a > 0, 0 < b < 1/2 < c < 1, with b + c > 1 and a, b, c

sufficiently close to 1/2, then for any v ∈ Y Tb,s, z ∈ XT

c,s, s = 0 or 1, we have

‖∂x(v2)‖Y T−a,s

≤ C‖v‖Y Tb,0‖v‖Y T

b,s,

‖∂x(vz)‖Y T−a,s

≤ C(‖v‖Y T

b,0‖z‖XT

c,s+ ‖v‖Y T

b,s‖z‖XT

c,0

),

and

‖∂x(z2)‖Y T−a,s

≤ C‖z‖XTc,s‖z‖XT

c,0.

Proof. These estimates are proved in Ref. [2], Proposition 2.2 and 2.3, for s = 0.

These are easily extended to s = 1. It suffices to add a factor 1+|ξ| in the expression

|〈f, ∂x(gh)〉|

=

∣∣∣∣∫

τ

∫

ξ

ξf(τ, ξ)

∫

τ1

∫

ξ1

g(τ − τ1, ξ − ξ1)h(τ1, ξ1)dτ1dξ1 dτdξ

∣∣∣∣ ,

and to use the fact that 1 + |ξ| ≤ (1 + |ξ − ξ1|) + (1 + |ξ1|).

Remark 2.4. It does not seem possible to get rid of the homogeneous Sobolev

space, i.e. of Xb,0,−3/8, to get the result of Proposition 2.3, when b < 1/2,

even in the case s = 1; indeed, a careful reading of the proof of Proposi-

tion 2.2 in Ref. [2] shows that the additional factor |ξ|−3/4 induced by the use

of Xb,0,−3/8 is necessary in a region of the integral where |ξ1| |ξ|, so that

|ξ − ξ1| ∼ |ξ|, and with moreover |ξ| ≤ 1; hence the supplementary factor

(1 + |ξ − ξ1|)(1 + |ξ1|)/(1 + |ξ|) is of no help there.



It remains to derive the estimates on the stochastic integrals in (2.3). In order

to be able to globalize the solutions in Section 3, we will need estimates on all the

moments of the stochastic integrals.

Proposition 2.5. Let m ∈ N, s = 0 or 1, and v ∈ L2m(Ω, XTb,s); then for any

0 ≤ b ≤ 1/2,

E

(∥∥∥∥∫ t

0

U(t− r) [v(r)ϕdW (r)]

∥∥∥∥2m

XTb,s

)≤ C‖k‖2m

Hs(R)E

(‖v‖2m

XT0,s

)

≤ CT bm‖k‖2mHs(R)E

(‖v‖2m

XTb,s

).

Moreover

E

(∥∥∥∥∫ t

0


∥∥∥∥2m

XTb,s−3/8

)≤ C

(‖k‖2m

Hs(R) + ‖k‖2mL1(R)

)E

(‖v‖2m

XT0,s

)

≤ CT bm(‖k‖2m

Hs(R) + ‖k‖2mL1(R)

)E

(‖v‖2m

XTb,s

).

Proof. We prove the result for s = 1, the proof is exactly the same for s = 0. We

set w(t) = 1l[0,T ](t)

∫ t

0

U(t− r)[v(r)ϕdW (r)]. Let

g(t) = 1l[0,T ](t)

∫ t

0

U(−r) [v(r)ϕdW (r)] ,

then w(t) = U(t)g(t), t ≥ 0. We have

E

(‖w‖2m

Xb,1

)= E

((∫ ∫

R2

(1 + |ξ|)2(1 + |τ |)2b|g(τ, ξ)|2dτdξ)m)

Choosing Brownian motions (βk)k∈N, defined on R, we have

(Fxg)(t, ξ) =

∞∑

k=0

1l[0,T ](t)

∫ t

0

eirξ3Fx (v(r)ϕek) (ξ)dβk(r)

=

∞∑

k=0

1l[0,T ](t)

∫ t

−∞1l[0,T ](r)e

irξ3Fx (v(r)ϕek) (ξ)dβk(r).

It follows

g(τ, ξ) =

∞∑

k=0

∫

R

1l[0,T ](t)

∫ t

−∞1l[0,T ](r)e

irξ3Fx (v(r)ϕek) (ξ)dβk(r)e−iτtdt

=

∞∑

k=0

∫

R

1l[0,T ](r)eirξ3Fx (v(r)ϕek) (ξ)

(∫ ∞

r

1l[0,T ](t)e−iτtdt

)dβk(r).



By Burkholder inequality, we deduce:

E

((∫ ∫

R2

(1 + |ξ|)2(1 + |τ |)2b|g(τ, ξ)|2dτdξ)m)

≤ CmE

(( ∞∑

k=0

∫ ∫ ∫

R3

(1 + |ξ|)2(1 + |τ |)2b1l[0,T ](r) |Fx (v(r)ϕek) (ξ)|2

×∣∣∣∣∫ ∞

r


∣∣∣∣2

drdξdτ

)m).

It is easy to see that

∣∣∣∣∫ ∞

r


∣∣∣∣2

≤ minT 2, 2τ−2.

Therefore, using Lemma 2.6 below,

E

((∫ ∫

R2

(1 + |ξ|)2(1 + |τ |)2b|g(τ, ξ)|2dτdξ)m)

≤ CE

((∫ ∫ ∫ ∫

R4

(1 + |ξ|)2(1 + |τ |)2b minT 2, 2τ−21l[0,T ](r) |(Fxv(r))(ξ + η)|2

×|k(η)|2dηdrdξdτ)m)

which in turn we bound from above, using the unitarity of U(t) in L2 and in H1,

by

CE

((∫ ∫ ∫

R3

[(1 + |ξ + η|)2 + (1 + |η|2)

]1l[0,T ](r) |(Fxv(r))(ξ + η)|2

×|k(η)|2dηdrdξ)m)

≤ C[‖k‖2m

L2(R)E

(‖1l[0,T ]v‖2m

X0,1

)+ ‖k‖2m

H1(R)E

(‖1l[0,T ]v‖2m

X0,0

)]

≤ C[‖k‖2m

L2(R)E

(‖v‖2m

XT0,1

)+ ‖k‖2m

H1(R)E

(‖v‖2m

XT0,0

)].

For the second statement, we proceed similarly. However, the extra |ξ|−3/4 implies

that a special treatment of the integral for |ξ| ≤ 1. On the region |ξ| ≥ 1, we simply



use |ξ|−3/4 ≤ 1. The following estimate is thus sufficient to conclude.

E

((∫

|ξ|≤1

∫ ∫

R2

(1 + |ξ|)2|ξ|−3/41l[0,T ](r) |(Fxv(r))(ξ + η)|2 |k(η)|2dηdrdξ)m)

≤ CE

((∫ ∫

R2

(∫

|ξ|≤1

|ξ|−3/4|k(η − ξ)|2dξ)

1l[0,T ](r) |Fx(v(r))(η)|2 dηdr)m)

≤ C‖k‖L∞(R)E

(‖v‖2m

XT0,0

)

≤ C‖k‖L1(R)E

(‖v‖2m

XT0,0

).

We now give the Lemma used in the above proof.

Lemma 2.6. Let v ∈ X0,0, then for any complete orthonormal system (ek)k∈N of

L2(R), we have

∞∑

k=0

|Fx (v(r)ϕek) (ξ)|2 =

∫

R

|Fx(v(r))(ξ + η)|2 |k(η)|2dη.

Proof. We have Fx (v(r)ϕek) (ξ) = Fx(v(r, x)〈k(x − y), ek(y)〉L2

y

)(ξ) =

〈Fx (v(r, x)k(x − y)) (ξ), ek(y)〉L2y. Therefore, by Parseval identity,

∞∑

k=0

|Fx (v(r)ϕek) (ξ)|2 = ‖Fx (v(r, x)k(x − y)) (ξ)‖2L2

y,

and by Plancherel theorem and an easy computation

∞∑

k=0

|Fx (v(r)ϕek) (ξ)|2 = ‖Fx,y (v(r, x)k(x − y)) (ξ, η)‖2L2

η= ‖Fx(v(r))(ξ + .)

¯k‖2

L2 ,

which gives the conclusion.

The proof of the next proposition is left to the reader. It only makes use of

classical arguments and ideas similar to those at the end of the proof of Proposition

2.5.

Proposition 2.7. Let s = 0 or 1. For any T0 > 0, any stopping time τ and any

predictable process v ∈ L2(Ω;C([0, T0 ∧ τ ];Hs(R))),∫ ·0 U(· − r)[ϕv(r)dW (r)] has

continuous paths with values in Hs(R) ∩ H−3/8(R) and for any integer m, there is

a constant Cm with

E

(sup

t≤T0∧τ

∥∥∥∥∫ t

0

U(t− r)[ϕv(r)dW (r)]

∥∥∥∥2m

Hs∩H−3/8

)≤ CmE

(sup

t≤T0∧τ‖v(t)‖2m

Hs

).



We are now able to prove the following existence theorem for the truncated

equation.

Theorem 2.8. Let s = 0 or 1 and assume that the convolution kernel of the operator

ϕ satisfies k ∈ Hs(R) ∩ L1(R); then for any u0 in Hs(R), equation (2.3) with

z(t) = U(t)u0 has a unique solution vR ∈ Y T0

b,s , for any b with 0 < b < 1/2, and any

T0 ≥ 0. Moreover vR ∈ L2(Ω;C([0, T0];Hs(R))).

Proof. We use a fixed point argument on equation (2.3). The following lemma,

whose first and third estimates were proved in Ref. [2], while the second one can be

proved in the same way, is useful.

Lemma 2.9.

• Let u0 ∈ Hs(R), s = 0 or 1. For any T > 0 and c > 1/2, z = U(·)u0 ∈ XTc,s

and

‖z‖XTc,s

≤ C(T )‖u0‖Hs(R).

• For any u0 ∈ Hs(R)∩H−3/8(R), and any b with 0 ≤ b < 1/2, z = U(·)u0 ∈ Y Tb,sand

‖z‖Y Tb,s

≤ C(T )(‖u0‖Hs(R) + ‖u0‖H−3/8(R)).

• For any a, b ∈ (0, 1) with a+ b ≤ 1, and any f ∈ Y T−a,s,∫ ·0 U(·− r)f(r)dr ∈ Y Tb,s

and∥∥∥∥∫ ·

0

U(· − r)f(r)dr

∥∥∥∥Y T

b,s

≤ CT 1−(a+b)‖f‖Y T−a,s

.

We first assume that the hypothesis of Theorem 2.8 hold with s = 0. Let us

fix a, b, c as in Proposition 2.3, with a + b < 1. We fix T0 and take T ≤ T0. Let

v1, v2 ∈ Y Tb,0, T being also fixed. We set vi(t) = θR

(|vi|Y t

b,0

)vi(t), i = 1, 2. Then,

recalling that TRvR is defined by the right hand side of (2.3), we have

E

(‖TRv1 − TRv2‖2

Y Tb,0

)

≤ CE

(∥∥∥∥∫ t

0

U(t− r)∂x

[(v1(r))

2 − (v2(r))2]dr

∥∥∥∥2

Y Tb,0

)

+CE

(∥∥∥∥∫ t

0

U(t− r)∂x [(v1(r) − v2(r)) z(r)] dr

∥∥∥∥2

Y Tb,0

)

+CE

(∥∥∥∥∫ t

0

U(t− r) [(v1(r) − v2(r))ϕdW (r)]

∥∥∥∥2

Y Tb,0

)



which, by Lemma 2.9 and Proposition 2.5, applied with m = 1, is bounded above

by

CT 2(1−(a+b))E

(∥∥∥∂x(

(v1)2 − (v2)

2)∥∥∥

2

Y T−a,0

)

+ CT 2(1−(a+b))E

(‖∂x ((v1 − v2) z)‖2

Y T−a,0

)+ CT bE

(‖v1 − v2‖2

XTb,0

).

By Proposition 2.3, it follows

E


Y Tb,0

)

≤ CT 2(1−(a+b))

E

(‖v1 − v2‖2

Y Tb,0

‖v1 + v2‖2Y T

b,0

)+ E

(‖v1 − v2‖2

Y Tb,0

)‖z‖2

XTc,0

+CT bE(‖v1 − v2‖2

XTb,0

).

By Lemma 2.2,

‖v1 + v2‖2Y T

b,0≤ C(R).

Moreover, it is not difficult to use the arguments of the proof of Lemma 2.2 and

prove

‖v1 − v2‖2Y T

b,0≤ C(R) ‖v1 − v2‖2

Y Tb,0.

We deduce that for some α > 0,

E


Y Tb,0

)≤ C(R, T0, ‖u0‖L2(R))T

αE ‖v1 − v2‖2Y T

b,0.

Thus, TR has a unique fixed point vR ∈ L2(Ω;Y Tb,0) for T ≤ T∗ where T∗ is chosen

such that

C(R, T0, ‖u0‖L2(R))Tα∗ ≤ 1/2.

Moreover, using arguments similar to the proof of Proposition 2.5, it can be seen that∫ t0U(−r) [(z(r) + vR(r))ϕdW (r)] is a square integrable martingale in L2(R) ∩

H−3/8(R). Since (U(t))t∈R is strongly continuous on L2(R) ∩ H−3/8(R), we de-

duce that

∫ t

0

U(t− r) [(z(r) + vR(r))ϕdW (r)] ∈ L2(Ω;C([0, T∗];L2(R) ∩ H−3/8(R))).

Using then Lemma 2.9 with b > 1/2 and similar estimates as above, we deduce that

vR is also in L2(Ω;C([0, T∗];L2(R) ∩ H−3/8(R))).



Then, we construct a solution in [T∗, 2T∗]. First, we write equation (2.3) with

t ≥ T∗ in the form

vR(T∗ + t) = U(t)vR(T∗) − 1

2

∫ t

0

U(t− r)[θ2R

(‖vR‖Y T∗+r

b,0

)∂x(v2

R(T∗ + r))]dr

−∫ t

0

U(t− r)[θR

(‖vR‖Y T∗+r

b,0

)∂x(z(T∗ + r)vR(T∗ + r))

]dr

−1

2

∫ t

0

U(t− r)[∂x(z2(T∗ + r))

]dr

+

∫ t

0

U(t− r)((z(T∗ + r) + vR(T∗ + r))ϕdW (r)), t ≥ 0.

(2.4)

Since vR(T∗) ∈ L2(Ω;L2(R)∩ H−3/8(R)), the first term is in L2(Ω;Y T∗b,0 ). It is then

easily seen that vR can be found on [T∗, 2T∗] as a fixed point in L2(Ω;Y T∗b,0 ) in the

same way as on the interval [0, T∗].

Iterating this, we get a solution on [0, T0] which is in fact in L2(Ω;Y T0

b,0 ) and also

in L2(Ω;C([0, T0];L2(R) ∩ H−3/8(R))). This proves the result for s = 0.

Now suppose that the assumptions hold with s = 1. Let v ∈ L2(Ω;Y Tb,1); we

have, setting v(t) = θR(‖v‖Y tb,0

)v(t), and using Lemma 2.9,

‖TRv‖Y Tb,1

≤ CT 1−(a+b)[‖v∂xv‖Y T

−a,1+ ‖∂x (vz)‖Y T

−a,1+∥∥∂x

(z2)∥∥Y T−a,1

]

+

∥∥∥∥∫ t

0

U(t− r) [z(r)ϕdW (r)]

∥∥∥∥Y T

b,1

+

∥∥∥∥∫ t

0


∥∥∥∥Y T

b,1

,

so that by Proposition 2.3,

‖TRv‖Y Tb,1

≤ CT 1−(a+b)[‖v‖Y T

b,0‖v‖Y T

b,1+ ‖v‖Y T

b,0‖z‖XT

c,1+ ‖v‖Y T

b,1‖z‖XT

c,0

+ ‖z‖XTc,0

‖z‖XTc,1

]+

∥∥∥∥∫ t

0


∥∥∥∥Y T

b,1

+

∥∥∥∥∫ t

0


∥∥∥∥Y T

b,1

.



We then make use of Lemma 2.2 and Lemma 2.9 to get

‖TRv‖Y Tb,1

≤ CT 1−(a+b)(‖v‖Y T

b,1+ ‖z‖

XT0c,1

)(C(R) + ‖z‖

XT0c,0

)

+

∥∥∥∥∫ t

0


∥∥∥∥Y T

b,1

+

∥∥∥∥∫ t

0


∥∥∥∥Y T

b,1

≤ C(R, T0, ‖u0‖L2(R))T1−(a+b)

(‖v‖Y T

b,1+ ‖u0‖H1(R)

)

+

∥∥∥∥∫ t

0


∥∥∥∥Y T

b,1

+

∥∥∥∥∫ t

0


∥∥∥∥Y T

b,1

.

Thus, by Proposition 2.5 and Lemma 2.9,

E

(‖TRv‖2

Y Tb,1

)≤[C(R, T0, ‖u0‖L2(R))T

2(1−(a+b)) + CT b] (

E

(‖v‖2

Y Tb,1

)+ ‖u0‖2

H1(R)

).

This shows that TR maps L2(Ω;Y Tb,1) into itself. Moreover, the ball in L2(Ω;Y Tb,1) of

radius R0 is invariant by TR if T ≤ T∗∗ such that C(R, T0, ‖u0‖L2(R))T2(1−(a+b))∗∗ +

CT b∗∗ ≤ 1/2 and R0 ≥ ‖u0‖H1(R).

Choosing T∗ ≤ T∗∗ in the construction of the solution vR of (2.3) in L2, it follows

that the solution vR is in L2(Ω;Y T∗b,1 ). We then use similar arguments as above to

prove that vR ∈ L2(Ω;C([0, T∗];H1(R)) and

E

(sup[0,T∗]

‖vR‖2H1(R)

)

≤ CT2(1−(a+b))∗

(E

(‖vR‖2

Y T∗b,1

)+ ‖u0‖2

H1(R)

)(C(R) + ‖u0‖L2(R)

)2

+E

(sup[0,T∗]

∥∥∥∥∫ t

0

U(t− r) [(z(r) + vR(r))ϕdW (r)]

∥∥∥∥2

H1(R)

)

≤ CT2(1−(a+b))∗

(E

(‖vR‖2

Y T∗b,1

)+ ‖u0‖2

H1(R)

)(C(R) + ‖u0‖L2(R)

)2

+CT∗E

(sup[0,T∗]

‖vR‖2H1(R)

)+ CT∗ ‖u0‖2

H1(R) ,

with b > 1/2 and a+ b < 1. Choosing a smaller T∗ if necessary, we deduce

E

(sup[0,T∗]

‖vR‖2H1(R)

)≤ R2

0,

if R0 ≥ ‖u0‖H1(R). On [T∗, 2T∗], we use equation (2.4) and obtain by similar

arguments

E

(‖vR(T∗ + ·)‖2

Y T∗b,1

)≤ C(T∗)E

(‖vR(T∗)‖2

H1(R)∩H−3/8(R)

)+ ‖u0‖2

H1(R),



and

E

(sup

[T∗,2T∗]

‖vR‖2H1(R)

)≤ CE

(‖vR(T∗)‖2

H1(R)∩H−3/8(R)

)+R2

0.

We know from the L2 construction that vR ∈ L2(Ω;C([0, T0]; H−3/8(R))), therefore

E

(sup

[T∗,2T∗]

‖vR‖2H1(R)

)≤ CE

(‖vR(T∗)‖2

H1(R)

)+R2

0 + E

(sup[0,T0]

‖vR‖2H−3/8(R)

).

It is now easy to iterate this argument and deduce that the solution vR is in

L2(Ω;Y T0

b,1 ) and also in L2(Ω;C([0, T0];H1(R))).

Theorem 2.8 gives the following local in time existence result for the non trun-

cated equation.

Corollary 2.10. Let s = 0 or 1 and assume that k ∈ Hs(R) ∩L1(R); then for any

u0 ∈ Hs(R), there is a stopping time τ∗(u0, ω) a.s. positive, such that (2.1) has an

adapted solution u, defined a.s. on [0, τ ∗(u0)[, unique in some class, and with paths

a.s. in C([0, τ∗(u0)[;Hs(R)). If s = 1, the uniqueness holds among solutions with

paths in C([0, τ∗(u0)[;H1(R)) a.s; moreover, the stopping time τ ∗(u0) satisfies

τ∗(u0) = +∞ or lim suptτ∗(u0)

‖u− U(·)u0‖Y tb,0

= +∞, a.s.

Remark 2.11. Let us explain what we mean by a solution on the random interval

[0, τ∗(u0)[. This means that u is defined on [0, τ∗(u0)[ and is an adapted process

such that for any stopping time τ < τ∗(u0) the following holds on [0, τ ]:

u(t) = U(t)u0 −1

2

∫ t

0

U(t− r)∂x(u2(r))dr +

∫ t∧τ

0

U(t− r)(u(r)ϕdW (r)).

Proof. Let z(t) = U(t)u0 and let vR ∈ Y T0

b,s for any b < 1/2 and any T0 > 0 be the

solution of (2.3) given by Theorem 2.8. We then set τR = inft ≥ 0, ‖vR‖Y tb,0

≥ R;

for t ∈ [0, τR], we have θR(‖vR‖Y tb,0

) = 1, hence vR is a solution of (2.2) on [0, τR].

It is not difficult to see that τR is non decreasing in R and that vR+1 = vR on

[0, τR]. Hence we may define u on [0, τ∗(u0)[ with τ∗(u0) = limR→∞ τR by setting

u(t) = vR(t) + z(t) for t ∈ [0, τR] and u is then a solution of (2.1) on [0, τ∗(u0)[.

The uniqueness for u holds in the class z + Y τR

b,0 for any R and it is not difficult to

see that any solution u with paths in C([0, τ∗(u0)[;H1(R)) is in this class. The last

property of the lemma is an immediate consequence of the definition of τ ∗(u0).

3. Global existence

As already seen, Theorem 2.8 gives a local in time existence result for the equation

without cut-off. In the present section, we end the proof of Theorem 1.1 by showing

that those solutions are globally defined in time. To that aim we need an estimate

on ‖v‖Y Tb,0

. We will use the following result.



Proposition 3.1. Assume that k ∈ L2(R). Let u ∈ C([0, τ ]);L2(R)) be a solution

of equation (2.1) with u0 ∈ L2(R), where τ is a stopping time. Then, for any m ≥ 1,

u ∈ L2m(Ω;C([0, τ ];L2(R))) and for any T > 0

E

(sup

t∈[0,τ∧T ]

|u(t)|2mL2(R)

)≤ C(T, ‖u0‖L2(R),m).

Proof. The result is a straightforward consequence of Ito formula. We prove it for

m = 2. For m ≤ 2 it then follows from Holder inequality. For m ≥ 2, the proof is

similar.

We apply Ito formula to M(u) = ‖u‖2L2(R) and obtain after a regularization

argument and easy computations (see Ref. [1] for more details in the case of an

additive noise or Ref. [10] for the case of the stochastic Schrodinger equation):

M(u(τ ∧ r)) = M(u0) + 2

∞∑

k=0

∫ τ∧r

0

∫

R

u2(σ, x)ϕek(x)dxdβk(σ)

+ ‖k‖2L2(R)

∫ τ∧r

0

M(u(σ))dσ.

We take the square of this identity and deduce:

E

(sup

r∈[0,τ∧T ]

M2(u(r))

)

≤ 2M2(u0) + 4E

supr∈[0,τ∧T ]

∣∣∣∣∣∞∑

k=0

∫ r∧T

0

∫

R

u2(σ, x)ϕek(x)dxdβk(σ)

∣∣∣∣∣

2

+2‖k‖4L2(R)E

(∫ τ∧T

0

M(u(σ))dσ

)2

≤ 2M2(u0) +(

4‖k‖2L2(R) + 2T‖k‖4

L2(R)

)E

(∫ τ∧T

0

M2(u(r))dr

),

thanks to Burkholder and Holder inequalities, and to Lemma 2.6. The result follows

from Gronwall Lemma.

Let vR be the solution given by Theorem 2.8, let T0 > 0 be fixed, and let

τR = inft ∈ [0, T0], ‖vR‖Y tb,0

≥ R; then on [0, τR], vR + z is a solution to (2.1)

which is a.s. in C([0, τR];L2(R)) and Proposition 3.1 applies:

E

(sup

t∈[0,τR∧T0]

|vR(t) + z(t)|2mL2(R)

)≤ C(T0, ‖u0‖L2(R),m). (3.1)

We now show that this implies an estimate on the Y Tb,0 norm of vR.

Lemma 3.2. Let vR be the solution of the truncated equation (2.3), then there

exists a constant C(T0, ‖u0‖L2(R)) independent of R such that

E

(‖vR‖Y τR

b,0

)≤ C(T0, ‖u0‖L2(R)).



Proof.

Step 1: Using similar arguments as in the beginning of the proof of Theorem 2.8,

taking into account the fact that vR satisfies equation (2.3), we prove using Lemma

2.2 that for T0 ≥ T ≥ 0,

‖vR‖Y T∧τRb,0

≤ CT 1−(a+b)[‖vR‖2

YT∧τR

b,0

+ ‖z‖2

XT0c,0

]

+

∥∥∥∥∫ t

0

U(t− r) [uR(r)ϕdW (r)]

∥∥∥∥Y

T0b,0

≤ CT 1−(a+b)[‖vR‖2

YT∧τR

b,0

+ C(T0) ‖u0‖2L2(R)

]

+

∥∥∥∥∫ t

0


∥∥∥∥Y

T0b,0

,

with uR(t) = vR(t) + U(t)u0. We set

K1 = K1(ω) = CT1−(a+b)0 C(T0) ‖uR‖2

C([0,T0];L2(R))+

∥∥∥∥∫ t

0

U(t− r) [uRϕdW (r)]

∥∥∥∥Y

T0b,0

,

then

CT 1−(a+b) ‖vR‖2

YT∧τR

b,0

− ‖vR‖Y T∧τRb,0

+K1 ≥ 0.

Therefore, if we choose T = T (ω) such that T 1−(a+b) =3

16CK1, we have

‖vR‖Y T∧τRb,0

≤ 2K1.

Note indeed that vR(0) = 0 and that ‖vR‖|Y tb,0 is a continuous function of t. Simi-

larly, for any k ≥ 0, we define

vkR(t) = uR(t) − U(t− kT )uR(kT ), t ∈ [kT, (k + 1)T ],

with T = T (ω) chosen above. Then the same argument shows that∥∥vkR

∥∥Y

[kT∧τR,(k+1)T∧τR]

b,0

≤ 2K1,

where we use the space Y[T1,T2]b,0 whose definition is exactly the same as Y Tb,0 but

[0, T ] is replaced by [T1, T2].

Step 2: Since uR is a solution of (2.1) on [0, τR], we may write for any t ∈ [0, τR],

uR(t) as

uR(t) = U(t)u0 − 12

∫ t

0

U(t− r)∂x(u2R(r))dr +

∫ t

0


= U(t)u0 −1

2

kt∑

k=0

∫ (k+1)T∧t

kT

U(t− r)∂x

[(vkR(r) + U(r − kT )uR(kT )

)2]dr

+

∫ t

0

U(t− r) [uR(r)ϕdW (r)] ,



where kt is the integer part of t/T . Using this decomposition and the unitarity of

U(σ) in H−3/8 for any σ, we deduce that for any t ∈ [0, τR],

‖uR(t) − U(t)u0‖H−3/8(R)

≤ 1

2

kt∑

k=0

∥∥∥∥∥∥∥

(k+1)T∧t∫

kT

U(t− r)∂x


)2]dr

∥∥∥∥∥∥∥H−3/8(R)

+

∥∥∥∥∫ t

0


∥∥∥∥H−3/8(R)

≤ 1

2

kt∑

k=0

∥∥∥∥∥∥∥

(k+1)T∧t∫

kT

U((k + 1)T ∧ t− r)∂x


)2]dr

∥∥∥∥∥∥∥H−3/8(R)

+

∥∥∥∥∫ t

0


∥∥∥∥H−3/8(R)

.

Now, suppose that a is fixed with 0 < a < 1/2 in such a way that Proposition 2.3

holds, and set b = 1 − a, so that b > 1/2. Then, using the fact that Y[T1,T2]

b,0⊂

C([[T1, T2];L2(R) ∩ H−3/8(R)) for any positive T1, T2, we have for t ∈ [0, τR] and

k = 0, . . . , kt,∥∥∥∥∥

∫ (k+1)T∧t

kT

U((k + 1)T ∧ t− r)∂x


)2]dr

∥∥∥∥∥H−3/8(R)

≤∥∥∥∥∫ ·

kT

U(· − r)∂x


)2]dr

∥∥∥∥C([kT∧τR,(k+1)T∧τR];H−3/8(R))

≤ C

∥∥∥∥∫ ·

kT

U(· − r)∂x


)2]dr

∥∥∥∥Y


b,0

.

By Lemma 2.9, the above term is majorized for each k ∈ 0, · · · , kt by

CT 1−(a+b)∥∥∥∂x

[(vkR + U(· − kT )uR(kT )

)2]∥∥∥Y


−a,0

≤ C∥∥vkR

∥∥2

Y[kT∧τR,(k+1)T∧τR]

b,0

+ ‖uR(kT )‖2L2(R)

,

by Proposition 2.3 and Lemma 2.9 again, since a+ b = 1. By the result of step 1,

we obtain

‖uR(t) − U(t)u0‖H−3/8(R) ≤ K2, t ∈ [0, τR],

with

K2 =1

2CT0T

−1[4K2

1 + ‖uR‖2C([0,T0];L2(R))

]

+

∥∥∥∥∫ ·

0

U(· − r) [uR(r)ϕdW (r)]

∥∥∥∥C([0,T0];H−3/8(R)

.



Step 3: By Lemma 2.9, and the unitarity of U(kT ) on L2(R) ∩ H−3/8(R) we have

‖U(t− kT )uR(kT ) − U(t)u0‖Y [kT∧τR,(k+1)T∧τR]

b,0

≤ C ‖U(−kT )uR(kT ) − u0‖L2(R)∩H−3/8(R)

≤ C ‖uR(kT ) − U(kT )u0‖L2(R)∩H−3/8(R) .

Therefore, using step 2,

‖U(· − kT )uR(kT ) − U(·)u0‖Y [kT∧τR,(k+1)T∧τR]

b,0

≤ K3

= C(K2 + 2 ‖uR‖C([0,T0];L2(R))

).

Finally, for t ∈ [kT ∧ τR, (k + 1)T ∧ τR], we have

vR(t) = vkR(t) + U(t− kT )uR(kT ) − U(t)u0,

and we may write, k0 being the integer part of T0/T ,

‖vR‖Y τRb,0

≤k0∑

k=0

‖vR‖Y [kT∧τR,(k+1)T∧τR]

b,0

≤k0∑

k=0

∥∥vkR∥∥Y


b,0

+ ‖U(· − kT )u(kT ) − U(·)u0‖Y [kT∧τR,(k+1)T∧τR]

b,0

≤(T0

T+ 1

)(2K1 +K3) .

Note that T−1 is proportional to K1/(1−(a+b))1 ; by Proposition 3.1 and Proposition

2.7, K1 and K3 have all moments finite, and it follows

E

(‖vR‖Y τR

b,0

)≤ c(T0, ‖u0‖L2(R))

which concludes the proof of Lemma 3.2.

It is now straightforward to achieve the proof of Theorem 1.1. Indeed, due to

Corollary 2.10, it suffices to see that lim supR→∞ τR ∧ T0 = T0 in probability as R

goes to infinity. But this is an easy consequence of Markov inequality and Lemma

3.2, since

P (τR < T0) = P

(‖vR‖Y T0∧τR

b,0

≥ R).

References

[1] A. de Bouard and A. Debussche, On the stochastic Korteweg-de Vries equation, J.Funct. Anal. 154(1), 215–251, (1998). ISSN 0022-1236.

[2] A. de Bouard, A. Debussche, and Y. Tsutsumi, White noise driven Korteweg-de Vriesequation, J. Funct. Anal. 169(2), 532–558, (1999). ISSN 0022-1236.

[3] A. De Bouard, A. Debussche, and Y. Tsutsumi, Periodic solutions of the Korteweg-deVries equation driven by white noise, SIAM J. Math. Anal. 36(3), 815–855 (elec-tronic), (2004/05). ISSN 0036-1410.



[4] R. L. Herman, The stochastic, damped KdV equation, J. Phys. A. 23(7), 1063–1084,(1990). ISSN 0305-4470.

[5] A. de Bouard and A. Debussche, Random modulation of solitons for the stochasticKorteweg-de Vries equation, Ann. Inst. H. Poincare Anal. Non Lineaire. 24, (2007).ISSN 0294-1449. To appear.

[6] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets andapplications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct.Anal. 3(3), 209–262, (1993). ISSN 1016-443X.

[7] C. E. Kenig, G. Ponce, and L. Vega, The Cauchy problem for the Korteweg-de Vriesequation in Sobolev spaces of negative indices, Duke Math. J. 71(1), 1–21, (1993).ISSN 0012-7094.

[8] C. E. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to theKdV equation, J. Amer. Math. Soc. 9(2), 573–603, (1996). ISSN 0894-0347.

[9] R. A. Adams, Sobolev spaces. (Academic Press [A subsidiary of Harcourt Brace Jo-vanovich, Publishers], New York-London, 1975). Pure and Applied Mathematics, Vol.65.

[10] A. de Bouard and A. Debussche, The stochastic nonlinear Schrodinger equation inH1, Stochastic Anal. Appl. 21(1), 97–126, (2003). ISSN 0736-2994.




Chapter 5

On Stochastic Burgers Equation Driven by a Fractional

Laplacian and Space-Time White Noise

Zdzis law Brzezniak and Latifa Debbi∗

Department of Mathematics, University of YorkHeslington, York YO10 5DD, UK

[email protected]

We prove existence and uniqueness of a mild global solution to the Cauchyproblem for the stochastic fractional Burgers equation on an interval with mul-tiplicative space time white noise and periodic boundary conditions when thepower of the Laplace operator is between 3/4 and 1.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

2 Existence of global solutions to approximating equations . . . . . . . . . . . . . . . . . 138

3 Global solutions to Burgers equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4 Proof of uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.1 Proof of Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156B.1 Gronwall Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

C.1 Some estimates on stopped stochastic convolutions . . . . . . . . . . . . . . . . . . . . 157

D.1 Pointwise multiplication in Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . 164

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

1. Introduction

It is widely accepted that the classical Burgers equation can be used to model

several physical phenomena and because of this it has been extensively studied.

Recently its random generalizations have been introduced either by considering

noisy force, see Refs. [1, 2] or by assuming randomness of the initial data, see a

recent paper [3]. Moreover other types of generalizations have been studied. For

example by considering an arbitrary algebraic nonlinearity instead of the quadratic,

see Ref. [4] or by replacing the Laplace operator by a non local operator, see Ref. [5]).

The first kind is called stochastic Burgers equation and the two latter ones are

called generalized Burgers equation. Generalized Burgers equations with fractional

differential operators are used to model some anomalous diffusions such as the far∗Institut Elie Cartan, Nancy 1 B.P 239, 54506 Vandoeuvre-Les-Nancy cedex, France & Departmentof Mathematics, Faculty of Sciences, University Ferhat Abbas, El-Maabouda Setif 19000, Algeria,[email protected]

135


136 Z. Brzezniak and L. Debbi

field (or long time) behavior of the acoustic waves propagating in a gas-filled tube

and the wave propagation in viscoelastic medium (see Refs. [6–8] for the generalized

Burgers equation and Refs. [9, 10] and the references therein for more details on

anomalous diffusions and fractional differential equations). Generalized Burgers

equations can also be found in works on continuum mechanics and hydrodynamics

(shallow water, bubbly liquid) and in molecular biology.

The equation dealt with in this paper is a generalization of Burgers equation

which is characterized by the coexistence of the stochastic noise and the fractional

power of the Laplacian. We are motivated by the tri-interaction between the wave

steepening given by the nonlinearity and by the small dissipation given by the

fractional power of the Laplacian and by the irregular random perturbation given

by the cylindrical noise. In particular, the critical power for the Laplacian is the

same as the one obtained in the deterministic fractional Burgers’ equation [5]. Some

other stochastic fractional partial differential equations have been studied in several

papers, see Refs. [11–15]. In this work, we are interested in the following Cauchy

problem for a Burgers equation driven by a fractional Laplacian and space-time

white noise: du(t) = [−Aαu(t) +Bu2(t)] dt + g(u(t)) dW (t), t > 0,

u(0) = u0,(1.1)

where 32 < α < 2, Aα = A

α2 , A = −∆, D(A) = H2,2(0, 1) ∩ H1,2

0 (0, 1). Here by

Hk,p(0, 1), for k ∈ N, p ∈ [1,∞) we denote the Banach space of all f ∈ Lp(0, 1) for

which Djf ∈ Lp(0, 1), j = 0, 1, . . . , k. The norm in Hk,p(0, 1) is given by

‖ f ‖Hk,p(0,1)

=

k∑

j=0

|Djf |Lp(0,1)

1p

. (1.2)

We define the fractional order Sobolev space Hβ,p(0, 1), β ∈ R+ \N by the complex

interpolation method, i.e.

Hβ,p(0, 1) = [Hk,p(0, 1), Hm,p(0, 1)]ϑ, (1.3)

where k,m ∈ N, ϑ ∈ (0, 1), k < m, are chosen to satisfy

β = (1 − ϑ)k + ϑm. (1.4)

One should bear in mind that the space on the LHS of formula (1.3) does not depend

on k,m, ϑ provided they satisfy condition (1.4).

In what follows by Hs,p0 (0, 1), s ≥ 0, p ∈ (1,∞) we will denote the closure of

C∞0 (0, 1) in the Banach space Hs,p(0, 1). It is well known, see Theorem 11.1 in

Ref. [16] and Theorem 1.4.3.2 on p.317 in Ref. [17] and Theorem 3.40 in Ref. [18]

that Hs,p0 (0, 1) = Hs,p(0, 1) iff s ≤ 1

p .

Since A is a self-adjoint operator in the Hilbert spaceH = L2(0, 1), the fractional

power, see Ref. [19], Aα = Aα2 is well defined and it follows from Theorem 1.15.3

on p. 103 in Ref. [17] that D(Aα) = [H,D(A)] α2

, where [H,D(A)] α2

is the complex


Stochastic Fractional Burgers Equation 137

interpolation space of order α2 , see Refs. [16], [17] and Theorem 4.2 in Ref. [20].

Moreover, by Seeley [21], see also Theorem in section 4.4.3 in the monograph [17],

we have that

D(Aα) =

Hα,2(0, 1) ∩H1,20 (0, 1), if 1 < α ≤ 2,

Hα,20 (0, 1), if 1

2 < α ≤ 1,

Hα,2(0, 1), if α = 12 ,

Hα,2(0, 1), if α < 12 ,

(1.5)

where Hα,2(0, 1) is the space of all f ∈ Hα,2(R) such that supp(f) ⊂ [0, 1].

Since the operator A is self-adjoint, A−1 exists and is compact, there exists an

ONB ejj∈N and a sequence λjj∈N such that λj→∞ and Aej = λjej , j ∈ N.

In fact, in our special case, ek =√

2 sin kπ· and λk = k2π2, k ∈ N. Then it is

well known that D(Aα) = D(Aα/2) = v ∈ L2(0, 1) :∑∞

k=1 λαk v

2k < ∞, where

vk = 〈v, ek〉 =√

2∫ 1

0v(x) sin kπx dx.

Notice that B is the first order differential operator defined by

Bu =∂u

∂x

and g : R→R, is a bounded Lipschitz continuous function on R. Let us note

here that the Lipschitz condition is not restrictive. In fact, it will be easily seen

from our proof that our method works when g is locally Lipschitz . W (t), t ≥ 0is a cylindrical Wiener process on the probability space (Ω,F , Ft≥0,P). The

initial condition u0 is a L2(0, 1)-valued F0-measurable function. Let us denote

by Sα(t), t ≥ 0 the semigroup generated by −Aα. The following is now widely

accepted definition, see Ref. [22].

Definition 1.1. Suppose that 32 < α < 2. An Ft-adapted L2(0, 1)-valued continu-

ous process u = u(t), t ≥ 0 is called a mild solution of equation (1.1) iff for some

p > 2αα−1

E supt∈[0,T ]

|u(t)|pL2 <∞, T > 0 (1.6)

and for all t ≥ 0, a.s. the following identity holds

u(t) = Sα(t)u0 +

∫ t

0

Sα(t− s)Bu2(s) ds+

∫ t

0

Sα(t− s)g(u(s)) dW (s). (1.7)

In the representation of the stochastic integral in (1.7), with a slight abuse

of notation, g is regarded as a nonlinear operator from H = L2(0, 1) to L(H),

the set of bounded linear operators on H , defined bya g(u)(h) = (0, 1) 3 x 7→g(u(x))h(x) ∈ R. In other words, the nonlinear operator g is the Nemytski map

associated with function g. For v ∈ H , g(v) is given as a multiplicative operator.

From the hypothesis that g is bounded we have ‖g(v)‖ ≤ b0, where b0 = supR |g(x)|.aNote that because g is bounded, the natural domain of this nonlinear operator is the whole spaceH.



Remark 1.2. We will see later, in Lemmas 2.9 and 2.13 , that if an Ft-adapted

L2(0, 1)-valued continuous process u = u(t), t ≥ 0 satisfies condition (1.6) then

all terms on the RHS of (1.7) make sense.

The main result of our paper is the following

Theorem 1.3. Let u0 be a L2(0, 1)-values F0-measurable function such that

E|u0|pL2 <∞

for some p > 2αα−1 and let T > 0. Then there exists a unique mild solution of

equation (1.1).

The paper is organized in the following way. In section 2 we prove existence and

uniqueness of a global solutions to the approximate versions of problem (1.1). In

section 3 we construct a local maximal solution to problem (1.1) and then prove

that it is in fact a global solution.

2. Existence of global solutions to approximating equations

We begin with the following definition used throughout the whole paper.

Definition 2.1. Let T > 0 and p ∈ [1,∞] be fixed and X is a separable Banach

space. By ZT,p(X) we denote the space of all X-valued continuous and Ft-adapted

processes u = u(t), t ∈ [0, T ] such that

‖u‖pT,X,p := E supt∈[0,T ]

|u(t)|pX <∞. (2.1)

When X = L2(0, 1) we will usually denote the space ZT,p(L2(0, 1)) by ZT,p and the

norm ‖ · ‖T,L2(0,1),p by ‖ · ‖T,p.

We have the following simple but useful observation.

Proposition 2.2. In the framework of Definition 2.1 the family spaces ZT,p(X) is

decreasing with respect to parameter p. To be precise, if 1 ≤ p1 ≤ p2 ≤ ∞, then

ZT,p2(X) ⊂ ZT,p1(X) and

‖u‖T,X,p1 ≤ ‖u‖T,X,p2 , for all u ∈ ZT,X,p2(X).

Proof. Follows immediately from Jensen’s Inequality.

We fix, for the time being, T > 0 and p > 2.

Let πn be the projection from H onto the ball B(0, n) ⊂ L2(0, 1) defined by

πn(v) =

v, if |v|L2 ≤ n,n

|v|L2v, if |v|L2 > n.

The following result is well known fact, see Refs. [1, 23, 24].



Lemma 2.3. The map πn : L2(0, 1)→L2(0, 1) is globally Lipschitz with Lipschitz

constant 1. Moreover,

|(πnu)2 − (πnv)2|L1 ≤ 2n|u− v|L2 , for all u, v ∈ L2, (2.2)

|(πnu)2|L1 ≤ n|u|L2 , for all u ∈ L2,

|(πnu)2|L1 ≤ |u|2L2 , for all u ∈ L2.

We fix n ∈ N and consider operators Hn, G : ZT,p → ZT,p defined by

Hnu(t) =

∫ t

0

Sα(t− s)B(πnu(s))2 ds (2.3)

and

Gu(t) =

∫ t

0

Sα(t− s)g(u(s)) dW (s). (2.4)

Let us recall that here g is regarded as a multiplicative operator. The above

constitute a special part of the following result.

Lemma 2.4. Suppose that q ∈ [2,∞] and z ∈ Lq(0, 1). Let Z denotes the multipli-

cation operator by z. Then,

‖Sα(t)Z‖2HS ≤ 2

2q |z|2Lq

∑

k

e−2παkαt = 22q |z|2Lq‖Sα(t)‖2

HS , t > 0,

where by ‖.‖HS we denote the Hilbert-Schmidt norm.

Proof. Let ek =√

2 sin kπ· and µk = λα/2k , where λk = k2π2, k ∈ N. Then, as

explained earlier, ek is an ONB of H and Aαek = µkek, k ∈ N. Then, because

both Sα(t) and Z are self-adjoint on H it follows from [25], section V.2.4,

‖Sα(t)Z‖2HS =

∑

k

|ZSα(t)ek|2L2 =∑

k

|ze−µktek|2L2 =∑

k

e−2µkt|zek|2L2 .

Let us observe that by the Holder inequality, |zek|L2 ≤ |z|Lq |ek|Lr , where 1r + 1

q = 12 .

Moreover, since |ek|L2 = 1 and |ek|L∞ = 21/2 it follows by applying the Holder

inequality that |ek|Lr ≤ 21/q . Therefore,

‖Sα(t)Z‖2HS ≤ 2

2q |z|2Lq

∑

k

e−2µkt,

what proves the result.

The above Lemma implies the following result.

Corollary 2.5. Suppose that z ∈ Lq(0, 1) for q ∈ [2,∞]. Let Z denotes the multi-

plication operator by z. Then, for β < 1 − 1α , one has

∫ t

0

s−β‖Sα(s)Z‖2HS ds≤

∫ ∞

0

s−β‖Sα(s)Z‖2HS ds

≤ 22q |z|2Lq(2πα)β−1Γ(1 − β)

+∞∑

k=1

k−α(1−β)<∞, t > 0. (2.5)



Proof. From the Fubini Theorem we have

∫ ∞

0

s−β‖Sα(s)Z‖2HS ds ≤ 2

2q |z|2Lq

∫ ∞

0

s−β+∞∑

k=1

e−2s(kπ)α

ds

= 2β−122q |z|2Lq

+∞∑

k=1

(kπ)αβ−α∫ ∞

0

τ−βe−τdτ

≤ 22q |z|2Lq(2πα)β−1Γ(1 − β)

+∞∑

k=1

k−α(1−β).

Since by our assumptions α(1 − β) > 1 the series on the RHS above is convergent

and the result follows.

Remark 2.6. Let us observe that the constant (2πα)β−1Γ(1 − β)∑+∞

k=1 k−α(1−β)

is equal to∫∞0 s−β‖Sα(s)‖2

HS ds.

Let us recall that for 0 < δ ≤ 1 the fractional power Aδα of Aα, see Ref. [19], is

defined as the inverse of operator A−δα defined by

A−δα =

1

Γ(δ)

∫ ∞

0

tδ−1Sα(t)dt.

Since, see Lemma 2.6.6 in Ref. [19], the operator A−δα is injective, Aδα :=

(A−δα

)−1is

well defined. Furthermore, it is known that Aδα is a closed densely defined operator

and that D(Aδα) = R(A−δα ), where R(Λ) denotes the range of an operator Λ.

The following result is just Lemma 3.3 from [26], see also [22], [27] and [28]. We

state and prove it here for the convenience of the reader.

Lemma 2.7. Provided that γ > p−1 + δ the operator Rγ : Lp(0, T ;L2(0, 1)) →C([0, T ];D(Aδα)) given by

Rγh(t) =

∫ t

0

(t− s)γ−1Sα(t− s)h(s) ds, h ∈ Lp(0, T ;L2(0, 1)),

is well defined, linear and bounded. Moreover, there exists a constant Cp,γ−δ > 0

such that for all h ∈ Lp(0, T ;L2(0, 1))

|Rγh|C([0,T ];D(Aδα)) ≤ Cp,γ−δT

γ−δ− 1p |h|Lp(0,T ;L2(0,1)). (2.6)

Proof. Let us fix h ∈ Lp(0, T ;L2(0, 1)). Then for t ∈ (0, T ) we have



|Rγh(t)|D(Aδα) = |Aδα

∫ t

0

(t− s)γ−1Sα(t− s)h(s) ds|L2

≤∫ t

0

(t− s)γ−1|AδαSα(t− s)h(s)|L2 ds

≤∫ t

0

(t− s)γ−1‖AδαSα(t− s)‖|h(s)|L2 ds.

Since Aα is self-adjoint it follows by Lemma 2.10 from [29] that ‖AδαSα(s)‖ ≤ s−δ,s > 0. Hence by applying the Holder inequality we get

|Rγh(t)|D(Aδα) ≤

∫ t

0

(t− s)γ−δ−1|h(s)|L2 ds

≤( ∫ T

0

s(γ−δ−1) pp−1 ds

) p−1p( ∫ T

0

|h(s)|pL2 ds) 1

p .

The integral∫ T0s(γ−δ−1) p

p−1 ds <∞ when γ > p−1 + δ. Hence for all t ≥ 0

|Rγh(t)|D(Aδα) ≤ (

1 − 1/p

γ − δ − 1/p)1−

1pT γ−δ−

1p |h|Lp(0,T ;L2(0,1)).

Remark 2.8. When δ = 0 then Rγ : Lp(0, T ;L2(0, 1))→C([0, T ];L2(0, 1)) is a

bounded linear operator and because the semigroup Sα(t)t≥0 is contractive,

Cp,1 = 1.

Lemma 2.9. If g : R→R is linear growth, i.e. for some b0, b1 ≥ 0

|g(x)| ≤ b0 + b1|x|, x ∈ R, (2.7)

α > 1 and p > 2αα−1 , then the nonlinear operator G : ZT,p → ZT,p is well defined

and for u ∈ ZT,p one has

‖Gu‖T,p ≤ [Cp,γb0 +√

2Cp,γb1‖u‖T,p]Cp,γT γ , (2.8)

where a positive number γ is such that 1p < γ < 1

2 (1 − 1α ) and Cp,γ is the constant

from inequality (2.6) and Cpp,γ := Cp((2πα)2γ−1Γ(1 − 2γ)

∑k≥1 k

−α(1−2γ)) p

2 .

Moreover, if g is a Lipschitz function, i.e. for some b2 ≥ 0

|g(x2) − g(x1)| ≤ b2|x2 − x1|, x1, x2 ∈ R, (2.9)

then G : ZT,p → ZT,p is Lipschitz continuous and for all u1, u2 ∈ ZT,p the following

inequality holds

‖Gu2 −Gu1‖T,p ≤√

2Cp,γCp,γb2Tγ‖u2 − u1‖T,p. (2.10)



Proof. We begin with an observation that a number γ such that 1p < γ < 1

2 (1− 1α )

exists thanks to one of our assumptions that p > 2αα−1 . We choose it and fix for the

remaining parts of the proof.

We first consider the case when g is bounded, i.e. g satisfies the condition (2.7)

with b1 = 0. Let u ∈ ZT,p. By Lemma 2.4 we have

∫ t

0

‖Sα(t− s)g(u(s))‖2HS ds ≤

∫ t

0

‖Sα(t− s)‖2HS‖g(u(s))‖2

L∞ ds

≤ b20

∫ t

0

‖Sα(s)‖2HS ds. (2.11)

Since α > 1 and so 0 < 1 − 1α by Corollary 2.5 we infer that for all t > 0, the

integral on the RHS of the inequality (2.11) is convergent. Hence the Ito stochastic

integral∫ t0Sα(t− s)g(u(s)) dW (s) exists and thus the process G(u) is well defined.

We shall prove that in fact Gu ∈ ZT,p.

Employing the factorization method (see Refs. [1, 22, 27]), we can rewrite Gu(t)

in the following form

Gu(t) = RγY (t) =

∫ t

0

(t− s)γ−1Sα(t− s)Y (s) ds, t ∈ (0, T ], (2.12)

where

Y (s) =sinπγ

π

∫ s

0

(s− r)−γSα(s− r)g(u(r)) dW (r), s ∈ [0, T ]. (2.13)

Arguing as above, we see that because 2γ < 1 − 1α , the RHS of (2.13) is a

well defined L2(0, 1)-valued stochastic Ito integral. Furthermore, since γ > p−1,

by Lemma 2.7, a process Gu(t) = [RγY ](t), t ∈ [0, T ] is continuous H-valued

provided we can show that a.s. Y ∈ Lp(0, T ;H). In fact, we will show below

that a stronger fact holds, namely that Y ∈ Lp(Ω;Lp(0, T ;H)). Indeed by the

Burkholder’s inequality and Corollary 2.5 there exists a constant Cp > 0 such that

∫ T

0

E∣∣Y (s)

∣∣pL2 ds≤Cp

∫ T

0

E

(∫ s

0

(s− r)−2γ‖Sα(s− r)g(u(r))‖2HSdr

) p2

ds

≤Cpbp0∫ T

0

(∫ s

0

r−2γ‖Sα(r)‖2HSdr

) p2

ds (2.14)

≤Cpbp0T(

(2πα)2γ−1Γ(1 − 2γ)

+∞∑

k=1

k−α(1−2γ))p

2

= Cpp,γbp0T.



Since 0 < γ < 12 − 1

2α what implies that α(1 − 2γ) > 1 we infer that the last term

is finite. Furthermore, by inequalities (2.6) and (2.14), we get

E(

supt∈[0,T ]

|Gu(t)|pL2

)= E

(supt∈[0,T ]

|RγY (t)|pL2

)= E

(|RγY |pC(0,T ;L2(0,1)

),

≤ Cpp,γTγp−1E|Y |pLp(0,T ;L2(0,1)

≤ Cpp,γCpp,γb

p0T

γp <∞.

This concludes the proof of inequality (2.8) in the special case considered. Next, we

should consider the case b0 = 0. However, the proof in this case is in fact a special

case of the proof of the Lipschitz property of the map G given below and hence will

be omitted. Moreover, the general case of inequality (2.8) is a consequence of the

just proved special case and the proof of the inequality (2.10).

Finally, we shall deal with the Lipschitz property of the map G. Take u2, u1 ∈ZT,p. From the 1st part of the proof we know that G(u2) and G(u1) both belong to

ZT,p. However, in order to estimate the norm in of G(u2) −G(u1) we need to use

Lemma 2.4 with q = 2 while in the 1st part of the proof we used it with q = ∞.

We choose γ such that p−1 < γ < 12 − 1

2α and introduce an auxiliary process h

by

h(s) =sinπγ

π

∫ s

0

(s− r)−γSα(s− r)(g(u2(r)) − g(u1(r)) dW (r), t ≥ 0.

Then we haveGu2(t)−Gu1(t) = Rγh(t), where Rγ is the operator defined in Lemma

2.7.

By the Burkholder inequality in view of Lemma 2.4 we have the following se-

quence of inequalities.

∫ T

0

E∣∣h(s)|pL2 ds (2.15)

≤ π−p∫ T

0

E∣∣∫ s

0

(s− r)−γSα(s− r)(g(u2(r)) − g(u1(r)) dW (r)∣∣pL2 ds

≤ Cp

∫ T

0

E

(∫ s

0

(s− r)−2γ‖Sα(s− r)(g(u2(r)) − g(u1(r))‖2HSdr

) p2

ds

≤ Cp2p2

∫ T

0

E

(∫ s

0

(s− r)−2γ‖Sα(s− r)‖2HS |(g(u2(r)) − g(u1(r))|2L2dr

) p2

ds

≤ Cp2p2 bp2

∫ T

0

E

(∫ s

0

(s− r)−2γ‖Sα(s− r)‖2HS |u2(r) − u1(r)|2L2 dr

) p2

ds

≤ Cp2p2 bp2E[sup

[0,T ]

|u2(r) − u1(r)|pL2 ]

∫ T

0

(∫ s

0

r−2γ‖Sα(r)‖2HS dr

) p2

ds

≤ Cpp,γ2p2 bp2T‖u− v‖pT,p <∞.



Therefore, by Lemma 2.7 we have that

‖Gu−Gv‖pT,p = E|Rγh|pC(0,T ;L2) ≤ Cpp,γTγp−1E|h|pLp(0,T ;L2)

≤ 2p2Cpp,γC

pp,γb

p2T

γp‖u− v‖pT,p.(2.16)

Remark 2.10. In the proof of the previous Lemma we used Lemma 2.7 with δ = 0.

If instead we used it with δ ≥ 0 such that δ+ 1p <

12 (1− 1

α ) then by choosing γ such

that

δ +1

p< γ <

1

2(1 − 1

α) (2.17)

we would prove that G maps the space ZT,p(L2) into the space ZT,p(D(Aδα)).

Moreover, under some appropriate assumptions the inequalities (2.8) and (2.10)

take the following forms,

‖Gu‖ZT,p(D(Aδα)) ≤ [Cp,γ−δb0 +

√2Cp,γ−δb1‖u‖T,p]Cp,γT γ−δ,

‖Gu2 −Gu1‖ZT,p(D(Aδα)) ≤

√2Cp,γ−δCp,γb2T

γ−δ‖u2 − u1‖T,p.

Lemma 2.11. For each α > 32 there exists a constant Cα > 0 such that for all

t > 0 and for any bounded and strongly-measurable function v : (0, t)→L1(0, 1) the

following inequality holds∫ t

0

|Sα(t− s)∂v

∂x(s)|L2 ds ≤ Cαt

1− 32α sup

s≤t|v(s)|L1 . (2.18)

Proof. We begin by showing that if v ∈ H1,2(0, 1) then

|Sα(s)∂v

∂x|L2 ≤

√2π(+∞∑

k=1

2πk2e−2s(kπ)α) 1

2 |v|L1 , s > 0. (2.19)

By the Parseval Identity, the fact that | cos(kπ·)|L∞ ≤ 1 and by integration by parts

formula we have

|Sα(s)∂v

∂x|2L2 =

+∞∑

k=1

e−2s(kπ)α〈 ∂∂xv, ek〉2

≤+∞∑

k=1

e−2s(kπ)α(√2π

∫ 1

0

∂v

∂x(x) sin(kπx) dx

)2

≤+∞∑

k=1

e−2s(kπ)α(k√

2π

∫ 1

0

v(x) cos kπx dx)2

≤ 2π2+∞∑

k=1

k2e−2s(kπ)α |v|2L1 .



This proves (2.19). Hence, if v ∈ L∞(0, t;L1(0, 1)), then

∫ t

0

|Sα(t− s)∂v

∂x(s)|L2 ds ≤

√2π

∫ t

0

( +∞∑

k=1

k2e−2(t−s)(kπ)α) 1

2 |v(s)|L1 ds

≤√

2π sups≤t

|v(s)|L1

∫ t

0

( +∞∑

k=1

k2e−2(t−s)(kπ)α) 1

2

ds. (2.20)

We need the following auxiliary result.

Lemma 2.12. Let us define a function h : (0,∞)→R by formula

h(s) =∞∑

k=1

k2e−2s(kπ)α

, s ∈ (0,∞).

Then there exists C > 0 such that

h(s) ≤ Cs−3/α, s > 0.

Proof. [Proof of Lemma 2.12] The function h is the Laplace transform of a purely

atomic measure µ =∑∞

k=1 k2δ2(kπ)α . Let U(σ) := µ((0, σ]) =

∑k≤ 1

π ( σ2 )

1αk2. Then,

by Ref. [30], we have

µ((0, σ]) =∑

k≤[ 1π (σ

2 )1α ]

k2 =1

6[1

π(σ

2)

1α ]([

1

π(σ

2)

1α ] + 1)(2[

1

π(σ

2)

1α ] + 1)

U(σ) ∼ 1

3[π−1(

σ

2)

1α ]3 =

2−3ασ

3α

3π3, as σ→∞,

where [·] denotes the integer part, we infer that for any y > 0, limσ→∞U(σy)U(σ) = y

3α .

Therefore the function L defined by L(x) = U(x)

x3α

, x > 0, varies slowly at infinity

and hence by the Abelian theorem, see Theorem XIII.5.2 in Ref. [31] we infer that

h(s) ∼ 1

Γ( 3α + 1)

s−3αL(s−1), as s→ 0.

In particular, since limx→∞ L(x) = 2− 3α

3π3 , there exists t0 > 0 and C0 >

0 such that h(s) ≤ C0s− 3

α , for s ∈ (0, t0]. Since the function h(s)s3α ,

s ∈ (0,∞) is continuous and the lims→0 h(s)s3α and lims→∞ h(s)s

3α =

lims→∞ s3α e−2s

∑∞k=1 k

2e−2s((kπ)α−1) exist, then it is bounded on (0,∞), hence

it is on [0, T ].

Combining Lemma 2.12 with inequality (2.20) we infer that Lemma 2.11 is valid

for v ∈ H1,2(0, 1). Hence the result follows by standard density argument.

Proof. [An alternative proof of Lemma 2.12.]

h(s) =

∞∑

k=1

k2e−2s(kπ)α

, s ∈ (0,∞).



Since for each N > 0 there exists CN > 0 such that

e−x ≤ CNx−N , x > 0

we have, for some N > 0,

h(s) ≤∞∑

k=1

k2(2s(kπ)α)−N =CNsN

∞∑

k=1

k2−Nα

Taking N > 3α (so that the series

∑∞k=1 k

2−Nα is convergent) we infer that there

exists CN > 0 such that

h(s) ≤ CNsN

, s ∈ (0,∞).

Inserting the last inequality into (2.20) yields

∫ t

0

|Sα(t− s)∂v

∂x(s)|L2 ds ≤ C sup

s≤t|v(s)|L1

∫ t

0

s−32α ds.

≤ C1

1 − 32α

sups≤t

|v(s)|L1 t1−32α .

Lemma 2.13. Assume that α > 32 and T > 0. Then the nonlinear operator Hn :

ZT,p → ZT,p is well defined and globally Lipschitz. Moreover, for all n ∈ N and

u, v ∈ ZT,p

E supt∈[0,T ]

∣∣∫ t

0

Sα(t− s)B(πnu(s))2 ds∣∣pL2 ≤ npCpαT

p(1− 32α )E sup

t∈[0,T ]

|u(t)|pL2 ,

E supt∈[0,T ]

∣∣∫ t

0

Sα(t− s)B(πnu(s))2 ds−∫ t

0

Sα(t− s)B(πnv(s))2 ds∣∣pL2

≤ (2n)pCpαTp(1− 3

2α )E supt∈[0,T ]

|u(t) − v(t)|pL2 ,

E supt∈[0,T ]

∣∣∫ t

0

Sα(t− s)B(u(s))2 ds∣∣p/2L2 ≤ Cp/2α T

p2 (1− 3

2α )E supt∈[0,T ]

|u(t)|pL2 ,

(2.21)

where Cα is the constant appearing in the inequality (2.18).

Proof. Take u, v ∈ ZT,p. It is enough to assume that u, v are deterministic

functions and u, v ∈ C([0, T ], L2). Then, by Lemma 2.11 and the second inequality



in Lemma 2.3 we have

supt∈[0,T ]

∣∣∫ t

0

Sα(t− s)B(πnu(s))2 ds∣∣pL2 ≤ sup

t∈[0,T ]

(∫ t

0

∣∣Sα(t− s)B(πnu(s))2∣∣L2 ds

)p

≤ CpαTp(1− 3

2α ) sups∈[0,T ]

|(πnu(s))2|pL1 ,

≤ CpαTp(1− 3

2α )np supt∈[0,T ]

|u(t)|pL2 ,

hence, we get the first inequality and that the operator Hn is well defined. To prove

the second inequality, we use again Lemma 2.11 and the first inequality in Lemma

2.3 and arguing as above we get

supt∈[0,T ]

∣∣∫ t

0

Sα(t− s)B((πnu(s))2 − (πnv(s))2

)ds∣∣pL2

≤ CpαTp(1− 3

2α ) sups∈[0,T ]

|(πnu(s))2 − (πnv(s))2|pL1 ,

≤ CpαTp(1− 3

2α )(2n)p supt∈[0,T ]

|u(t) − v(t)|pL2 .

Hence we conclude that Hn is globally Lipschitz. We finish with the proof of

the third inequality. As before it is enough to assume that u ∈ C([0, T ], L2) is a

deterministic function. Then, by Lemma 2.11 we have

supt∈[0,T ]

∣∣∫ t

0

Sα(t− s)B(u(s))2 ds∣∣p/2L2 ≤ sup

t∈[0,T ]

(∫ t

0

∣∣Sα(t− s)B(u(s))2∣∣L2 ds

)p/2

≤ Cp/2α Tp2 (1− 3

2α ) sups∈[0,T ]

|(u(s))2|p/2L1 ,

≤ Cp/2α Tp2 (1− 3

2α ) supt∈[0,T ]

|u(t)|pL2 .

It follows from Lemmas 2.9 and 2.13 that a map

Φn : ZT,p 3 u = Sα(·)u0 + [Hnu](·) + [Gu](·) ∈ ZT,p (2.22)

is well defined.

We are now ready to formulate the main result of this section.

Theorem 2.14. Assume that α > 32 , p > 2α

α−1 and that the function g : R→R is

Lipschitz. Then the problemdu(t) =

−Aαu(t) +B[(πnu(t))2]

dt+ g(u(t)) dW (t), t > 0,

u(0) = u0,(2.23)

has a unique global mild solution un = un(t), t ≥ 0 such that

E supt∈[0,T ]

|un(t)|pL2 <∞, for each T > 0.



Proof. Let us fix n ∈ N. We will prove that the operator Φn is a strict contraction

provided T > 0 is small enough. Then by the Banach Fixed Point Theorem there

exists a unique un ∈ ZT,p such that Φn(un) = un. It is standard to show that then

un is a mild solution to problem 2.23 on the time interval [0, T ]. It is also standard,

see Ref. [26] that this solution can be extended to the whole half-line [0,∞) and

that the extension, denoted also by un is a fixed point of Φn for each T > 0 and in

consequence is a mild solution to problem 2.23 on the time interval [0,∞).

Let us take and fix u, v ∈ ZT,p. Then by Lemma 2.13 we have

‖Hnu−Hnv‖pT,p = E[

supt∈[0,T ]

∣∣∫ t

0

Sα(t− s)(B(πnu(s))2 −B(πnv(s))2) ds∣∣pL2

]

≤ 2pCpαnpT p(1−

32α )‖u− v‖pT,p. (2.24)

Taking T ≤ (6Cαn)2α

3−2α , we get

‖Hnu−Hnv‖T,p ≤1

3‖u− v‖T,p.

Next, from inequality (2.10) in Lemma 2.9 we infer thatG is Lipschitz continuous

with Lipschitz constant√

2Cp,γCp,γb2Tγ, where b2 is the Lipschitz constant of the

function g.

Taking T ≤ (3√

2Cp,γCp,γb2)−1γ , we infer that ‖Gu − Gv‖T ≤ 1

3‖u − v‖T .

From (2.24) and (2.16) and for T ≤ min(3√

2Cp,γCp,γb2)−1γ , (6nCα)

2α3−2α , we get

‖Φnu− Φnv‖T ≤ 23‖u− v‖T . Thus Φn is a contraction on ZT,p and so there exists

a unique fixed point un ∈ ZT,p of Φn.

Concerning regularity of solutions to problem (2.23) we have the following result,

a standard proof of which will be omitted.

Proposition 2.15. The solution un(t), t ≥ 0 of equation (2.23) belongs to the

space C((0, T ];D(Aδα)) ∩ C([0, T ];L2(0, 1)) provided δ + 32α < 1 and δ + p−1 <

12 (1 − 1

α ).

3. Global solutions to Burgers equations

In this section we assume that the function g is bounded and Lipschitz, i.e. condition

(2.7) with b1 = 0 and condition (2.9) are satisfied.

Let us define a stopping time τn by τn = inf t > 0, |un(t)|L2 ≥ n. Let n ≥ m,

for t ≤ τm, P a.s. um(s), s ≤ t is solution of Equation (2.23). The same argument

as in Ref. [26], compare with Lemma 4.11 and the proof of Theorem 4.10, based

on the uniqueness part of Theorem 2.14, we getb P ⊗ ds a.s, for all 0 ≤ t ≤ τn,

un(t) = um(t). Hence the sequence τnn>0 is increasing. Let τ∞ := limn→∞

τn. We

define the function u(t) for t ≤ τ∞ by u(t) = un(t), t ≤ τn. Then we also have that

τn = inf t > 0, |u(t)|L2 ≥ n, n ∈ N. (3.1)

bThe equality is in L2(0, 1).



Then, see Theorem 1.5 in Ref. [32], Theorem 4.10 in Ref. [26] , [33] or [34], u(t),

t ≤ τ∞ is a unique maximal local (mild) solution to (1.1) with lifespan τ∞, i.e. τ∞is an accessible, strictly positive P-almost surely stopping time, (u(t), t < τ∞) is a

progressively measurable process satisfying

lim suptτ∞

|u(t)|L2 = +∞ P-almost surely on τ∞ <∞,

and

u(t∧τn) = Sα(t ∧ τn)u(0)+

∫ t∧τn

0

Sα(t ∧ τn − r)B(u(r)2) dr+Iτn(g(u))(t∧τn), t ≥ 0

(3.2)

for all n ∈ N, where we define

Iτn(g(u))(t) =

∫ t

0

1[0,τn)(r)Sα(t− r)g(u(r ∧ τn)) dW (r), t ≥ 0.

For the choice of the process Iτn(g(u)) see Proposition C.1.1. Note that the

process u has continuous paths, u ∈ C([0, τ∞) ;L2) P-almost surely.

To prove that it is the global solution it is sufficient to prove that τ∞ = ∞.

Let us consider, for n ∈ N, a process zn defined byc

zn(t) =

∫ t

0

1[0,τn)(s)Sα(t− s)g(un(s)) dW (s), t ≥ 0, (3.3)

=

∫ t

0

1[0,τn)(s)Sα(t− s)g(u(s)) dW (s), t ≥ 0.

Let vn be a process defined pathwise as a solution to the following initial value

problem

dvn(t)dt = −Aαvn(t) +B(vn(t) + zn(t))2, t > 0,

v(0) = u0.(3.4)

It follows from Corollary C.1.9 that the processes zn(t), t ≥ 0 defined by (3.3)

satisfy the following. If q ∈ [2,∞), p ∈ [1,∞) and

1

p< β <

1

2(1 − 1

α) − δ, (3.5)

then there exists C > 0 such that for all T > 0 and all n ∈ N,

E supt∈[0,T ]

|zn(t)|pHαδ,q (0,1)

≤ CT pβ. (3.6)

The following result follows directly from the definitions of the local solution

and of the process zn.cThe second equality below follows as P-a.s. un(s) = u(s) for s ∈ [0, τn).



Lemma 3.1. If n ∈ N, then

u(t ∧ τn) = zn(t ∧ τn) + vn(t ∧ τn), t ≥ 0. (3.7)

Let us also quote the following consequence of a much more general result from

a monograph by Runst & Sickel [35], see Theorem 2 in section 4.4.4, Theorem 7.48

in Ref. [36] and section 2.3.5 and Remark 4.4.2/2 p. 324 in Ref. [37].

Proposition 3.2. Assume that 0 < s2 < 12 and s1 > maxs2, 1

q1, q1 ∈ [2,∞).

Then the pointwise multiplication between spaces Hs1,q1(0, 1) and Hs2,2(0, 1) is well

defined and moreover, there exists C > 0 such that

|uv|Hs2,2 ≤ C|u|Hs1,q1 |v|Hs2,2 , u ∈ Hs1,q1(0, 1), v ∈ Hs2,2(0, 1). (3.8)

We will also need the following results.

Proposition 3.3. Assume that β ∈ ( 12 , 1). Then there exists a constant C > 0

such that for each u ∈ Hβ,20 (0, 1) and each v ∈ H1−β,2(0, 1) the following inequality

is satisfied

|∫ 1

0

u(x)Dv(x) dx| ≤ C|u|Hβ,20

|v|H1−β,2 . (3.9)

Lemma 3.4. Suppose that T > 0, a function z : [0, T ]→Hs,q(0, 1), with s and q

such that 1q < s < 1

2 , is continuous, and that a function v ∈ C(0, T ;Hα2 ,20 ) is a

solution to dv(t)dt = −Aαv(t) +B(v(t) + z(t))2, t > 0,

v(0) = u0.(3.10)

Then, there exists a generic constant C > 0 such that

ln+ |v(t)|L2 ≤ ln+ |v(0)|L2 + C(1 + ln+ t) + ln+ sup0≤τ≤t

|z(τ)|α

α−1

Hs,q

+ ln+ sup0≤τ≤t

[|z(τ)|4Hs,q + |z(τ)|4H1−α/2,2

]+ sup

0≤τ≤t|z(τ)|

αα−1

Hs,q , (3.11)

where ln+ x = max0, lnx.

Proof. [Proof of Proposition 3.3.]

Let en be the ONB of L2(0, 1) as before and let fn, n ≥ 0 be an ONB

in L2(0, 1), given by fn(x) =√

2 cosnπx for n ≥ 1 and f0(x) = 1. Assume that

u ∈ Hβ,20 (0, 1) and v ∈ C1

0 (0, 1), where C10 (0, 1) is the set of differentiable functions

on (0, 1) with compact support. So that 〈en, Dv〉 = −nπ〈fn, v〉.Let us denote by A the -Laplacian with the Neumann boundary conditions, i.e.

a linear self-adjoint operator defined by the following

D(A) = H2,2(0, 1)

A(u) = −∆u, u ∈ H2,2(0, 1).



It is known that A(fk) = (kπ)2fk, k ∈ N and that D(A12−β/2) = H1−β,2. Further,

from formula (1.5) we have D(Aβ2 ) = Hβ,2

0 (0, 1). Then, by the Plancherel Theorem

we have the following sequence of inequalities

|∫ 1

0

u(x)Dv(x) dx| = |∞∑

k=1

〈u, ek〉〈ek, Dv〉| = |∞∑

k=1

−kπ〈u, ek〉〈v, fk〉|

= |∞∑

k=1

(kπ)β〈u, ek〉(kπ)1−β〈v, fk〉|

≤ (

∞∑

k=1

(kπ)2β〈u, ek〉2)12 (

∞∑

k=1

(kπ)2−2β〈v, fk〉2)12

≤ C|u|Hβ,20

|v|H1−β,2 (0, 1).

Using the density of C10 (0, 1) in H1−β,2

0 = H1−β,2, we get the result.

Proof. [Proof of Lemma 3.4.] We begin with an observation that in view of

Proposition 3.3 (B(v2), v)L2 = 0 for all v ∈ Hα2 ,20 . Indeed this is true for v ∈

C10 (0, 1), the space C1

0 (0, 1) is dense in Hα2 ,20 and, see Ref. [38], the space H

α2 ,20 is

an algebra with pointwise multiplication.

Next, applying Lemma III.1.2 from [39] the solution v of problem (3.10) we get

1

2

d

dt|v(t)|2L2 = −〈Aαv(t), v(t)〉 + 2〈B(z(t)v(t)), v(t)〉 + 〈Bz2(t), v(t)〉. (3.12)

Since Aα = Aα/2, where A is positive self-adjoint we infer that

〈Aαv, v〉 = 〈Aα/2v, v〉 = |Aα/4v|2L2 , v ∈ D(Aα).

Since α ∈ ( 32 , 2) so that α

4 ∈ ( 38 ,

12 ) in view of equality (1.5) we infer that there

exists C > 0 such that |Aα/4v|2L2 ≥ C|v|2H

α2

,2

0

, for v ∈ D(Aα/2). Therefored,

〈Aαv, v〉 ≥ν12|v|2

Hα2

,2

0

, v ∈ D(Aα). (3.13)

Applying Proposition 3.3 with β = α/2 we infer that

|〈B(zv), v〉| ≤ C|v|H

α/2,20

|zv|H1−α/2,2

Let us choose positive numbers s and q such that 1q < s < 1

2 . Since α ∈ ( 32 , 2),

we infer that 1 − α2 < 1

2 and therefore by Proposition 3.2, formula (3.8) we infer

that that there exists a constant C > 0 such that

4|〈B(zv), v〉| ≤ C|v|H

α/2,20

|z|Hs,q |v|H1−α/2,2

dOne can choose an equivalent norm on Hα2

,2

0 such that the constant ν1 > 0 in inequality (3.13)is equal to 1.



Moreover, since 0 < 1 − α2 <

α2 and H0,2(0, 1) = L2(0, 1), by the interpolation and

definition (1.3) we infer that

|v|H1− α

2,2 ≤ |v|2−

2α

L2 |v|2α−1

Hα/2,20

, v ∈ Hα2 ,20 (0, 1).

Since α > 1 we infer by applying the classical Young inequality that for some

generic constant C > 0

4|〈B(zv), v〉| ≤ ν14|v|2

Hα/2,20

+ C|z|α

α−1

Hs,q |v|2L2 . (3.14)

Applying again Proposition 3.3 with β = α/2 we infer that

|〈B(z2), v〉| ≤ C|v|H

α/2,20

|z2|H1−α/2,2 .

Arguing as before and with the same choice of constants we have

2|〈B(z2), v〉| ≤ C|v|H

α/2,20

|z|Hs,q |z|H1−α/2,2

≤ ν14|v|2

Hα/2,20

+ C|z|4Hs,q + C|z|4H1−α/2,2 . (3.15)

Combining inequalities (3.12) and (3.13) with inequalities (3.14) and (3.15) we

infer that

d

dt|v(t)|2L2 ≤ −ν1

2|v(t)|2Hα/2,2

+C|z(t)|α

α−1

Hs,q |v(t)|2L2 + C|z(t)|4Hs,q + C|z(t)|4H1−α/2,2 , t ≥ 0. (3.16)

Since there exists a constant ν2 > 0 such that |v|2H

α/2,20

≥ ν2|v|2L2 , u ∈Hα/2,20 (0, 1), we infer that with ν := ν1ν2

2 ,

d

dt|v(t)|2L2 ≤ (−ν + C|z(t)|

αα−1

Hs,q )|v(t)|2L2

+ C|z(t)|4Hs,q + C|z(t)|4H1−α/2,2 , t ≥ 0. (3.17)

Applying finally the Gronwall Lemma we obtain that for t ≥ 0,

|v(t)|2L2 ≤ |v(0)|2L2e∫ t0(−ν+C|z(τ)|

αα−1Hs,q ) dτ

+ C

∫ t

0

[|z(σ)|4Hs,q + C|z(σ)|4H1−α/2,2

]e∫

tσ(−ν+C|z(τ)|

αα−1Hs,q ) dτ dσ, (3.18)

≤ |v(0)|2L2e−νt+Ct sup0≤τ≤t |z(τ)|α

α−1Hs,q

+ Ct sup0≤τ≤t

[|z(τ)|4Hs,q + |z(τ)|4H1−α/2,2

]eCt sup0≤τ≤t |z(τ)|

αα−1Hs,q .

Applying the increasing function ln+ x = max0, lnx to both sides of the above

inequality and then using the classical inequalities ln+(a+b) ≤ ln+(a)+ln+(b)+ln 2

and ln+(ab) ≤ ln+(a) + ln+(b), we get the result.



Proof. [Completion of the proof of Theorem 1.3.] To finish the proof we will

employ the Hasminski’s method as in Ref. [34], see Ref. [40], Theorem III.4.1, for

the finite-dimensional case. Let us recall that

τk =t ≥ 0; |u(t)|L2 ≥ k

, k ∈ N.

In order to prove that τ∞ = +∞ P-almost surely it suffices to find a Lyapunov

function V : L2(0, 1) −→ R and a function C : [0,∞)→(0,∞) satisfying

V ≥ 0 on L2(0, 1), (3.19)

qR := inf|u|L2≥R

V (u)→∞ as R→∞, (3.20)

EV (u(0)) < ∞ (3.21)

and

EV (u(t ∧ τk)) ≤ C(t)(1 + EV (u(0))

)for all t ≥ 0, k ∈ N. (3.22)

Inequality (3.22) implies easily that

Pτk < t

≤ 1

qkE1τk<tV (u(t ∧ τk))

≤ 1

qkC(t)

(1 + EV (u(0))

),

so that

limk→∞

Pτk < t

= 0

for each fixed t ≥ 0, and Pτ∞ < t = 0 follows.

Set

V (u) = ln+ |u|L2 , u ∈ L2(0, 1).

Obviously V is uniformly continuous on bounded sets and satisfies (3.19), (3.20).

Moreover, (3.22) follows from (3.18) and (3.21) is obviously satisfied for initial

condition u0 such that E ln+ |u0|L2 <∞.

4. Proof of uniqueness

Suppose that two processes u1(t), t ≥ 0 and u2(t), t ≥ 0 are solutions to problem

(1.1) with the same initial data u0. We may suppose that both satisfy condition

(1.6) with the same constant p > 2αα−1 . Let us define a stopping time τR = τ1

R ∧ τ2R,

where for i = 1, 2 and R > 0,

τ iR = inf t > 0, |ui(t)|L2 ≥ R, n ∈ N. (4.1)

Then for all R > 0 each ui satisfies the stopped version of equation (1.7), i.e.



ui(t ∧ τn) = Sα(t ∧ τR)u0 +

∫ t∧τR

0

Sα(t ∧ τR − r)B(ui(r))2 dr

+ IτR(g(ui))(t ∧ τR), a.s., t ≥ 0 (4.2)

IτR(g(ui))(t) =

∫ t

0

1[0,τR)(r)Sα(t− r)g(ui(r ∧ τR)) dW (r), a.s., t ≥ 0.

Subtracting the equations (4.2) for i = 2, 1 and denoting by u the difference between

u2 and u1 we infer that for t ≥ 0, a.s.

u(t ∧ τR) =

∫ t∧τR

0

Sα(t ∧ τR − r)B[(u2(r))2 − (u1(r))2

]dr

+ JτR(t ∧ τR),

JτR(t) := IτR(g(u2) − g(u1))(t)

=

∫ t

0

1[0,τR)(r)Sα(t− r)g(u2(r ∧ τR)) − g(u1(r ∧ τR)) dW (r).

Because |u(t)|L2 ≤ R for t ∈ [0, τR], the proof can be easily concluded by

applying Gronwall Lemma in conjunction with Lemmata 2.11 and 2.13. Details

of this idea are presented below. We shall prove that E|u(t ∧ τR)|pL2 = 0. i.e

u(t) = 0 a.s on t ≤ τR. In fact, using Lemma A.1 from [34] and the semigroup

property, we get, for some Cp > 0

E|u(t ∧ τR)|pL2 ≤ CpE|∫ t0 1(0,τR)Sα(t− r)B

[(u2(r ∧ τR))2 − (u1(r ∧ τR))2

]dr|pL2

+CpE|∫ t0 1[0,τR)(r)Sα(t− r)g(u2(r ∧ τR)) − g(u1(r ∧ τR)) dW (r)|pL2 . (4.3)

Put the following locally integrable functions:

ϕ1(s) := h12 (s) =

(∑

k≥1

k2e−2s(kπ)α) 1

2

, s > 0,

ϕ2(r) :=∑

k≥1

∫ r

0

ξ−2γe−2ξ(kπ)α

dξ, r > 0

ϕ := maxϕ1, ϕ2.

Arguing as above, using Lemmata 2.6, 2.7 and 2.8 and thanks to the assumptions

above, we get the following sequence of inequalities



E|∫ t

0

1[0,τR)Sα(t− r)B[(u2(r ∧ τR))2 − (u1(r ∧ τR))2

]dr|pL2

≤ E

(∫ t

0

|1[0,τR)Sα(t− r)B[(u2(r ∧ τR))2 − (u1(r ∧ τR))2

]|L2 dr

)p

≤ 2p2 πpE

(∫ t

0

(∑

k≥1

k2e−2(t−r)(kπ)α) 1

2 [1[0,τR)|u2(r ∧ τR)) − u1(r ∧ τR)|L2

|u2(r ∧ τR)) + u1(r ∧ τR)|L2

]dr)p

(4.4)

≤ 23p2 πpRpE

(∫ t

0

ϕ1(t− r)|u2(r ∧ τR)) − u1(r ∧ τR)|L2dr)p

≤ 23p2 πpRp

( ∫ t

0

ϕ1(r)dr)p−1

∫ t

0

ϕ1(t− r)E|u(r ∧ τR)|pL2dr

≤ C23p2 πpRpt(1−

32α )(p−1)

∫ t

0

ϕ1(t− r)E|u(r ∧ τR)|pL2dr,

and

E|∫ t

0

1[0,τR)Sα(t− r)[g(u2(r)) − g(u1(r))] dW (r)|pL2

≤ E( ∫ t

0

(t− r)γ−1|Y1(r)|L2 dr)p,

≤(∫ t

0

r(γ−1)( pp−1 )

)p−1∫ t

0

E|Y1(r)|pL2 dr,

where

Y1(r) =sinπγ

π

∫ r

0

(r−s)−γSα(r − s)1[0,τR)(s)(g(u2(s∧τR))−g(u1(s∧τR))

)dW (s).

Furthermore,

E|Y1(r)|pL2

≤ Cpbp2E

(∫ r

0

(r − s)−2γ∑

k≥1

e−2(r−s)(kπ)α |u2(s ∧ τR) − u1(s ∧ τR)|L2

) p2

ds

≤ Cpbp2

(∑

k≥1

∫ ∞

0

s−2γe−2s(kπ)α

ds) p

2q

(∫ r

0

(r − s)−2γ∑

k≥1

e−2(r−s)(kπ)α

E|u(s ∧ τR)|pL2 ds).



Hence

E|∫ t

0

1[0,τR)Sα(t− r)(g(u2(r)) − g(u1(r))

)dW (r)|pL2

≤ C

∫ t

0

∫ r

0

(r − s)−2γ∑

k≥1

e−2(r−s)(kπ)α

E|u(s ∧ τR)|pL2 ds dr

≤ C

∫ t

0

(∫ t−s

0

ξ−2γ∑

k≥1

e−2ξ(kπ)α

dξ)

E|u(s ∧ τR)|pL2 ds (4.5)

≤ C

∫ t

0

ϕ2(t− s)E|u(s ∧ τR)|pL2 ds.

Replacing (4.5) and (4.4) in (4.3) we get

E|u(t ∧ τR)|pL2 ≤ CR,p

∫ t

0

ϕ(t− r)E|u(r ∧ τR)|pL2 dr for all t ≥ 0.

Hence the result follows from the application of the Gronwall Lemma and Jensen

inequality. This method can work for any exponent q ≥ p.

A.1. Proof of Lemma 2.3

Lemma A.1.1. The map πn : L2→L2 is globally Lipschitz with Lipschitz constant

1. Moreover,

|(πnu)2 − (πnv)2|L1 ≤ 2n|u− v|L2 , for all u, v ∈ L2(0, 1), (A.1)

|(πnu)2|L1 ≤ n|u|L2 , for all u ∈ L2(0, 1),

|(πnu)2|L1 ≤ |u|2L2 , for all u ∈ L2(0, 1).

Proof. Let us first consider the case when u, v ∈ L2(0, 1) are such that |u|L2 > n,

|v|L2 > n. Then we have

|πnu− πnv|2L2 = |πnu|2L2 + |πnv|2L2 − 2〈πnu, πnv〉

= n2 + n2 − 2n2〈 u|u| ,v

|v| 〉 = n2 2|u| |v| − 2〈u, v〉|u| |v|

≤ n2

|u|L2 |v|L2

(|u|2L2 + |v|2L2 − 2〈u, v〉

)≤ |u− v|2L2 .

In the case when |u|L2 > n and |v|L2 ≤ n, we have

|πnu− πnv|2L2 =n

|u|L2

(n|u|L2 +

|v|L2

n|v|L2 |u|L2 − 2〈u, v〉

)

≤ n

|u|L2

(|u|2L2 + |v|2L2 − 2〈u, v〉

)≤ |u− v|2L2 .



The case |u|L2 < n, |v|L2 < n is trivial.

The inequality (A.1) then easily follows. Indeed, by using the trivial identity

a2 − b2 = (a+ b)(a− b), the Holder inequality and the fact that the projection πn is

both Lipschitz continuous and bounded we have, for u, v ∈ L2(0, 1), the following

inequalities,

|(πnu)2 − (πnv)2|L1 ≤ |(πnu) + (πnv)|L2 |(πnu) − (πnv)|L2

≤ 2n|(πnu) − (πnv)|L2 .

In a similar way we can prove the second and the third inequalities.

B.1. Gronwall Lemma

Lemma B.1.1. Assume that ϕ, ψ ∈ L1(a, b) and y : [a, b]→R is an absolutely

continuous function. Suppose that these functions satisfy

dy(t)

dt≤ ϕ(t)y(t) + ψ(t) for a.a. t ∈ (a, b). (B.1)

Then we have

y(t) ≤ y(a)e∫

taϕ(τ)dτ +

∫ t

a

ψ(s)e∫

tsϕ(τ)dτds, for all t ∈ [a, b]. (B.2)

C.1. Some estimates on stopped stochastic convolutions

The main technical tool is the following result stated in Ref. [41] and proven in

Ref. [33], Theorem 4.2.7. We formulate the result in the Hilbert space framework

as we do not require in this paper Banach space valued Ito integrals. For a notion of

γ-radonifying operator see Refs. [26] and [41] or Remark 6.1 in a recent paper [42].

The operator ideal of all gamma-radonifying operators from a separable Hilbert

space H into a separable Banach space X will be denoted by R(H,X). Let us

recall that with a naturally defined norm, R(H,X) is a separable Banach space, see

Ref. [43].

Proposition C.1.1. Assume that H is a separable Hilbert space. Assume that X

is an 2-smoothable Banach. Assume that a linear operator −A is an infinitesimal

generator of an analytic semigroup (S(t))t≥0 on X. Assume finally that ξ is a

progressively measurable operator-valued stochastic process such that S(t− r)ξ(r) ∈R(H,X) for t > r ≥ 0, where R(H,X) is the Banach space of all γ-radonifying

operators from H to X and there exists p > 2 and β0 ∈ ( 1p ,

12 ) such that for some

T ∈ (0,∞),

|||ξ|||pβ,p,T :=

∫ T

0

(∫ s

0

(s− r)−2β‖S(s− r)ξ(r)‖2R(H,X) dr

)p/2ds <∞, β < β0.



Assume that δ ∈ [0, β0− 1p ). Then there exists an D(Aδ)-valued continuous stochas-

tic process v(t), t ∈ [0, T ], such that

x(t) =

∫ t

0

S(t− s) ξ(s) dW (s), a.s. for each t ∈ [0, T ]. (C.1)

Moreover, if δ + 1p < β < β0 then there exists a constant C > 0 independent of ξ

and T such that for any accessible stopping time σ,

E sup0≤t≤T

|x(t ∧ σ)|pD(Aδ)

≤ CpT p(β−δ)−1 |||1[0,σ)ξ|||pβ,p,T (C.2)

= CpT p(β−δ)−1

∫ T

0

(∫ s∧σ

0

(s− r)−2β‖S(s− r)ξ(r)‖2R(H,X) dr

)p/2ds.

Corollary C.1.2. Assume that a progressively measurable L(H)-valued stochastic

process ξ satisfies, for some M > 0 and T ∈ (0,∞),

|ξ(t)|L(H) ≤M, t ∈ [0, T ],P − a.s.,

where L(H) is the space of all bounded linear operators in H. Assume that S(t) ∈R(H,X) for t > 0 and there exists p > 2 and β0 ∈ ( 1

p ,12 ) such that for T > 0 and

β < β0,

∫ T

0

(∫ s

0

r−2β‖S(r)‖2R(H,X) dr

)p/2ds <∞.

Then the assumptions of Proposition C.1.1 are satisfied and therefore the process

x(t), t ≥ 0, satisfies the following inequality

E sup0≤t≤T

|x(t ∧ σ)|pD(Aδ)

(C.3)

≤ CpMpT p(β−δ)−1

∫ T

0

(∫ s∧σ

0

(s− r)−2β‖S(s− r)‖2R(H,X) dr

)p/2ds.

Proof. [Proof of Corollary C.1.2.] It is enough to use the fact, see Ref. [44], that

R(H,X) an operator ideal and so for a constant C independent of ξ and S(t)t≥0

one has for all 0 ≤ r < s <∞

‖S(s− r)ξ(r)‖R(H,X) ≤ C‖S(s− r)‖R(H,X)|ξ(r)|L(H) ≤ CM‖S(s− r)‖R(H,X).

Proof. [Proof of Proposition C.1.1.] Let us choose β ∈ ( 1p +δ, β0). First we define

a process y by

y(t) =1

Γ(1 − β)

∫ t

0

(t− s)−βS(t− s) ξ(s) dW (s), t ∈ [0, T ].

Then E∫ T0|y(t)|pX dt <∞. Indeed, by the Fubini Theorem and the Burkholder

and Young inequalities



E

∫ T

0

∣∣y(s)∣∣pXds =

∫ T

0

E∣∣y(s)

∣∣pXds

≤ Cp

∫ T

0

E

(∫ s

0

(s− r)−2β‖S(s− r) ξ(r)‖2R(H,X) dr

) p2

ds (C.4)

≤ Cp|||ξ|||pβ,p,T <∞.

Hence, there exists a set of full measure Ω such that y(·, ω) ∈ Lp(0, T ;X). For

ω ∈ Ω we set

x(t, ω) =1

Γ(β)

∫ t

0

(t− s)β−1S(t− s) y(s, ω) ds, t ∈ [0, T ].

Let ω ∈ Ω be fixed. Let us recall without proof the following classical result.

Lemma C.1.3. Assume that δ ≥ 0, p ∈ (1,∞) and 1 > β > 1p + δ. Then the map

Rβ : Lp(0, T ;X) → C([0, T ];D(Aδ)) given by

Rβh(t) =

∫ t

0

(t− s)β−1S(t− s)h(s) ds, h ∈ Lp(0, T ;X),

is well defined, linear and bounded. Moreover, there exists a constant Cp,β−δ > 0

such that the norm of ‖Rβ‖ ≤ Cp,β−δTβ−δ− 1

p .

Since β > 1p + δ. By the above Lemma there exists a constant CT (independent

of y (and hence ω)) such that

| x(t, ω) |pD(Aδ)

≤ CT

∫ t

0

| y(s, ω) |pX ds, t ∈ [0, T ].

In particular,

| x(t, ω) |pD(Aδ)

≤ CT

∫ T∧σ(ω)

0

| y(s, ω) |pX ds, t ∈ [0, T ∧ σ(ω)].

It follows that

sup0≤t≤T∧σ(ω)

| x(t, ω) |pD(Aδ)

≤ CT

∫ T∧σ(ω)

0

| y(s, ω) |pX ds. (C.5)

Noting that

sup0≤t≤T

| x(t ∧ σ) |pD(Aδ)

= sup0≤t≤T∧σ

| x(t) |pX ,

we infer that

sup0≤t≤T

| x(t ∧ σ) |pD(Aδ)

≤ CT

∫ σ∧T

0

| y(s) |pX ds a.s..

Taking expectations gives



E sup0≤t≤T

| x(t ∧ σ) |pD(Aδ)

≤ CTE

∫ σ∧T

0

| y(s) |pX ds. (C.6)

Note that

E

∫ σ∧T

0

| y(s) |pX ds = E

∫ T

0

| 1[0,σ)(s)y(s) |pX ds

=

∫ T

0

E | 1[0,σ)(s)y(s) |pX ds. (C.7)

Define next a process

y(s) =1

Γ(1 − β)

∫ s

0

(s− r)−βS(s− r)1[0,σ)(r)ξ(r) dW (r).

y is well defined and for s ∈ [0, T ]

1[0,σ)(s)y(s) =

y(s) if 1[0,σ)(s) = 1,

0 if 1[0,σ)(s) = 0.

Therefore,

E | 1[0,σ)(s)y(s) |pX =

∫

1[0,σ)(s)=1| 1[0,σ)(s, ω)y(s, ω) |pX dP(ω)

≤∫

1[0,σ)(s)=1| 1[0,σ)(s, ω)y(s, ω) |pX dP(ω)

+

∫

1[0,σ)(s)=0| y(s, ω) |pX dP(ω)

=

∫

1[0,σ)(s)=1| y(s, ω) |pX dP(ω)

+

∫

1[0,σ)(s)=0| y(s, ω) |pX dP(ω) = E | y(s) |pX .

Thus, integrating from 0 to T and then applying as earlier the Fubini Theorem

and the Burkholder and Young inequalities we infer that

E

∫ T

0

| 1[0,σ)(s)y(s) |pX ds ≤ E

∫ T

0

| y(s) |pX ds

≤ Cp

∫ T

0

E

(∫ s

0

(s− r)−2β1[0,σ)(r)‖S(s− r) ξ(r)‖2R(H,X) dr

) p2

ds (C.8)

≤ Cp|||1[0,σ)ξ|||pβ,p,T <∞.



Corollary C.1.4. Assume that the assumptions of Proposition C.1.1 are satisfied.

Let 0 < T < ∞ and τ : Ω → [0, T ] be a stopping time. Furthermore assume that

ξ(t), t < τ is an admissible R(H,X)-valued process with

|||1[0,τ)ξ|||β,p,T <∞, if β < β0.

Set

z(t) =

∫ t

0

S(t− s) 1[0,τ)(s)ξ(s) dW (s).

Then, there exists a modification z of z such that

z ∈ Lp(Ω;C(0, T ;X)).

Moreover, there exists C(T ) > 0, independent of ξ and τ , such that for each

t ∈ [0, T ],

E sup0≤s≤t

| z(s ∧ τ) |pD(Aδ)

≤ C(T )|||1[0,τ)ξ|||pβ,p,t. (C.9)

Proof. Define the process η : [0, T ] × Ω → R(H,X) by

η(s, ω) = 1[0,τ(ω))(s)ξ(s, ω).

As ξ is admissible and 1[0,τ) is right continuous and adapted, then η right contin-

uous and adapted. In particular, η has a progressively measurable modification.

Moreover, as

|||η|||β,p,T = |||ξ|||β,p,T <∞,

it follows, by Proposition C.1.1 that the process z is well defined and has a continu-

ous modification z with z ∈ Lp(Ω;C(0, T ;X)). Furthermore, we can find C(T ) > 0,

independent of ξ and τ , such that

E sup0≤s≤t

| z(s ∧ τ) |pD(Aδ)

≤ C(T )|||1[0,τ)(s)ξ|||pβ,p,t, t ∈ [0, T ]. (C.10)

This completes the proof of Corollary C.1.4.

Finally, we shall prove

Corollary C.1.5. Assume that the assumptions of Proposition C.1.1 are satisfied.

Let x(t), t ∈ [0, T ], be an D(Aδ)-valued continuous stochastic process such that

x(t) =

∫ t

0


Let σ be an accessible [0, T ]-valued stopping time.



Let y(t) = yσ(t), t ∈ [0, T ], be an X-valued continuous stochastic process such

that

y(t) =

∫ t

0

S(t− s) 1[0,σ)(r)ξ(r ∧ σ) dW (s), a.s. for each t ∈ [0, T ] (C.12)

Then, for each t ∈ [0, T ],

x(t ∧ σ) = y(t ∧ σ), a.s..

Proof. If η(r) := 1[0,σ)(r)ξ(r ∧ σ), r ∈ [0, T ], then y depends on η in the same

way as x depends on ξ and y − x on η − ξ. Therefore, by Proposition C.1.1

E sup0≤t≤T

|x(t ∧ σ) − y(t ∧ σ)|p ≤ C(T ) |||1[0,σ)(ξ − η)|||pβ,p,T (C.13)

Hence, for each t ∈ [0, T ], E|x(t ∧ σ) − y(t ∧ σ)|p = 0. In particular, x(t ∧ σ) −y(t ∧ σ) = 0 a.s. as claimed.

We finish this section with describing applications of the results obtained here

to the stochastic evolution equation driven by the fractional Laplace operator or,

as called by some authors, stochastic fractional heat equation.

The following result is stated Remark 6.1 in a recent paper [42]. It can be proved

by applying Theorem 2.3 from Ref. [45].

Proposition C.1.6. Assume that K is a linear self-adjoint compact operator in

L2(0, 1). Let ej∞j=1 and λj∞j=1 be the corresponding sequence of eigenvectorse

and eigenvalues of K. Let q ∈ (1,∞). Then K : L2(0, 1)→Lq(0, 1) is γ-radonifying

if and only if

∫ 1

0

∞∑

j=1

λ2j |ej(x)|2

q/2

dx <∞.

Moreover, the γ-radonifying norm of K, denoted by ‖K‖R(L2,Lq), is equivalent to[∫ 1

0 [∑j λ

2j |ej(x)|2]q/2 dx

]1/q.

In particular, if supj |ej |L∞ < ∞, then there exists a constant Mq independent of

K, such that

‖K‖R(L2,Lq) ≤Mq‖K‖R(L2,L2) = Mq‖K‖HS(L2,L2),

where as before ‖K‖HS(L2,L2) denotes the Hilbert-Schmidt norm of the operator K.

Example C.1.7. Assume that q ∈ (1,∞) and let Aq,α be the fractional power of

order α2 of the operator Aq defined by

D(Aq) = H2,q(0, 1) ∩H1,q

0 (0, 1),

Au = −∆u, u ∈ D(Aq).(C.14)

eLet us here recall that ej∞j=1 is an ONB of L2(0, 1) and Kej = λjej , j ∈ N∗.



Briefly, Aq,α = Aα2q . Let us denote the semigroups generated by the operators Aq,α

by one symbol Sα(t)t≥0.

Since |ej |L2 = 1 By Proposition C.1.6 we infer that there exists a constant C > 0

such that

‖Sα(t)‖2R(L2,Lq) ≤ C‖Sα(t)‖2

HS(L2,L2), t > 0. (C.15)

Assume that β < 12 − 1

2α =: β0(α). Then, because ‖Sα(t)‖2HS(L2,L2) =

∑k e

−2παkαt

we infer that for s > 0∫ s

0

r−2β‖Sα(r)‖2R(L2,Lq) dr ≤ C2

∫ ∞

0

r−2β‖Sα(r)‖2HS(L2 ,L2) dr

= C2

∫ ∞

0

r−2β∞∑

k=1

e−2παkαr dr = C2∞∑

k=1

∫ ∞

0

r−2βe−2παkαr dr

= C2Γ(1 − 2β)(2πα)2β−1∞∑

k=1

kα(2β−1) =: C2α,β <∞.

Therefore∫ T

0

(∫ s

0

r−2β‖Sα(r)‖2R(L2,Lq) dr

)p/2ds ≤ Cpα,βT <∞

and so the semigroup Sα(t)t≥0 satisfies the assumptions of Corollary C.1.2 with

β0 = β0(α) = 12 − 1

2α .

Hence we proved the following result.

Proposition C.1.8. Consider the framework of Example C.1.7 with α ∈ (1, 2] and

q ∈ [2,∞). Assume that a progressively measurable L(L2)-valued stochastic process

ξ satisfies, for some M > 0,

|ξ(t)|L(L2) ≤M, t ∈ [0,∞),P − a.s.. (C.16)

Assume that p ∈ (2,∞) and δ ≥ 0 satisfy the following inequality

δ +1

p< β <

1

2− 1

2α=: β0(α). (C.17)

Then there exists an D(Aδq,α)-valued continuous stochastic process x(t), t ∈ [0, T ]

such that for t ≥ 0

x(t) =

∫ t

0


and there exists a constant C > 0 independent of ξ and T such that for any accessible

stopping time σ,

E sup0≤t≤T

|x(t ∧ σ)|pD(Aδ

q,α)≤ CpMpT p(β−δ) (C.19)



Since D(Aδα,q) = D(Aαδ/2q ) ⊂ Hαδ,q(0, 1) we infer the following corollaryllary.

Corollary C.1.9. Consider the framework of Example C.1.7 with α ∈ (1, 2] and

q ∈ [2,∞). Assume that p ∈ (2,∞) and δ ≥ 0 satisfy condition (C.17). Then

there exists a constant C > 0 such that for any T > 0, any accessible stopping time

σ, any progressively measurable L(L2)-valued stochastic process ξ satisfying (C.16),

the process x(t), t ≥ 0 from Proposition C.1.8 satisfies the following

E sup0≤t≤T

|x(t ∧ σ)|pHαδ,q ≤ CpMpT p(β−δ). (C.20)

Let us finish this section with the following two observations. The first is that

given positive numbers α and δ satisfying inequality (C.17) we can always find q

such that αδ > 1q . The second is that because of Jensen’s inequality, inequalities

(C.19) and (C.20) is also valid for all p2 ≤ p.

D.1. Pointwise multiplication in Sobolev spaces

Since our proof relies on Theorems 1 and 2 from section 4.4.4 of Ref. [35] and the

formulations of these theorems contain some errors we have decided to present them

here for the convenience of the reader.

Theorem 1. Assume that

0 < s1 ≤ s2 (1)

1

p≤ 1

p1+

1

p2, (2)

n

p− s1 >

( np1 − s1)+ + ( np2 − s2)+, if maxi(

npi

− si) > 0,

maxi(npi

− si) > 0, otherwise(3)

s1 + s2 >n

p1+n

p2− n, (4)

and either

q ≥ q1 if s1 < s2 (5)

or

q ≥ max(q1, q2) if s1 = s2. (6)

(i) Then

F s1p1,q1 · F s2p2,q2 → F s1p,q (7)

(ii) Also we have

Bs1p1,q1 · Bs2p2,q2 → Bs1p,q (8)



Theorem 2. Assume that the conditions (1), (2), (4) and (5) are satisfied. Assume

also thatf

n

p− s1 =

( np1 − s1)+ + ( np2 − s2)+, if maxi(

npi

− si) > 0,

maxi(npi

− si) > 0, otherwise(16)

(i) If

i ∈ 1, 2 : si =n

piand pi > 1 = ∅, (17)

then (7) remains true.

(ii) Suppose in addition thatg

q ≥ q2, if s1 −n

p1= s2 −

n

p2. (18)

If

i ∈ 1, 2 : si =n

piand qi > 1

∪i ∈ 1, 2 : si <n

piand qi >

nnpi

− si = ∅, (19)

then (8) remains true.

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Chapter 6

Stochastic Control Methods for the Problem of Optimal

Compensation of Executives

Abel Cadenillas, Jaksa Cvitanic∗ and Fernando Zapatero†

Department of Mathematical and Statistical Sciences, University of AlbertaEdmonton, Alberta T6G 2G1, Canada.

[email protected].

We consider the problem of an executive who receives call options as com-pensation. He can influence the mean return of the stock with his effort, andthe level of volatility through his choice of projects. The executive wants to se-lect the optimal effort and choice of projects to maximize the expected utilityfrom the call option minus the disutility associated with the effort. We modelthis as a stochastic control problem, and present an analytical solution. In thisframework, we also introduce the problem of the company that simultaneouslywants to minimize overtime volatility and maximize final expected value of theprice of the stock. We characterize (and compute numerically for the logarithmiccase) the optimal strike price the company should choose. When the executivecan affect the mean return of the stock, we find that options should be grantedout-of-the-money. In a context of perfect information, the executive would notbe able to affect the mean return of the stock independently of the volatility, andit would be optimal to grant options in-the-money. The mathematical techniquesare those of stochastic control, convex analysis, and martingale theory.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

2 The Executive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

2.1 Stock Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

2.2 The Problem of the Executive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

2.3 Optimal Effort and Choice of Projects . . . . . . . . . . . . . . . . . . . . . . . 173

3 The Company . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

3.1 The Problem of the Company . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

3.2 Optimal Strike Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4 Numerical Computations of the Strike Price . . . . . . . . . . . . . . . . . . . . . . . . 179

5 Price of the Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6 The Case of Additional Cash Compensation . . . . . . . . . . . . . . . . . . . . . . . . 186

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

∗Humanities and Social Sciences, Caltech, M/C 228-77, 1200 E. California Blvd. Pasadena, CA91125, [email protected].†FBE, Marshall School of Business, University of Southern California, Los Angeles, CA 90089-1427, [email protected].

169


170 A. Cadenillas, J. Cvitanic, and F. Zapatero

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

1. Introduction

The optimal compensation of executives is currently one of the most active areas

of research in mainstream finance, although it has not received much attention yet

in the mathematical finance literature. The problem of the company consists in

selecting the optimal compensation package to give the executive, and the problem

of the executive consists in selecting the optimal strategy to take advantage of the

compensation package.

A call option gives its owner the right to buy one share of an underlying stock

at a predetermined price. Options are similar to a leveraged portfolio invested in

the underlying security. For that reason, options have become the main choice for

companies to compensate executives in order to align their interests with those of

the company stockholders. Hall and Leibman [1] and Murphy [2] report statistics

about the number of companies that use stock options as the main component of the

compensation of their executives. Carpenter [3] and Hall and Murphy [4] consider

the problem of valuation of executive stock options, Detemple and Sundaresan [5]

develop a pricing model for options that can be applied to executive options, and

Hall and Murphy [6] study the problem of choosing the strike price of the options.

Stoughton and Wong [7] consider the optimality of stock versus options in a context

of industry competition. Aseff and Santos [8] address the broader problem of the

optimality of stock options versus other contracts. Kadan and Swinkels [9, 10]

compare options vs. stock when there is a possibility of bankruptcy.

We develop a dynamic model in order to address some of the problems mentioned

above. In particular, the existent literature considers the incentives of options in a

static framework, which does not seem appropriate to address some of the questions

considered. One of the potential problems of stock options compensation (see, for

instance, Johnson and Tian [11, 12]) is the incentive to the executive to increase

volatility. However, volatility is a dynamic concept and it is appropriate to consider

problems like these in a dynamic setting. Another element that is missing in the

current literature is the computation of the effort of the executive and the tradeoff

between the amount of that effort and the disutility resulting from it.

We assume that the executive receives call options as compensation and we

study the optimal strike price of the options. The strike price of zero means that

the compensation is in stock. In our setting, the effort of the executive affects

the mean return of the stock. Besides, the executive can choose among a menu of

projects with a tradeoff between expected return and volatility: projects with higher

volatility offer a higher expected return (see Cadenillas, Cvitanic and Zapatero [13]

for a similar model in a different application).

The executive will choose effort and volatility in order to maximize expected

utility coming from compensation minus the effort disutility. The company cares

both about the final value of the stock and the overtime volatility. The company


Optimal Compensation of Executives 171

chooses the strike price of the option so as to induce the executive to achieve an

optimal mix of effort and volatility. Carpenter [14] studies, also in a dynamic

setting, the optimal choice of volatility when the effort of the agent cannot affect

the stock dynamics and considers several exogenously given strike prices and more

general benchmark functions. Bolton and Harris [15] also study a model where the

agent has control of the dynamics of the asset, for a different purpose. In Cadenillas

Cvitanic and Zapatero [16] we extend the model of this paper to the situation in

which the type of the executive δ is unknown by the company, which has a prior

distribution for it.

For the executive that maximizes logarithmic utility from the options, minus

the quadratic disutility from the effort, we compute the optimal level of effort and

volatility in closed form. Based on that, we can compute the optimal strike price to

be set by the company. We find that the strike price should be higher, the higher

the type (quality) of the executive, and the more the company is interested in

maximizing the expected value of the price of the stock (as opposed to minimizing

overtime volatility). The relationship between the optimal strike price and the

quality of the projects (defined as the additional expected return resulting from

accepting an extra unit of risk) is not monotonic.

We also compute the no-arbitrage (complete markets) price of the option. This

would be the price of the option for the company and, therefore, would represent

the cost the company incurs by granting options. The executive, on the other hand,

faces incomplete markets since he cannot trade in the underlying. We compute the

value he assigns to the option as the certainty equivalent associated with his utility.

The price of the options is lower for the executive (facing incomplete markets) than

for the company. We show that options seem to be more efficient when the executive

is of a high type relative to the quality of the projects.

Finally, we also compute the optimal effort of the executive when cash is a part

of the compensation package, but we are not able to get explicit expressions for the

optimal strike price in this case.

The paper is structured as follows. In section 2 we present and solve explicitly the

problem of the executive, and in section 3 we present and solve the problem of the

company. In section 4 we perform some numerical computations and comparative

statics. In Section 5 we compute the option prices from the perspectives of both the

company and the executive. Section 6 considers the case of additional compensation

in cash. We close the paper with some conclusions.

2. The Executive

2.1. Stock Dynamics

Consider a probability space (Ω,F , P ) endowed with a filtration (Ft), which is the

P -augmentation of the filtration generated by a one-dimensional Brownian motion

W . Our benchmark stock has a price that follows a geometric Brownian motion



process,

dSt = µStdt+ σStdWt

with starting value S0. Here, µ ∈ [0,∞) and σ ∈ (0,∞) are exogenous constants.

Definition 2.1. The level of effort is an (Ft)-adapted stochastic process u that

satisfies

E

[∫ T

0

|ut|2dt]<∞.

The value ut is the amount of effort that the executive puts in the management of

the company at time t. The choice of projects is an (Ft)-adapted stochastic process

v that satisfies

E

[∫ T

0

|vtSt|2dt]<∞.

We assume that at each moment in time, the executive can choose different projects

(or strategies) vt, that are characterized by a level of risk and expected return. We

will denote by A the pairs (u, v) which satisfy the above technical conditions, and

say that A is the class of admissible pairs of effort and choice of projects.

When the company is managed by the executive, the dynamics of the stock price S

is given by

dSt = µStdt+ δutdt+ αvtStdt+ vtStdWt. (2.1)

Here, δ ∈ [0,∞) and α ∈ (0,∞). The higher the value of u, the higher the expected

value of the stock. On the other hand, the choice of projects v is equivalent to the

choice of the volatility of the stock, although it also has an impact on the expected

value. We assume that the executive can choose different projects or strategies that

are characterized by a level of risk and expected return. Since α > 0, the higher the

risk of a project, the higher its expected return. The parameter α is a characteristic

of the type of the company. One possible interpretation of α (and, potentially, a

way to estimate it empirically) would be the slope of the equivalent of the “Capital

Market Line” resulting from all the projects available to the company (more about

the distinction between firm-specific and market risk below). On the other hand, δ

is a measure of the impact of the effort of the executive on the value of the company.

It can be interpreted as an indicator of the quality of the executive.

We emphasize that this is a partial equilibrium setting and we do not compare

the dynamics of the stock of this company with the dynamics of other stocks. In

the simplest setting, with complete markets, the coexistence of different stocks

with different drifts (and maybe same standard deviations) would be consistent

with incomplete information about the actions of the agent and its effects on the

dynamics of the stock. In theory, in that setting, complete information would be



consistent with δ = 0, which is a special case of the model that we consider in this

paper.

Without loss of generality, we will assume that µ = 0. We then rewrite equation

(2.1) as

dSt = δutdt+ αvtStdt+ vtStdWt. (2.2)

2.2. The Problem of the Executive

In our model, the executive is risk-averse and experiences disutility as a result of

the effort.

Problem 2.2. The objective of the executive is to select (u, v) ∈ A that maximizes

the functional J defined by

J(u, v) := E

[logn(ST −K)+

− 1

2

∫ T

0

u2tdt

]. (2.3)

Here, n is the number of call options the executive receives as a part of his com-

pensation package. The second term of the objective function of the executive

represents the disutility from effort. That disutility might be, for example, the re-

sult of spending more time working for the company. The other control, v, only

involves the choice of projects the company will undertake and has no effect on the

disutility of the executive, since it does not require any effort: the executive has a

menu of projects and decides the level of risk to undertake.

With this parameterization, the number of options n becomes irrelevant for the

executive’s problem, because log n(ST −K)+ = logn + log(ST −K)+. We will

then assume that n = 1. It is obvious that a more general parameterization (such

as power utility) would not have this property.

Problem 2.1 of the executive of this paper is similar to the problem of the

manager considered by Cadenillas, Cvitanic and Zapatero [13]. In that paper, the

manager controls the value of unlevered assets.

2.3. Optimal Effort and Choice of Projects

First we introduce the auxiliary exponential martingale Z, defined by

Zt := exp

−1

2α2t− αWt

. (2.4)

Also, consider the function of time T , defined by

Tt :=eα

2(T−t) − 1

α2. (2.5)

Using the previous notation and given the following quadratic equation in z,

δ2T0z2 + (S0 −K)z − 1 = 0, (2.6)



where δ is the parameter that measures the type of the executive and K is the strike

price of the options, we denote by z the positive solution of (2.6):

z =1

2δ2T0

((K − S0) +

√(K − S0)2 + 4δ2T0

)(2.7)

We now present the optimal controls of the executive:

Theorem 2.3. Consider the problem of the executive described in Section 2.2. Con-

sider also the exponential martingale of (2.4), the positive number z given by (2.7),

and T , the time function of (2.5). The optimal effort u of the executive is

ut = δzZt. (2.8)

The optimal choice of projects v is given by

vtSt =α

zZt+ αzδ2ZtTt. (2.9)

The optimal effort and volatility determine that the price of the stock of the firm be

given by

St =1

zZt+K − zδ2ZtTt. (2.10)

If S0 > K the above formulas remain valid even for δ = 0, but with

z =1

S0 −K.

If S0 ≤ K and δ = 0, then the utility of the executive is minus infinity, since it is

impossible in this case to guarantee ST > K with probability one.

Proof. This theorem can be proved applying the duality method to solve stochastic

control problems. We omit the proof, because it is a special case of the proof of

Theorem 6.1.

We observe that the optimal effort and volatility can also be written as functions

of the price of the stock. Indeed, from equations (2.8)-(2.10), we obtain that when

δ > 0,

ut =1

2δTt

(K − St) +

√(K − St)2 + 4δ2Tt

(2.11)

and

vtSt =αδ

ut+ αδutTt

=2αδ2Tt

(K − St) +√

(K − St)2 + 4δ2Tt

+α

2

[(K − St) +

√(K − St)2 + 4δ2Tt

]. (2.12)



With respect to the optimal effort, as expected, u is increasing in the strike

price K: as K goes to infinity, the effort goes to infinity, as well. Besides, we note

that z is decreasing in T , the time to maturity of the option (T0 is increasing in

T and z is decreasing in T0). Therefore, the larger the maturity of the option, the

lower the effort of the executive. The intuition is clear: a larger T has a similar

effect on the executive as a reduction of the strike price. The effect of δ (the “type”

of executive) depends on whether the option is in-, out-, or at-the-money. When

the option is at-the-money, the optimal effort is independent of δ, as we can see by

substituting (2.7) in (2.8). We can also check that when the option is in-the-money

the effort is increasing in δ, and when the option is out-of-the-money the optimal

effort decreases with δ.

Since Z is a martingale, the initial expected value of the effort at any point in

time is

E[ut] = δz. (2.13)

With respect to the effect of α, we note that T0 is increasing in α and, therefore, z is

decreasing in α. Expected effort is, then, decreasing in α (everything else constant):

the better the menu of projects the executive can choose, the lower the expected

effort of the executive.

The analysis of the volatility is more complicated. Since T· is a decreasing

function of t and TT = 0, the second term of (2.9) decreases in expected value

as we approach maturity, and will tend to be negligible relative to the first term.

Therefore, for short maturities, optimal volatility will tend to decrease with higher

strike price. For large maturities, the relation will tend to be the opposite. We also

see that the volatility is increasing in the type δ of executive (z is decreasing in

δ, and zδ2 is increasing in δ). The economic intuition is straightforward: a high-

type executive can afford more volatility because his effort will be more effective

to counteract drops in the value of the stock. It is straightforward to see that the

expected value of the volatility at a future date t is

E[vtSt] =α

zeα

2t + αzδ2Tt. (2.14)

Since Tt is increasing in α, the expected volatility is increasing in α. In other words,

the higher the expected return-risk tradeoff, the higher the risk the executive will

be willing to undertake.

Some of these comparative statics are illustrated in the results of Table 6.1 (that

we analyze in detail in section 4). However, in that table the strike price is always

the optimal one, and some of the previous conclusions hold for changes in a given

parameter with constant strike price.

It is also interesting to study the correlation between optimal effort and optimal

choice of projects. By Ito’s lemma, and equation (2.9), the dynamics of the optimal

volatility are

d(vtSt) = (. . .)dt+ α2

(1

zZt− δ2zZtTt

)dWt. (2.15)



Equation (2.8) yields

dut = (. . .)dt− αutdWt. (2.16)

It is clear that their correlation can be either positive or negative. Considering

only the instantaneous correlation and ignoring the drift terms, we see that for a

short maturity of the option they tend to be negatively correlated, and increases

in optimal effort will be typically associated with decreases of the optimal level of

volatility.

3. The Company

3.1. The Problem of the Company

The company is interested in two things: the expected value and the volatility of

the stock price. Specifically, we suppose that the objective of the company is to

select the strike price K that solves the following problem:

maxK∈[0,∞)

λE[ST ] −

∫ T

0

Var[St]dt

. (3.1)

Here, T is the maturity of the European call options and λ is an exogenous constant

that represents the tradeoff in the preferences of the company between the expected

final value and the variance of the stock. The objective of the company has some

similarity with quadratic preferences (in which the agent only cares about expected

return and variance), but in our model the company cares about overtime variance.

The objective function of the company is justified as a way to incorporate risk

aversion in a tractable way. Another possible criterion is to select the number of

options and the strike price to maximize the expected value of the price of the stock

at maturity minus the price of the compensation package. In that case, the number

of options and the strike price must be restricted so that the expected utility of the

executive be greater or equal than a positive number. A similar problem has been

studied by Cadenillas, Cvitanic and Zapatero [13].

3.2. Optimal Strike Price

We assume that the company has full information about the parameters that char-

acterize the dynamics of the stock, as well as the preferences of the executive. The

objective of the company is given by (3.1). In order to characterize the optimal

choice of the company we introduce the following auxiliary notation:

A :=e3α

2T − 1

3α2− e2α

2T − 1

2α2, (3.2)

B :=δ4

α4

1

2α2e2α

2T +

(2

α2− 2T

)eα

2T − 5

2α2− T

, (3.3)



and the function h defined by

h(z) := λ1

z(eα

2T − 1) − 1

z2A− z2B + λzδ2T0, (3.4)

where λ is the parameter of (3.1). We also define

η :=

−S0+

√S2

0+4δT0

2δT0if δ > 0

1S0

if δ = 0.

We will be interested in maximizing the function h in the interval [η,∞). We now

present the optimal strike price for the company.

Theorem 3.1. Consider the company whose objective is given by (3.1). Let z ∈[η,∞) be the quantity that maximizes (3.4), Then the optimal strike price is given

by

K = S0 −1

z+ zδ2T0. (3.5)

Proof. Our first objective is to compute the variance and the expectation of St.

Some preliminary computations give

E[Z2(t)] = eα2t, E[Z(t)] = 1, E[Z−2(t)] = e3α

2t, E[Z−1(t)] = eα2t.

Thus, according to equation (2.10),

E[St] =1

zeα

2t +K − zδ2Tt

and

V ar[St] = V ar

[1

zZt− zδ2ZtTt

]

= E

[(1

zZt− zδ2ZtTt

)2]−(E

[1

zZt− zδ2ZtTt

])2

=1

z2(e3α

2t − e2α2t) + 2δ2Tt(e

α2t − 1) + z2δ4T 2t (eα

2t − 1).

Noting that TT = 0, K = S0 − 1/z + zδ2T0 from (2.6)-(2.7), we see that

λE[ST ] = λ

[1

zeα

2T +K

]

= λ

[1

zeα

2T + S0 − 1/z + zδ2T0

]

= λ

[1

zeα

2T − 1/z + zδ2T0

]

+ an expression that involves neither z nor K.



Furthermore,∫ T

0

Var[St]dt =1

z2

∫ T

0

(e3α2t − e2α

2t)dt+ 2δ2∫ T

0

Tt(eα2t − 1)dt

+ z2δ4∫ T

0

T 2t (eα

2t − 1)dt =1

z2

e3α

2T − 1

3α2− e2α

2T − 1

2α2

+z2 δ4

α4

1

2α2e2α

2T +

(2

α2− 2T

)eα

2T − 5

2α2− T

+ an expression that involves neither z nor K.

Disregarding the terms that do not depend on z or K, we see that (3.1) is equivalent

to finding z ∈ (0,∞) that maximizes the function h defined in (3.4). Here, A and

B are given in (3.2) and (3.3). The maximization has to be performed under the

constraint that the corresponding K is non-negative. We can find K from (2.6) as

K = S0 − 1/z + zδ2T0,

the equation given in (3.5). We observe that z ∈ (0,∞) satisfies the restriction

S0 − 1/z + zδ2T0 ≥ 0

if and only if

δT0z2 + S0z − 1 ≥ 0.

That is equivalent to

z ≥

−S0+√S2

0+4δT0

2δT0if δ > 0

1S0

if δ = 0= η.

We do not have a closed-form solution for the constant z. However, it can be

found easily by applying the following necessary condition of optimality:

2Bz4 − λδ2T0z3 + λ(eα

2T − 1)z − 2A = 0 if z > η (3.6)

2Bz4 − λδ2T0z3 + λ(eα

2T − 1)z − 2A ≤ 0 if z = η, (3.7)

which is the result of differentiating (3.4) and the constraint z ∈ [η,∞) (which is

equivalent to K ∈ [0,∞)). We will perform some numerical exercises in the next

section. Next, we consider the case in which the company is only interested in

variance minimization.

Corollary 3.2. Consider a company that chooses the strike price that minimizes

overtime variance, i.e.,

minK≥0

∫ T

0

Var[St]dt. (3.8)

Suppose also that δ = 0, i.e., that the executive cannot influence the drift indepen-

dently of volatility. Then it is optimal to issue at the money options, i.e.,

K = S0. (3.9)



Proof. This corresponds to the case λ = 0. Since also δ = 0, then B = 0, and (3.4)

implies we have to find z that minimizes 1z2A. Hence optimal z = ∞, and (3.5)

gives K = S0.

If the company cares only about the variance and the executive can affect the

drift of the stock only through the choice of the level of volatility, the corollary

says that options should be optimally issued at-the-money (which is the customary

practice). We will see later that in general we get a large range of possible optimal

values for the strike price .

Corollary 3.3. Consider a company that chooses the strike price that maximizes

the expected value of the stock, i.e.,

maxK≥0

E[ST ]. (3.10)

Suppose also that δ = 0. Then, the optimal strike price is K = 0.

Proof. We note that in the case δ = 0, equation (2.10) gives

E[ST ] =eα

2T

z+K.

Thus, according to (3.5), the company wants to find the value of K that maximizes

E[ST ] = K(1 − eα2T ) + S0e

α2T .

Since α > 0, the optimal K is zero.

When the company only cares about the expected final price of the stock and

the executive can influence the drift of the stock only through the choice of projects,

this corollary says that it is optimal for the company to give all the compensation

in stock.

4. Numerical Computations of the Strike Price

In the previous section we derived the optimal exercise price for a company that

cares about both the expected final value of the stock and the overtime volatility. It

is expressed in equation (3.5). It depends on the solution to equations (3.6)-(3.7),

which do not have an explicit solution. In this section we perform some numerical

exercises in order to derive some properties of the optimal strike price. The results

are included in Table 6.1.

We study the strike price as a function of the parameters of the model. The

effects of the time to maturity are obvious and we fix it at T = 5. The initial price

of the stock is 100. Besides the optimal strike price, we also record the optimal

initial effort level u0 and the optimal initial choice of projects v0 (expressed in %).

The last two columns are helpful for intuitive purposes.

The first general observation is that for strictly positive values of the parameters,

the optimal strike price is out-of-the money. Next, we analyze the type of executive.



Table 6.1. Optimal strike price, effort, and volatility.

S0 = 100; T = 5

δ α λ K u0 v0(%)

1 0.05 0.5 140.01 9.56 2.411 0.1 0.5 111.77 2.38 1.261 0.25 0.5 103.78 0.85 1.541 0.5 0.5 108.70 0.98 5.381 0.75 0.5 143.14 1.57 33.311 0.25 0.25 103.40 0.79 1.481 0.25 0.5 103.78 0.85 1.541 0.25 1 104.70 0.99 1.701 0.25 2 107.88 1.46 2.311 0.25 5 120.16 3.28 5.181 0.5 0.1 108.60 0.97 5.331 0.5 0.25 108.64 0.97 5.351 0.5 0.5 108.70 0.98 5.351 0.5 1 108.85 0.99 5.431 0.5 2 109.16 1.02 5.561 0.5 5 110.35 1.13 6.065 0.05 0.5 146.35 1.94 2.575 0.1 0.5 116.24 0.85 2.795 0.25 0.5 116.02 0.76 7.265 0.5 0.5 142.98 0.95 26.665 0.25 0.1 115.52 0.75 7.195 0.25 0.25 115.70 0.76 7.225 0.25 0.5 116.02 0.76 7.265 0.25 1 116.67 0.78 7.355 0.25 2 118.12 0.82 7.565 0.25 5 123.88 0.98 8.505 0.5 0.1 142.87 0.96 26.625 0.5 0.25 142.91 0.96 26.645 0.5 0.5 142.98 0.96 26.665 0.5 1 143.11 0.97 26.725 0.5 2 143.39 0.97 26.835 0.5 5 144.24 0.99 27.17

The higher the type of the executive (higher δ), the higher the optimal strike price.

A higher type can have more impact with lower effort and will also tend to choose

more volatile projects, since she will be able to counteract negative shocks more

easily.

The quality of the projects α is very important and has an effect on the optimal

strike price that is not monotonic: the optimal strike price is very high for low

quality projects, then it is decreasing as α increases, up to a certain point. Beyond

that point, the optimal strike price is increasing in α. For very low α, a high strike

price is needed to force the executive to choose a high effort level. As the quality

of the projects increases, less effort is required. However, when volatility becomes

very effective (that is, large α), it is necessary again to set a higher strike price to

induce higher effort in order to guarantee that the executive keeps an adequate mix



0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Alpha

K

Delta = 0.1 Delta = 1 Delta = 5 Delta = 10

(a)

80

90

100

110

120

130

140

150

160

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Alpha

K

(b)

Fig. 6.1. The optimal strike price as a function of the quality of projects (a) Different values of δ

(b) δ = 0.1

of effort u and choice of projects v, and does not rely on the choice of projects v

exclusively. It is clear from our argument that the type of the executive plays a

major role in this trade-off. This point is illustrated in Figures 6.1(a) and 6.1(b). In

Figure 6.1(a) we observe that the pattern described above holds for any level of type

of the executive, but the inflection point at which it is necessary to start raising the

strike price again comes faster the higher the type of the executive. Figure 6.1(b)

is the lowest of the four lines in Figure 6.1(a) measured in a different scale.

The effect of λ in this range is clear: the less the company cares about overtime

volatility, the higher the strike price it should set, regardless of the values of other

parameters. In order to get a higher expected price of the stock, the executive

can either exercise more effort or choose riskier projects. Since the company is

concerned about volatility it will prefer more effort, but the only way to induce it

is by setting a higher strike price. Of course, the impact of additional volatility on

expected return (α) affects the effectiveness of the strike price to attain company

goals.

As we mentioned above, an important conclusion of our results is that for strictly

positive values of the parameters, the optimal strike price is always out-of-the-

money. Our results shed some light on the debate about whether options should be

at-the-money or out-of-the-money (Hall and Murphy [6]). As we see in Table 6.2,

options in-the-money would be optimal in the case of an executive whose effort

does not have any effect on the stock return (δ = 0) and when the company is

primarily concerned about the price of the stock (high λ). We also recall from our

discussion about the dynamics of the stock of equation (2.1) that the case δ = 0

could correspond to the situation of perfect information about the effect of the

actions of the executive on the dynamics of the stock. Options at-the-money or

close to at-the-money are optimal (as we see asymptotically in Table 6.2) when the

executive is of very low type (low δ) and the company is mainly interested in price



volatility (low λ). We also see that for δ = 0 the optimal strike price gets lower as

the company cares more about the expected return.

Table 6.2. Optimal strike price for “very small” δ.

S0 = 100; T = 5

δ α λ K

0 0.25 1 99.860 0.25 10 98.630 0.25 100 86.310 0.5 100 96.230 0.5 1000 62.26

0.005 0.25 100 510.210.005 0.25 10 141.020.005 0.25 0.1 100.41

5. Price of the Options

The executive of the problem considered in this paper faces incomplete markets as a

result of the fact that he cannot sell the options and cannot trade in the underlying

security. In practice, the executive will not be allowed to sell the option or take a

short position in the underlying stock (as he would optimally do for diversification

purposes). On the other hand, we can assume that the company faces complete

markets and use, therefore, arbitrage arguments to price the option. There is no

obvious way of computing the exact difference between the value of the option for

the executive and for the company. The approach taken in the literature (that we

follow here) is to compute the certainty equivalent of the executive: a constant

amount of money instead of the option that would leave the executive at the same

utility level as the option. We do not interpret the difference in the value of the no-

arbitrage price and the certainty equivalent as the actual dollar amount. Rather,

we consider the ratio of those two numbers as meaningful: a higher ratio of the

price for the executive to the price for the company indicates that the options are

more appropriate as compensation. A low ratio will question the optimality of using

options as incentive.

Our setting allows us to compute the price of the option, both for the company

and for the executive. We first introduce the price of the option for the company.

Proposition 5.1. Consider the problem of the company described in section 3.

Assume that the company faces complete markets, i.e., it can replicate the option

with the underlying security and a risk-free asset that pays a constant interest rate

r and whose price satisfies

dBt = Btrdt and B0 = 1. (5.1)



The price for the company of the call option granted to the executive as compensation

is given by

C0 =1

zE

[exp

∫ T

0

(α− θs)dWs +1

2

∫ T

0

(α2 − 2r − θ2s)ds

]. (5.2)

Here, θ is the market price of risk given by

θt := (µt − r)/νt, (5.3)

where r ∈ [0,∞) is the interest rate of (5.1), and µ and ν are defined by

µt :=Htα

2 + zδ2Zteα2(T−t)

Ht +K − zδ2ZtTt(5.4)

νt :=αHt + αzδ2ZtTtHt +K − zδ2ZtTt

. (5.5)

In addition, Z and T are defined in (2.4) and (2.5), z is defined in Theorem 3.1,

and H is defined by

Ht :=1

zZt. (5.6)

Proof. The company faces complete markets and prices the option by arbitrage.

We introduce the notation

Ht =1

zZt

where Z is given by equation (2.4) and z is defined in Theorem 3.1. Applying Ito’s

formula, we see that

1

Zt= 1 +

∫ t

0

α2

Zsds+

∫ t

0

α

ZsdWs,

or equivalently

Ht =1

z+

∫ t

0

α2Hsds+

∫ t

0

αHsdWs.

Furthermore,

ZtTt =eα

2T − 1

α2−∫ t

0

Zseα2(T−s)ds−

∫ t

0

αTsZsdWs.

Thus, according to (2.10),

dSt = d(Ht +K − zδ2ZtTt) = (Htα2 + zδ2Zte

α2(T−t))dt+ (αHt + αδ2zZtTt)dWt,

where T is given by equation (2.5). According to equations (5.4)-(5.5), we have

dSt/St = µtdt+ νtdWt.



The market price of risk is then given by (5.3), where µ and ν are defined in (5.4)-

(5.5). The risk neutral density is

Z∗T = exp

−∫ T

0

θsdWs −1

2

∫ T

0

θ2sds

.

The no-arbitrage price is then

C0 = E[e−rTZ∗T (ST −K)+] = E[e−rTZ∗

T HT ]

=1

zE[e∫

T0

(α−θs)dWs+12

∫T0

(α2−2r−θ2s)ds].

The executive, however, cannot replicate the option given his trading constraints.

In order to compute the value of the option for him, we use the certainty equivalent

suggested in Hall and Murphy [6]. This is the fixed amount of money, paid instead

of the option, that would provide the executive the same level of utility as the

option.

Proposition 5.2. Consider the problem of the executive described in section 2. The

fixed amount of money CE that would provide the executive the same utility level

as the option is

CE =1

zexp

1

2

[α2T − δ2α−2(eα

2T − 1)z2]

. (5.7)

Proof. We compute the fixed amount x that provides the executive with the same

utility level as the option. The certainty equivalent CE = x is computed from

logx = E

[log(ST −K)+ − 1

2

∫ T

0

u2tdt

].

Substituting ST = HT +K, ut = δzZt, and computing expectations, we obtain

logx = − log z +1

2[α2T − δ2α−2(eα

2T − 1)z2].

We now perform some comparative statics. The numerical results are presented

in Table 6.3. It is similar to Table 6.1, but it also includes the prices of the option (we

do not report the optimal effort level in Table 6.3). We assume that the company

chooses the optimal strike price, recorded in the column K. C0 is the price of the

option for the company, that we compute using equation (5.2). The expression

of equation (5.2) does not have a closed form solution, but is easy to compute

numerically using Monte Carlo simulation (see Boyle, Broadie and Glasserman [17]

for a review of the method). Column CE is the price of the option for the executive,

the certainty equivalent, that we compute using equation (5.7). We compare the

values in the columns C0 and CE. The options are worth less for the executive

than for the company. However, the ratio will vary depending on the values of the

parameters. When the ratio of the price of the option for the executive and the price



of the option for the company is too low, options might not be the optimal form of

compensation. Finally, in the column denoted BS we record the Black and Scholes

price (Black and Scholes [18]). For volatility, we use the optimal initial volatility,

that is, v0 as given by equation (2.9) and reported in the column “Vol.”The only

extra parameter required for Table 6.3 is the interest rate of equation (5.1). For

concreteness, we assume that the interest rate is zero.

Table 6.3. Option price.

S0 = 100; T = 5

δ α λ K Vol C0 CE BS

0.1 0.25 0.5 102.02 0.52 0 0 0.0200.1 0.5 0.5 101.35 0.61 0.061 0 0.1180.1 0.75 0.5 104.40 3.39 0.173 0 1.3751 0.25 0.5 103.78 1.54 0.589 0.165 0.251 0.5 0.5 108.70 5.38 0.713 0.016 1.8071 0.75 0.5 143.14 33.31 1.720 0 17.521 0.25 0.25 103.40 1.48 0.323 0.237 0.2751 0.25 1 104.70 1.70 0.120 0.068 0.2061 0.25 2 107.88 2.31 0.216 0.002 0.1701 0.5 0.5 108.70 5.38 0.713 0.016 1.8071 0.5 2 109.16 5.56 0.685 0.010 1.8395 0.25 0.5 116.02 7.26 1.800 1.372 1.7045 0.5 0.5 142.98 26.66 3.655 0.110 11.7455 0.25 0.1 115.52 7.19 1.854 1.453 1.7405 0.25 0.25 115.70 7.22 1.762 1.412 1.7305 0.25 1 116.67 7.35 1.607 1.221 1.658

Table 6.3 complements nicely our conclusions from Table 6.1. It shows that

when the quality of the projects is very high (large α) and the executive is of a low

quality (small δ), the optimal strike price will be high in order to force the executive

to exercise more effort. As a result, the value of the option for the executive will be

low relative to the complete market price of the option. In that case it might not

be optimal to grant options as an incentive mechanism. In summary, it seems that

options are more appropriate the higher the quality of the executive and the lower

the quality of the projects.

We also observe that in the cases in which the ratio of the certainty equivalent

with respect to the price of the option is high, the Black and Scholes price is a

relatively good approximation to the true price of the option, with our choice of

volatility. However, in general, the quality of the approximation using Black and

Scholes is questionable.



6. The Case of Additional Cash Compensation

In this section we consider the case in which the executive receives, in addition to

the stock options, some cash compensation. The problem of the executive is to

select (u, v) ∈ A that maximize the criterion J defined by

J(u, v) := E

[logw + n(ST −K)+

− 1

2

∫ T

0

u2tdt

]. (6.1)

Here w > 0 represents some cash compensation independent of the price of the

stock of the company. The rest of the parameters are as in section 2.

Alternatively, this case can be interpreted as the situation in which the executive

has some wealth whose value is independent of the price of the stock. The main

difference between this case and the case considered in section 2 is that now the

number of options received in the compensation package becomes relevant. The

problem becomes considerably more difficult, but we can still compute numerically

the optimal effort and volatility. We include in this section the main result and

some numerical examples.

We start by considering the auxiliary function g defined by

g(t, y) :=(K − w

n

)N(d2(t, y)) + yN(d1(t, y)), (6.2)

where

d1(t, y) :=log yncw + 1

2α2(T − t)

α√T − t

, (6.3)

d2(t, y) :=log yncw − 1

2α2(T − t)

α√T − t

, (6.4)

and

N(x) :=

∫ x

−∞

1√2π

exp

−z

2

2

dz. (6.5)

We observe that

∂

∂yg(t, y) =

(K − w

n

) 1√2π

exp

−1

2(d2(t, y))2

1

y

1

α√T − t

+1√2π

exp

−1

2(d1(t, y))2

1

α√T − t

+N(d1(t, y)).

Let us consider the function f : [1,∞) 7→ R defined by

f(x) := logx − 1 −(nK

w− 1

)1

x.

We observe that f(1) = −nKw < 0, f(∞) = ∞, and f is strictly increasing (indeed,

for every y > 1: f ′(y) = 1y + (nKw − 1) 1

y2 > 1/y − 1/y2 > 0). Thus, there exists a

constant c = c(n,K,w) ∈ (1,∞) which satisfies the following two properties:

f(c) = 0 and y > c ⇐⇒ f(y) > 0. (6.6)



Let z be the positive solution of the nonlinear equation

S0 = g

(0,

1

z

)− zδ2T0. (6.7)

Theorem 6.1. Consider an executive who wants to solve (6.1). Consider the

stochastic process Z and the function T defined in section 2. Then the optimal

effort is given by

ut = δzZt (6.8)

and the optimal choice of projects is given by

vtSt = αHt∂

∂yg(t, Ht) + αzδ2ZtTt. (6.9)

Here, H is defined by

Ht =1

zZt. (6.10)

Proof. We want to find u and v that solve the problem

max(u,v)∈A

E

[F (ST ) −

∫ T

0

G(us)ds

],

where

F (s) = logw + n(s−K)+ and G(u) =u2

2.

We observe that the stochastic process Z of (2.4) satisfies

dZt = −αZtdWt and Z(0) = 1.

Applying the formula of integration by parts, we see that

ZtSt = S0 +

∫ t

0

Zsδusds+

∫ t

0

SsZs(vs − α)dWs.

Let us consider the stochastic process M defined by

Mt := ZtSt − δ

∫ t

0

Zsusds = S0 +

∫ t

0

SsZs(vs − α)dWs.

Obviously, M is a local martingale, but we would like to prove that M is also a

martingale. For that purpose, it is sufficient to verify the condition

E

[sup

0≤t≤T|Mt|

]<∞. (6.11)

According to the Burkholder-Davis-Gundy inequality (see, e.g., Theorem 3.3.28 of

Karatzas and Shreve [19]), it is sufficient to verify

E

(∫ T

0

(vt − α)2S2tZ

2t dt

)1/2 <∞. (6.12)



We observe that, according to Theorem 1.6.16 of Yong and Zhou [20],

E[sup0≤t≤T Z

2t

]< ∞. Since E[

∫ T0 |vtSt|2dt] < ∞ by Definition 2.1, that the-

orem applied to equation (2.1) or (2.2), gives E[sup0≤t≤T S

2t

]< ∞. Applying

Holder’s inequality (see, e.g., Theorem 4.2 of Chow and Teicher [21]) and again the

condition E[∫ T0 |vtSt|2dt] <∞, we note that

E

(∫ T

0

(vtStZt)2dt

)1/2 ≤ E

(

sup0≤t≤T

Z2t

∫ T

0

(vtSt)2dt

)1/2

= E

(

sup0≤t≤T

Z2t

)1/2(∫ T

0

(vtSt)2dt

)1/2

≤(E

[sup

0≤t≤TZ2t

])1/2(E

[∫ T

0

(vtSt)2 dt

])1/2

< ∞. (6.13)

This implies that

E

(∫ T

0

((vt − α)StZt)2dt

)1/2 <∞, (6.14)

and therefore M is a martingale. Thus,

E[MT ] = E[M0] = S0. (6.15)

Next, consider the dual function F : (0,∞) 7→ R defined by

F (z) = maxs≥0

[F (s) − sz].

The maximum is attained at the points of the form

s = s(z, a)

=

(K + max

(1

z− w

n, 0

))Ilogw+max( n

z −w,0)−zK−max(1−wzn ,0)>logw

+aIlogw+max( nz −w,0)−zK−max(1−wz

n ,0)=logw

=

(K + max

(1

z− w

n, 0

))Ilogw+nmax( 1

z −wn ,0)−zK−zmax( 1

z −wn ,0)>logw

+aIlogw+nmax( 1z −w

n ,0)−zK−zmax( 1z −w

n ,0)=logw

where a is either 0 or K + max(

1z − w

n , 0). Consider also the dual function G :

(0,∞) 7→ R defined by

G(z) = maxu

[−G(u) + δuz],

where the maximum is attained at

u = u(z) = δz.



By definitions, we get

E

[F (ST ) −

∫ T

0

G(us)ds

]≤ E

[F (zZT ) +

∫ T

0

G(zZs)ds

]+ zE[MT ].

Since M is a martingale, E[M(T )] = S0. Therefore the above inequality gives an

upper bound for our maximization problem. The upper bound will be attained if

the maximums are attained, and if E[MT ] = S(0). In other words, the optimal

solution is given by

ST = s(zZT , A) and ut = δzZt,

where A and z are chosen so that A is any FT measurable random variable taking

only two possible values, 0 and K + max(

1z − w

n , 0), and so that E[MT ] = S(0).

The optimal v is obtained from the martingale representation of

Mt = E[MT |Ft] = E

[ZT ST − δ

∫ T

0

Zsusds|Ft]

= S0 +

∫ t

0

(vs − α)SsZsdWs.

For the case α > 0 that we are considering, we will set A ≡ 0. Introduce the

stochastic process W defined by

Wt = Wt + αt.

We observe that W is a Brownian motion under the measure P defined by dP /dP =

ZT . Introduce also the notation

H(t) :=1

zZtand Mt := St − zδ2

∫ t

0

Zsds.

We observe that

H(t) =1

zexp

α2

2t+ αW (t)

=

1

zexp

−α

2

2t+ αW (t)

.

Thus,

dHt = HtαdW .

Hence, for every 0 ≤ t ≤ s:

Hs = Ht exp

−1

2α2(s− t) + α(Ws − Wt)

.

We also note that

St−zδ2∫ t

0

Zsds = S0+

∫ t

0

δusds+

∫ t

0

vsSsdWs−zδ2∫ t

0

Zsds = S0+

∫ t

0

vsSsdWs.

Thus, the stochastic process M is a P−martingale. Hence,

Mt = E[MT | Ft],



or equivalently

St = E

[s(zZT , 0) − zδ2

∫ T

t

Zsds | Ft].

We note that for every 0 ≤ t ≤ s:

Zs = exp

−α

2

2s− αW (s)

= exp

α2

2s− αW (s)

,

so for every 0 ≤ t ≤ s:

E[Zs| Ft] = E

[Zt exp

α2

2(s− t) − α(W (s) − W (t))

| Ft

]

= Zt expα2(s− t)

.

Denoting

Tt =eα

2(T−t) − 1

α2,

we obtain

St = E[s(zZT , 0)| Ft] − zδ2ZtTt.

Here,

s(zZT , 0) =

(K + max

(1

zZT− w

n, 0

))

× Ilogw+nmax( 1zZT

−wn ,0)−zZTK−zZT max( 1

zZT−w

n ,0)>logw

=(K + max

(H(T ) − w

n, 0))

× Ilogw+nmax(H(T )−wn ,0)− 1

H(T )K− 1

H(T )max(H(T )−w

n ,0)>logw.

Thus,

E[s(zZT , 0)| Ft]

= KP

logw + nmax(H(T ) − w

n, 0) − 1

H(T )K

− 1

H(T )max(H(T ) − w

n, 0) > logw| Ft

+E[

max(H(T ) − w

n, 0)

×Ilogw+nmax(H(T )−wn ,0)− 1

H(T )K− 1

H(T )max(H(T )−w

n ,0)>logw|Ft].



To continue with the computations, we are going to use c = c(n,K,w) ∈ (1,∞)

defined in (6.6). We note that

logw + nmax

(H(T ) − w

n, 0)

− 1

H(T )K

− 1

H(T )max

(H(T ) − w

n, 0)> logw

=

logw + nmax

(H(T ) − w

n, 0)

− 1

H(T )K

− 1

H(T )max

(H(T ) − w

n, 0)> logw

∩H(T ) − w

n> 0

⋃logw + nmax

(H(T ) − w

n, 0)

− 1

H(T )K

− 1

H(T )max

(H(T ) − w

n, 0)> logw

∩H(T ) − w

n≤ 0

=

logw + n

(H(T ) − w

n

)− 1

H(T )K − 1

H(T )

(H(T ) − w

n

)> logw

∩H(T ) − w

n> 0

⋃logw + n(0) − 1

H(T )K − 1

H(T )(0) > logw

∩H(T ) − w

n≤ 0

=

lognH(T ) − 1

H(T )K − 1 +

w

nH(T )> logw

∩H(T ) − w

n> 0

⋃logw − 1

H(T )K > logw

∩H(T ) − w

n≤ 0

=

log

nH(T )

w

>

nK

w(nH(T )w

) + 1 − w

nH(T )

∩

H(T ) − w

n> 0

=

f

(nH(T )

w

)> 0

∩H(T ) − w

n> 0

=

nH(T )

w> c

∩H(T ) − w

n> 0

=H(T ) > c

w

n

.



Thus,

E[s(zZT , 0)| Ft] = KPHT > c

w

n| Ft

+ E

[(HT − w

n

)IHT>c

wn | Ft

]

= K

∫ ∞

1α log cw

n1

Ht+ 1

2α(T−t)

1√2π(T − t)

exp

−1

2

x2

(T − t)

dx

+

∫ ∞

1α log cw

n1

Ht+ 1

2α(T−t)

(Ht exp

−1

2α2(T − t) + αx

− w

n

)

1√2π(T − t)

exp

−1

2

x2

(T − t)

dx.

To continue with the computations, we are going to return to the original probability

measure P . We can write

dP

dP= exp

αWT − 1

2α2T

=

H(T )

H(0)

and

Wt = Wt − αt.

We note that according to Bayes’ formula,

E[HT IHT>c

wn | Ft

]= exp

αWt −

1

2α2t

× E

HT

1

expαWT − 1

2α2TIHT>c

wn | Ft

= HtE[IHT>c

wn | Ft

]

= HtPHT > c

w

n| Ft

= HtP

Ht exp

α(WT −Wt) +

1

2α2(T − t)

> c

w

n| Ft

= HtP

α(WT −Wt) +

1

2α2(T − t) > log

1

Ht

cw

n

| Ft

.

We know that under P , the random variable WT − Wt has, given Ft, a normal

distribution with mean zero and variance T − t. Hence,

E[s(zZT , 0)| Ft] =(K − w

n

)PHT > c

w

n| Ft

+ E

[HT IHT>c

wn | Ft

]

=(K − w

n

)N(d2(t, Ht)) + HtN(d1(t, Ht)),

where

d1(t, Ht) :=log Htn

cw + 12α

2(T − t)

α√T − t

d2(t, Ht) :=log Htn

cw − 12α

2(T − t)

α√T − t

,



and

N(x) :=

∫ x

−∞

1√2π

exp

−z

2

2

dz.

This means that we can write

E[s(zZT , 0)| Ft] = g(t, Ht),

where the function g is defined by

g(t, y) := K

∫ ∞

1α log cw

n1y + 1

2α(T−t)

1√2π(T − t)

exp

−1

2

x2

(T − t)

dx

+

∫ ∞

1α log cw

n1y + 1

2α(T−t)

(y exp−1

2α2(T − t) + αx − w

n

)

1√2π(T − t)

exp

−1

2

x2

(T − t)

dx

=(K − w

n

)N(t, d2(y)) + yN(t, d1(y)).

Thus,

St = g(t, Ht) − zδ2ZtTt. (6.16)

If we take t = 0 in the above equation, we get

S0 = g(0, H0) − zδ2Z0T0 = g

(0,

1

z

)− zδ2T0. (6.17)

Although this does not give an explicit solution for z, it is possible to obtain it

numerically. Applying Ito’s formula in (6.16), and comparing equations (2.2) and

(6.16), we obtain

vtSt = αHt∂

∂yg(t, Ht) + αzδ2ZtTt. (6.18)

Here,

∂

∂yg(t, y) =

(K − w

n

) 1√2π

exp

−1

2(d2(t, y))2

1

y

1

α√T − t

+1√2π

exp

−1

2(d1(t, y))2

1

α√T − t

+N(d1(t, y)).

We observe that the u and v defined above are adapted stochastic processes with

E[∫ T0 |ut|2dt] < ∞ and E[

∫ T0 |vtSt|2dt] < ∞. Therefore, if w > 0 and α > 0, the

optimal effort u and optimal choice of projects v are given by equations (6.8)-(6.9).

From the discussion above, it is clear that both the optimal drift u and the

optimal choice of projects v are completely determined by z. This has to be found

numerically from equation (6.7). We present several examples in Table 6.4. We see

that the optimal effort increases with the number of options granted; it does not

change much when the strike price changes; and it may be decreasing or increasing

with respect to the time to maturity, depending on the number of options granted.



Table 6.4. Optimal effort.

S0 = 100, w = 50, δ = 0.5, α = 0.5

n K T z

1 90 1 0.00627435 90 1 0.01148991 90 5 0.00712965 90 5 0.01068221 100 1 0.00617455 100 1 0.01130051 100 5 0.00710885 100 5 0.01069291 110 1 0.00607845 110 1 0.01110761 110 5 0.00708675 110 5 0.0106937

7. Conclusions

We present a model where an executive is granted stock options as compensation.

His decisions can affect the dynamics of the stock of the company in two ways: first,

he can increase the expected return of the stock through his effort; second, he can

choose the level of risk of the stock price by selecting different projects. The effort

produces disutility, and higher level of risk will result in higher expected return. The

executive is risk averse. We obtain closed form solutions for both the optimal effort

and choice of projects. The company chooses the strike price of the call options.

The company cares both about the mean and the volatility of the stock price. We

find that there is a large range of optimal strike prices, depending on the values of

the parameters of the model, but for most interesting cases it will be optimal to

issue options out-of-the-money. We find that the optimal strike price is increasing

with the type of the executive and the emphasis on the mean stock price, rather

than volatility. The relationship between the optimal strike price and the quality

of the projects is not monotonic.

In our setting, we can price the options for the company (assuming complete

markets) and for the executive (through the certainty equivalent). Although options

are always worth less for the executive than for the company, we show that the ratio

is more favorable when the projects are of low quality and the executive is of high

quality.

In order to derive closed form solutions for the optimal policies of the executive

we have to assume that the utility is logarithmic and the only source of wealth of

the executive is the package of options granted as compensation. As a result, the

number of options is irrelevant. It is possible to extend our method to more general

utility functions, as done in the proof of Theorem 6.1. In that case, the optimal

solution would typically depend on the number of options. That would allow us to



address some important issues such as policies for resetting the option contracts.

However, it seems that in the general case we cannot get results as explicit as for

the logarithmic utility. We can also extend some of the results to the case in which

the executive receives cash, besides options. In this case, we can verify that the

number of options becomes relevant. However, a procedure to derive the optimal

strike price does not seem to be explicit. Another possible extension of the present

work is to include tax effects in the model.

Acknowledgements

The research of A. Cadenillas was supported by the Natural Sciences and Engi-

neering Research Council of Canada grant 194137. The research of J. Cvitanic

was supported in part by the National Science Foundation, under NSF grant DMS

04-03575.

References

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[15] P. Bolton and C. Harris. The continuous-time principal-agent problem: Frequent-monitoring contracts, (2001). Working Paper, Princeton University, Princeton.

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Chapter 7

The Freidlin-Wentzell LDP with Rapidly Growing Coefficients

Pavel Chigansky and Robert Liptser∗

Department of Mathematics, The Weizmann Institute of ScienceRehovot 76100, Israel

[email protected]

The Large Deviations Principle (LDP) is verified for a homogeneous diffusionprocess whose coefficients are locally Lipschitz functions with super linear growth.It is assumed that the drift is directed towards the origin and the growth ratesof the drift and diffusion terms are properly balanced. Nonsingularity of thediffusion matrix is not required.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

2 Notations and the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4 The proof of C-exponential tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

4.1 Auxiliary lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

4.2 The proof of (3.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

4.3 The proof of (3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

5 Local LDP upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

6 Local LDP lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

6.1 Nonsingular a(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

6.2 General a(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

A.1 Exponential estimates for martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

A.2 Pseudoinverse of nonnegative definite matrices . . . . . . . . . . . . . . . . . . . . . . 214

A.3 Exponential negligibility of Xε,βt − Xε

t . . . . . . . . . . . . . . . . . . . . . . . . . . 214

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

1. Introduction

In this paper we extend the set of conditions, under which Freidlin-Wentzell’s Large

Deviation Principle (LDP) for a homogeneous diffusion process remains valid. We

consider a family (Xεt )t≥0ε→0 of diffusions, where Xε

t ∈ Rd, d ≥ 1 is defined by

∗Department of Electrical Engineering Systems, Tel Aviv University, 69978 Tel Aviv, Israel,[email protected]

197


198 P. Chigansky and R. Liptser

the Ito equation

Xεt = x0 +

∫ t

0

b(Xεs )ds+ ε

∫ t

0

σ(Xεs )dBs, (1.1)

with respect to a standard Brownian motion Bt, where b(x) and σ(x) are vector

and matrix valued continuous functions of dimensions d and d × d respectively,

guaranteeing existence of the unique weak solution.

The classical Freidlin-Wentzell setting [1] (see also Dembo and Zeitouni [2]) is

applicable to the model (1.1) with bounded b(x) and σ(x) and uniformly positive def-

inite diffusion matrix a(x) = σσ∗(x). Various LDP versions can be found in Dupuis

and Ellis [3], Feng [4], Feng and Kurtz [5], Friedman [6], Liptser and Pukhalskii [7],

Mikami [8], Narita [9], Stroock [10], Ren and Zhang [11]. In the recent paper [12],

Puhalskii extends LDP to (1.1) with continuous and unbounded coefficients and sin-

gular a(x), assuming b(x) and a(x) are Lipschitz continuous functions (concerning

singular σ(x) see also Liptser et al, [13]). Being Lipschitz continuous, the entries of

b, σ grow not faster than linearly and, thereby, automatically guarantee one of the

necessary conditions for LDP (‖ · ‖ denotes the Euclidean norm in Rd)

limC→∞

limε→0

ε2 log P

(supt≤T

‖Xεt ‖ > C

)= −∞, ∀ T > 0. (1.2)

Relinquishing the linear growth condition for b, σ would require additional assump-

tions providing (1.2).This paper is inspired by Puhalskii’s remark in Ref. [12]:

If the drift is directed towards the origin, then no restrictions are needed on thegrowth rate of the drift coefficient.

In particular, in this case the LDP holds, regardless of the growth rate of b(x), for

a constant diffusion matrix (not necessarily nonsingular).

In this paper, we show that in fact LDP remains valid for (1.1) with non-constant

diffusion term, if its growth rate is properly balanced relatively to the drift (see

(2.1) of Theorem 2.1 below). Our result is formulated in terms of Khasminskii-

Veretennikov’s condition (2.1) (see Refs. [14–16])

The rest of the paper is organized as follows. In Sections 2 and 3, the main

result, notations and preliminary facts on the LDP are given. Sections 4 - 6 contain

the proof of the main result. Auxiliary technical details are gathered in Appendices

A.1 - A.3.

2. Notations and the main result

The following notations and conventions are used throughout the paper.

- ∗ denotes the transposition symbol

- all vectors are columns (unless explicitly stated otherwise)

- |x| and ‖x‖ denote the `1 and `2 (Euclidean) norms of x ∈ Rd


The Freidlin-Wentzell LDP with Rapidly Growing Coefficients 199

- (x, y) denotes the scalar product of x, y ∈ Rd

- ‖x‖2Γ = (x,Γx) with an nonnegative definite matrix Γ

- a(x) = σ(x)σ∗(x)

- a⊕(x) denotes the Moore-Penrose pseudoinverse matrix of a(x) (see Ref. [17])

- ∇V (x) is the gradient (row) vector of V (x):

∇V (x) :=(∂V (x)

∂x1, . . . ,

∂V (x)

∂xd

)

- 〈M,N〉t is the joint quadratic variation process of continuous martingales Mt

and Nt; for brevity 〈M,M〉t = 〈M〉t- a.s. abbreviates “almost surely”; when the corresponding measure is not spec-

ified the Lebesgue measure on R+ is understood

- % is the locally uniform metric on C[0,∞)(Rd)

- I denotes d× d identity matrix

- the convention 0/0 = 0 is kept throughout

- Xε = (Xεt )t≥0

- inf∅ = ∞.

We study the LDP for the family Xεε→0 in the metric space (C[0,∞)(Rd), %) with

%(x, y) =∑∞

k=1 2−k(1 ∨ supt≤k ‖xt − yt‖

), x, y ∈ C[0,∞)(R

d). Recall that Xεε→0

satisfies the LDP with the good rate function J(u) : C[0,∞)(Rd) 7→ [0,∞] and the

rate ε2, if the level sets of J(u) are compact and for any closed set F and open set

G in C[0,∞)(Rd),

limε→0

ε2 log P(Xε ∈ F

)≤ − inf

u∈FJ(u),

limε→0

ε2 log P(Xε ∈ G

)≥ − inf

u∈GJ(u).

Our main result is

Theorem 2.1. Assume:

(H-1) the entries of b(x) and σ(x) are locally Lipschitz continuous functions,

(H-2) lim‖x‖→∞(x, b(x))

‖x‖ = −∞,

(H-3) for some positive constants K and L,(x, a(x)x)

‖x‖ |(x, b(x))| ≤ K, ∀ ‖x‖ > L.

Then Xεt ε→0 obeys the LDP in the metric space (C[0,∞)(R

d), %) with the rate ε2

and the rate function

J(u) =

12

∫∞0 ‖ut − b(ut)‖2

a⊕(ut)dt, u ∈ Γ

∞, u 6∈ Γ,

where

Γ =u ∈ C[0,∞) :

u0 = x0, dut dt,∫∞0 ‖ut‖2dt <∞

a(ut)a⊕(ut)[ut − b(ut)] = [ut − b(ut)] a.s.

.



Remark 2.2. In the scalar case (recall 0/0=0)

J(u) =

1

2

∫∞0

(ut − b(ut))2

σ2(ut)dt, dut = utdt, u0 = x0,

∫∞0 u2

tdt <∞

∞, otherwise.

Example 2.3. A typical example within the scope of Theorem 2.1 is

Xεt = x0 −

∫ t

0

(Xεs )3ds+ ε

∫ t

0

|Xεs |3/2dBs.

3. Preliminaries

We follow the framework, set up by A. Puhalskii (see Refs. [18, 19]):

Exponential tightness

Local LDP

⇐⇒ LDP

The exponential tightness in the metric space (C[0,∞), %) is convenient to verify in

terms of, so called, C-exponential tightness conditions introduced by A. Puhalskii

(see e.g. Ref. [7]), based on the stopping times technique introduced by D. Aldous

in Refs. [20, 21]). To this end, let us assume that the diffusion processes are defined

on a stochastic basis (Ω,F ,Fε = (F εt )t≥0,P), satisfying the usual conditions, where

the filtration Fε may depend on ε.

Recall (see Ref. [7]) that the family of diffusion processes is C-exponentially tight

if for any T > 0, η > 0 and any Fε-stopping time θ,

limC→∞

limε→0

ε2 log P

(supt≤T

‖Xεt ‖ > C

)= −∞, (3.1)

lim4→0

limε→0

ε2 log supθ≤T

P

(supt≤4

‖Xεθ+t −Xε

θ‖ > η)

= −∞. (3.2)

The family of diffusion processes obeys the local LDP in (C[0,∞)(Rd), %) if for any

T > 0 there exists a local rate function JT (u) such that

limδ→0

limε→0

ε2 log P

(supt≤T

‖Xεt − ut‖ ≤ δ

)≤ −JT (u) (3.3)

limδ→0

limε→0

ε2 log P

(supt≤T


)≥ −JT (u). (3.4)

Under the conditions (3.1)-(3.4), the family of diffusion processes obeys the LDP

with the rate ε2 and the good rate function

J(u) = supTJT (u), u ∈ C[0,∞)(R

d),

where

JT (u) =

12

∫ T0 ‖ut − b(ut)‖2

a⊕(ut)dt, u ∈ ΓT

∞, u 6∈ ΓT ,



with

ΓT =u ∈ C[0,T ] :

u0 = x0, dut dt,∫ T0‖ut‖2dt <∞

a(ut)a⊕(ut)[ut − b(ut)] = [ut − b(ut)] a.s.

.

Thus the proof of Theorem 2.1 reduces to establishing (3.1)–(3.4).

4. The proof of C-exponential tightness

4.1. Auxiliary lemma

Let D be a nonlinear operator acting on continuously differentiable functions V (x) :

Rd→R as follows:

DV (x) = (∇V (x), b(x)) +1

2(∇V (x), a(x)∇V (x)).

Lemma 4.1. Assume there exists twice continuously differentiable nonnegative

function V (x) such that

(a-1) limC→∞ inf‖x‖≥C V (x) = ∞(a-2) for some L > 0, DV (x) ≤ 0, ∀ ‖x‖ > L.

Then (3.1) holds.

Proof. Notice that (3.1) is equivalent to

limC→∞

limε→0

ε2 log P(ΘC ≤ T

)= −∞, (4.1)

where

ΘC = inft : ‖Xεt ‖ ≥ C, C > 0 (4.2)

are stopping times relative to Fε.

We use (A.1.1) of Proposition A.1.1 to estimate log P(ΘC ≤ T ). An appropriate

martingale M εt is constructed with the help of function V (x). Let Ψ(x) be the

Hessian of V , namely a matrix with the entries Vij(x) = ∂2V (x)∂xi∂xj

. By the Ito formula

ε−2V (XεΘC∧t) = ε−2V (x0) +

∫ ΘC∧t

0

ε−2(∇V (Xεs ), b(Xε

s ))ds

+

∫ ΘC∧t

0

ε−1(∇V (Xεs ), σ(Xε

s )dBs) +

∫ ΘC∧t

0

1

2trace

(Ψ(Xε

s )a(Xεs ))ds.

We choose M εt =

∫ t0ε−1(∇V (Xε

s ), σ(Xεs )dBs), which has the variation process

〈Mε〉t =∫ t0ε−2(∇V (Xε

s ), a(Xεs )∇V (Xε

s ))ds. Clearly

MεΘβ

C∧t = ε−2V (XεΘC∧t) − ε−2V (x0)

−∫ ΘC∧t

0

ε−2(∇V (Xεs ), b(Xε

s ))ds−∫ ΘC∧t

0

1

2trace

(Ψ(Xε

s )a(Xεs ))ds.



Hence, by the definition of D, one gets

MεΘC∧T − 1

2〈Mε〉ΘC∧T = ε−2V (Xε

ΘC∧T ) − ε−2V (x0)

−∫ ΘC∧T

0

1

2trace

(Ψ(Xε

s )a(Xεs ))ds −

∫ ΘC∧T

0

ε−2DV (Xεs )ds. (4.3)

On the set ΘC ≤ T, we have

ε−2V (XεΘC∧T ) − ε−2V (x0) ≥ ε−2 inf

‖x‖≥CV (x) − ε−2V (x0),

and∣∣∣∫ ΘC∧T

0

1

2trace

(Ψ(Xε

s )a(Xεs ))ds∣∣∣ ≤ T

2sup

‖x‖≤C

∣∣ trace(Ψ(x)a(x)

)∣∣,

and, by (a-2),

−∫ Θ∧T

0

ε−2DV (Xεs )ds

≥ −∣∣∣∫ ΘC∧T

0

ε−2I‖Xεs‖≤LDV (Xε

s )ds∣∣∣ ≥ −ε2T sup

‖x‖≤L|DV (x)|.

These inequalities and (4.3) imply

MεΘC

− 1

2〈Mε〉ΘC ≥ ε−2 inf

‖x‖≥CV (x) − ε−2V (x0)

− T

2sup

‖x‖≤C


)∣∣− ε−2T sup‖x‖≤L

∣∣DV (x)∣∣

on the set ΘC ≤ T. Hence, due to (A.1.1) of Proposition A.1.1

ε2 log P(ΘC ≤ T

)≤

− inf‖x‖≥C

V (x) + V (x0) +Tε2

2sup

‖x‖≤C


)∣∣+ T sup‖x‖≤L

∣∣DV (x)∣∣

−−−→ε→0

− inf‖x‖≥C

V (x) + V (x0) + T sup‖x‖≤L

∣∣DV (x)∣∣

and it remains to recall that by (a-1) limC→∞ inf‖x‖≥C V (x) = ∞.

4.2. The proof of (3.1)

We apply Lemma 4.1 to

V (x) =c‖x‖2

1 + ‖x‖ ,

with a positive parameter c ≤ 1K for K from (2.1) of Theorem 2.1. The function

V (x) is twice continuously differentiable and satisfies (a-1). It is left to show that

V (x) satisfies (a-2) as well.



Direct computations give ∇V (x) = c (2+‖x‖)‖x‖(1+‖x‖)2

x‖x‖ . Denote

r(x) :=(2 + ‖x‖)‖x‖

(1 + ‖x‖)2

and notice that r(x) ≤ 1. By assumption (2.1) of Theorem 2.1, one can choose

L > 0 sufficiently large so that (x, b(x)) < 0 for any ‖x‖ ≥ L. On the other hand,

by assumption (2.1) of Theorem 2.1, −1 + c2

(x,a(x)x)‖x‖ |(x,b(x))| ≤ − 1

2 for ‖x‖ ≥ L and

DV (x) =

(cr(x)

‖x‖ (x, b(x)) +c2r2(x)

2

(x, a(x)x)

‖x‖2

)

=

(− c

r(x)

‖x‖∣∣(x, b(x))

∣∣+c2r2(x)

2

(x, a(x)x)

‖x‖2

)

= cr(x)|(x, b(x))|

‖x‖

(− 1 +

c

2r(x)

(x, a(x)x)

‖x‖ |(x, b(x))|

)

≤ cr(x)|(x, b(x))|

‖x‖

(− 1 +

c

2

(x, a(x)x)

‖x‖ |(x, b(x))|

)

≤ −1

2cr(x)

|(x, b(x))|‖x‖

and (a-2) follows.

4.3. The proof of (3.2)

The obvious inclusion

supt≤4

‖Xεθ+t −Xε

θ‖ > η⊆

supt≤4

‖Xεθ+t −Xε

θ‖ > η, ΘC = ∞⋃

ΘC ≤ T

reduces the proof to verifying

lim4→0

limε→0

ε2 log supθ≤T

P

(supt≤4

‖Xεθ+t −Xε

θ‖ > η, ΘC = ∞)

= −∞ (4.4)

for any fixed C. Indeed if (4.4) holds, then

lim4→0

limε→0

ε2 log supθ≤T

P

(supt≤4

‖Xεθ+t −Xε

θ‖ > η)

≤ lim4→0

limε→0

ε2 log supθ≤T

P

(supt≤4

‖Xεθ+t −Xε

θ‖ > η, ΘC = ∞)

∨limC→∞

limε→0

ε2 log P(ΘC ≤ T)

and, thus, (3.2) is implied by (4.4) and (4.1). So, it is left to check (4.4) for any

entry xεt of Xεt :

lim4→0

limε→0

ε2 log supθ≤T

P

(supt≤4

|xεθ+t − xεθ| > η, ΘC = ∞)

= −∞.



An entry of Xεt satisfies

xεt = xε0 +

∫ t

0

γεsds+ εmεt ,

where γεt is Fε-adapted continuous random process and mt is Fε-continuous martin-

gale with 〈mε〉t =∫ t0µεsds. Since b and σ are locally Lipschitz continuous functions,

there is a constant lC , such that |γεΘC∧t| ≤ lC and µεΘC∧t ≤ lC . Taking into account

that

supt≤4

∣∣∣∫ θ+t

θ

γεsds∣∣∣ ≥ η, ΘC = ∞

⊆lC4 ≥ η

= ∅, for 4 < η/lC ,

it is left to verify

lim4→0

limε→0

ε2 log supθ≤T

P

(supt≤4

|εmεθ+t − εmε

θ| > η, ΘC = ∞)

= −∞.

Due to the obvious inclusion

supt≤4

|εmεθ+t − εmε

θ| > η, ΘC = ∞

=

supt≤4

|εmεΘC∧(θ+t) − εmε

ΘC∧θ| > η, ΘC = ∞

⊆

supt≤4


ΘC∧θ| > η,

we shall verify

lim4→0

limε→0

ε2 log supθ≤T

P

(supt≤4


ΘC∧θ| > η)

= −∞.

Notice that nεt := εmεΘC∧(θ+t) − εmε

ΘC∧θ is a continuous martingale relative to

(F εΘC∧θ+t)t≥0 (see e.g. Ch. 4, §7 in Ref. [22]) with 〈nε〉t = ε2

∫ ΘC∧(θ+t)

ΘC∧θ µεsds ≤ε2lCt. By the statement (A.1.1) of Proposition A.1.1, P

(supt≤4 |nεt | ≥ η

)≤

2e−η2/(2lCε

24), so that limε→0 ε2 log P

(supt≤4 |nεt | ≥ η

)≤ − η2

2lC4 −−−→4→0

−∞.

5. Local LDP upper bound

We start with the observation that (3.3) holds if for any T > 0

limδ→0

limε→0

ε2 log P

(supt≤T

‖Xεt − ut‖ ≤ δ, ΘC = ∞

)≤ −JT (u), (5.1)

since by the inclusion

supt≤T


⊆

supt≤T

‖Xεt − ut‖ ≤ δ, ΘC = ∞

⋃ΘC ≤ T



we have

limδ→0

limε→0

ε2 log P

(supt≤T


)

≤ limδ→0

limε→0

ε2 log P

(supt≤T

‖Xεt − ut‖ ≤ δ, ΘC = ∞

)∨limε→0

ε2 log P(ΘC ≤ T),

and, by (4.1), the last term goes to −∞ as C→∞.

The proof for u0 6= x0 or dut 6 dt is standard (see e.g. Ref. [2]) and is omitted.

The rest of the proof is split into two steps.

Step 1: u0 = x0, dut dt,∫ T

0‖us‖2ds <∞. Define the set

A =

supt≤T

‖Xεt − ut‖ ≤ δ, ΘC = ∞

.

With a continuously differentiable vector-valued function λ(s) of dimension d, let

us introduce a continuous local martingale Ut =∫ t0

(λ(s), εσ(Xεs )dBs) and its mar-

tingale exponential zt = eUt−0.5〈U〉t , where

〈U〉t =

∫ t

0

ε2(λ(s), a(Xεs )λ(s))ds.

It is well known that zt is a continuous positive local martingale, as well as a

supermartingale. Consequently, EzT ≤ 1 and, therefore,

1 ≥ EIAzT . (5.2)

The required upper bound for P(A) is obtained by estimating zT from below on A.

Since Ut =∫ t0 (λ(s), dXε

s − b(Xεs )ds),

UT − 0.5〈U〉T =∫ T

0

[(λ(s), dXε

s − b(Xεs )ds) − ε2

2(λ(s), a(Xε

s )λ(s))ds]

=

∫ T

0

[(λ(s), us − b(us)) −

ε2

2(λ(s), a(us)λ(s))

]ds

+

∫ T

0

(λ(s), dXεs − usds) +

∫ T

0

(λ(s), b(us) − b(Xεs )ds

+

∫ T

0

ε2

2(λ(s), [a(us) − a(Xε

s )]λ(s))ds.

(5.3)

We derive lower bounds on the set A for each term in the right hand side of (5.3).

Applying the Ito formula to (λ(t), Xεt − ut), and taking into account that Xε

0 = u0,

we find that

(λ(T ), XεT − uT ) =

∫ T

0

(λ(s), dXεs − usds) +

∫ T

0

(λ(s), Xεs − us)ds.



Therefore,

∫ T

0

(λ(s), dXεs − usds)

≥ −∣∣∣(λ(T ), Xε

T − uT )∣∣∣−∣∣∣∫ T

0

(λ(s), Xεs − us)ds

∣∣∣ ≥ −r1δ,

with r1 := r1(λ, T, C) ≥ 0, independent of ε.

Further, with ri := ri(λ, T, C) ≥ 0, i = 2, 3, due to the local Lipschitz continuity

of σ and a, we find that∫ T

0

(λ(s), b(us) − b(Xεs ))ds ≥ −r2(λ,C, T )δ

∫ T

0

ε2

2(λ(s), [a(us) − a(Xε

s )]λ(s))ds ≥ −ε2r3(λ,C, T )δ.

Hence with r := r1 + r2 + ε2r3,

log zT ≥∫ T

0

[(λ(s), us − b(us)) −

ε2

2(λ(s), a(us)λ(s))

]ds− r(λ, T, C)δ.

Set ν(s) = ε2λ(s) and rewrite the above inequality as:

log zT ≥ 1

ε2

∫ T

0

[(ν(s), us − b(us)) −

1

2(ν(s), a(us)ν(s))

]ds− r

( νε2, T, C

)δ.

This lower bound, along with (5.2), provides the following upper bound

ε2 log P(A)≤ −

∫ T

0

[(ν(s), us − b(us)) −

1

2(ν(s), a(us)ν(s))

]ds

+ ε2r( νε2, T, C

)δ.

Clearly limε→0 ε2r(νε2 , T, C

)<∞ and, hence,

limδ→0

limε→0

ε2 log P(A)≤ −

∫ T

0

[(ν(s), us − b(us)) −

1

2(ν(s), a(us)ν(s))

]ds. (5.4)

Since the left hand side of (5.4) is independent of ν(s), (5.1) is derived by minimizing

the right hand side of (5.4) with respect to ν(s). Two difficulties arise on the way

to direct minimization:

- the matrix a(us) may be singular

- the entries of ν(s) should be continuously differentiable functions.

Assume first a(us) is a positive definite matrix, uniformly in s, and write

(ν(s), us − b(us)) −1

2(ν(s), a(us)ν(s)) =

1

2‖us − b(us)‖2

a−1(us)

− 1

2

∥∥∥a1/2(us)(ν(s) − a−1(us)[us − b(us)]

)∥∥∥2

.



If the entries of a−1(us)[us − b(us)] are continuously differentiable functions, then,

by taking ν(s) ≡ −a−1(us)[us − b(us)] we find that

limδ→0

limε→0

ε2 log P(A)≤ −1

2

∫ T

0

‖us − b(us)‖2a−1(us)ds. (5.5)

In the general case, due to∫ T0 ‖us‖2ds < ∞, the entries of a−1(us)[us − b(us)]

are square integrable with respect to the Lebesgue measure on [0, T ]. Choose a

maximizing sequence νn(s), n ≥ 1, of continuously differentiable functions such

that limn→∞∫ T0

∥∥νn(s)− a−1(us)[us− b(us)]∥∥2ds = 0. Since all the entries of a(us)

are uniformly bounded on [0, T ]

limn→∞

∫ T

0

∥∥a1/2(us)(νn(s) − a−1(us)[us − b(us)]

)∥∥2ds = 0

and (5.5) holds too.

Now we drop the uniform nonsingularity assumption of a(us). The upper bound

in (5.5) remains valid with a(us) replaced by aβ(us) ≡ a(us) + βI, where β is a

positive number and I is (d× d)-unit matrix:

limδ→0

limε→0

ε2 log P(A)≤ −1

2

∫ T

0

‖us − b(us)‖2[a(us)+βI]−1ds.

For any fixed s, the function ‖us − b(us)‖2[a(us)+βI]−1 increases with β ↓ 0 and

by Lemma A.2.1 possesses the limit

limβ→0

‖us − b(us)‖2[a(us)+βI]−1 =

‖us − b(us)‖2a⊕(us),

a(us)a⊕(us)[us − b(us)]

= [us − b(us)]

∞, otherwise.

Thus the required upper bound

limδ→0

limε→0

ε2 log P(A)≤

−∫ T0

12‖us − b(us)‖2

a⊕(us)ds,a(us)a

⊕(us)[us − b(us)]

= [us − b(us)], a.s.

∞, otherwise

follows by the monotone convergence theorem.

Step 2. u0 = x0, dut dt,∫ T

0‖us‖

2ds = ∞.

We emphasize that dut dt on [0, T ] implies∫ T0‖us‖ds < ∞ and return to

the upper bound from (5.4). Since b and σ are locally Lipschitz, one can choose a

constant L (depending on u(s)), so that, |(ν(s), b(us))| ≤ ‖b(us)‖‖ν(s)‖ ≤ L‖ν(s)‖and (ν(s), a(us)ν(s)) ≤ L‖ν(s)‖2. Then, (5.4) implies

limδ→0

limε→0

ε2 log P(A)≤ −

∫ T

0

[(ν(s), us) − L‖ν(s)‖ − L

2‖ν(s)‖2

]ds.



Let νn(s) be a sequence of continuously differentiable functions, approximating

the bounded (for each fixed p > 0) function L−1usI‖us‖≤p in the sense that

limn→∞∫ T0 ‖ 1

L usI‖us‖≤p − νn(s)‖2ds = 0. Thus,

limδ→0

limε→0

ε2 log P(A)≤ − 1

2L

∫ T

0

∥∥us∥∥2I‖us‖≤pds

︸︷︷︸↑∞ as p ↑ ∞

+

∫ T

0

∥∥us∥∥ds

︸︷︷︸<∞

−−−→p→∞

−∞

6. Local LDP lower bound

If limδ→0 limε→0 ε2 log P

(supt≤ΘC∧T |Xε

t − ut| ≤ δ)≤ −JT (u) = −∞, then the

corresponding local LDP lower bound is −∞ as well and hence only the case JT (u) <

∞ is to be considered, i.e. we may restrict ourselves to analyzing test functions with

the properties:

(i) u0 = x0

(ii) dut dt

(iii) a(ut)a⊕(ut)[ut − b(ut)] = [ut − b(ut)] a.s.

(iv)

∫ T

0

‖ut − b(ut)‖2a⊕(ut)

dt <∞, ∀ T > 0

(v)

∫ T

0

‖ut‖2dt <∞.

(6.1)

Another helpful observation is that (3.4) holds if for any C > 0

limδ→0

limε→0

ε2 log P

(sup

t≤ΘC∧T‖Xε

t − ut‖ ≤ δ)≥ −JT (u) (6.2)

due to

supt≤ΘC∧T


⊆

supt≤T


⋃ΘC ≤ T

and (4.1).

6.1. Nonsingular a(x)

In this section, the matrix a(x) is assumed to be uniformly nonsingular in x ∈ R, in

the sense that a(x) ≥ βI for a positive number β. Let λ(s) := σ−1(Xεs )[us− b(Xε

s )]

and introduce a martingale Ut =∫ ΘC∧t0

1ε (λ(s), dBs) and its martingale exponential

zt = eUt−0.5〈U〉t , t ≤ T , where 〈U〉t =∫ ΘC∧t0

1ε2 ‖λ(s)‖2ds.

By (iv) and (v) of (6.1), 〈U〉T ≤ const. and hence EzT = 1. We use this fact in

order to define a new probability measure Qε by dQε = zT dP. Since zT is positive

P-a.s., P Qε as well and dP = z−1T dQε.



We proceed with the proof of (6.2) by applying

P(A) =

∫

A

z−1T dQε (6.3)

to the set A =

supt≤ΘC∧T ‖Xεt − ut‖ ≤ δ

, and estimating from below the right

hand side in (6.3). In order to realize this program, it is convenient to have a

semimartingale description of the process XεΘC∧t under Qε. Recall that the random

process BΘC∧t is a martingale under P with the variation process 〈B〉ΘC∧t ≡ (ΘC ∧t)I. It is well known (see e.g. Theorem 2, Ch. 4, §5 in Ref. [22]) that BΘC∧t is

a continuous semimartingale under Qε with the decomposition BΘC∧t = Bt + ABt ,

where Bt is a martingale (under Qε) with 〈B〉t ≡ 〈B〉ΘC∧t and, by the Girsanov

theorem,

ABt =

∫ ΘC∧t

0

1

εσ−1(Xε

s )[us − b(Xεs )]ds.

In particular,

XεΘC∧t = uΘC∧t + ε

∫ ΘC∧t

0

σ(Xεs )dBs, t ≤ T, Qε-a.s.

As the next preparatory step we derive the semimartingale decomposition of Utunder Qε. As before, the continuous martingale Ut under P is transformed to a

semimartingale under Qε:

Ut = Ut +AUt

with continuous Qε-martingale Ut, having the variation process 〈U〉t ≡ 〈U〉t, P- and

Qε-a.s., and a continuous drift AUt ≡ 〈U〉t.Thus, Ut = Ut + 〈U〉t, t ≤ T, Qε-a.s. and, thereby, z−1

T = e−UT − 12 〈U〉T . Conse-

quently, (6.3) is transformed to

P(A) =

∫

A

exp(− UT − 1

2〈U〉T

)dQε

=

∫

A

exp(− UT − 1

2ε2

∫ ΘC∧T

0

‖us − b(Xεs )‖2

a−1(Xεs )ds

)dQε.

We are now in the position to derive a lower bound for the right hand side. Replacing

A with a smaller set A ∩ B, where B =∣∣ε2UT

∣∣ ≤ η

, write

P(A) ≥∫

A∩B

exp(− η

ε2− 1

2ε2

∫ ΘC∧T

0

‖us − b(Xεs )‖2

a−1(Xεs )ds

)dQε.

By the local Lipschitz continuity of b, σ and the uniform nonsingularity of a(x),

∣∣∣‖us − b(Xεs )‖2

a−1(Xεs ) − ‖us − b(us)‖2

a−1(us)

∣∣∣ ≤ lC(‖us‖ + 1)2δ, δ ≤ 1,



on the set A ∩ B for any s ≤ ΘC ∧ T . Then,

P(A) ≥∫

A∩B

exp(− η

ε2− δlC

ε2

∫ T

0

(‖us‖ + 1)2ds

− 1

2ε2

∫ ΘC∧T

0

‖us − b(us)‖2a−1(us)ds

)dQε

≥∫

A∩B

exp(− η

ε2− δlC

ε2

∫ T

0

(‖us‖ + 1)2ds

− 1

2ε2

∫ T

0

‖us − b(us)‖2a−1(us)ds

)dQε.

Consequently,

limε→0

ε2 log P(A) ≥ −η − δlC

∫ T

0

(‖us‖ + 1)2ds− JT (u) + limε→0

ε2 log Qε(A ∩ B

).

We prove now that limε→0 ε2 log Qε

(A ∩ B

)= 0 by showing

limε→0

Qε(Ω \ A

)= 0 and lim

ε→0Qε(Ω \ B

)= 0.

To this end, recall that

Ω \ A =ε supt≤T

∥∥∥∫ ΘC∧t

0

σ(Xεs )dBs

∥∥∥ > δ

Ω \ B =ε∥∥∥∫ ΘC∧T

0

σ−1(Xεs )[us − b(Xε

s )]dBs

∥∥∥ > η.

(6.4)

We verify (6.4) componentwise. Let Lεt denote any entry of∫ ΘC∧t0 σ(Xε

s )dBs or∫ ΘC∧t0 σ−1(Xε

s )[us − b(Xεs )]dBs. We show that

limε→0

Qε(ε supt≤T

∣∣Lεt∣∣ > δ

)= 0 and lim

ε→0Qε(ε∣∣LεT

∣∣ > δ)

= 0. (6.5)

In both cases, Lεt is a continuous Qε-martingale with 〈Lε〉t =∫ t0g(s)ds and∫

Ω

∫ T0g(s)dsdQε <∞. Then (6.5) holds by Doob’s inequality:

limε→0

Qε(ε supt≤T

∣∣Lεt∣∣ > δ

)≤ 4ε2

δ2

∫

Ω

∫ T

0

g(s)dsdQε −−−→ε→0

0.

Now, for any fixed δ and η,

limε→0

ε2 log P(A) ≥ −η − δlC

∫ T

0

(‖us‖ + 1)2ds− JT (u).

The required lower bound

limδ→0

limε→0

ε2 log P(A) ≥ −JT (u)

follows by taking limη→0 limδ→0.



6.2. General a(x)

This part of the proof requires perturbation arguments. The idea is to use the

already obtained local LDP lower bound for the uniformly nonsingular a(x). Let

Wt be a standard d dimensional Brownian motion, independent of Bt, defined on the

same stochastic basis. Since b and σ are assumed to be locally Lipschitz continuous,

one can introduce the perturbed diffusion process controlled by a free parameter

β ∈ (0, 1]:

Xε,βt = x0 +

∫ t

0

b(Xε,βs )ds+ ε

∫ t

0

[σ(Xε,βs )dBs +

√βdWs]. (6.6)

The processXε,βt , defined in (6.6), solves the Ito equationXε,β

t = x0+∫ t0 b(X

ε,βs )ds+

ε∫ t0 [a(Xε,β

s ) + βI]1/2dBβs with respect to a standard Brownian motion Bβt =∫ t0 [a(Xε,β

s )+βI]−1/2[σ(Xε,βs )dBs+

√βdWs]. Then the family (Xε,β

t )t≤T ε→0 satis-

fies the local LDP lower bound. Indeed, the matrix aβ(x) is uniformly nonsingular,

its entries are locally bounded and satisfy the assumption (2.1) of Theorem 2.1 since

(x, aβ(x)x)

‖x‖ |(x, b(x))| =(x, a(x)x)

‖x‖ |(x, b(x))| + β‖x‖

|(x, b(x))|

and ‖x‖|(x,b(x))| converges to zero as ‖x‖→∞ by (2.1). In particular, with ΘβC = inft :

‖Xε,βt ‖ ≥ C and u0 = x0, dut dt,

∫ T0 ‖ut‖2dt <∞, we have

limδ→0

limε→0

ε2 log P

(sup

t≤ΘβC∧T

‖Xε,βt − ut‖ ≤ δ

)≥

− 1

2

∫ T

0

‖us − b(us)‖2(a(us)+βI)−1ds. (6.7)

Further, we will use (6.7) to establish

limδ→0

limε→0

ε2 log P

(supt≤T


)≥ −1

2

∫ T

0

‖us − b(us)‖2a⊕(us)ds. (6.8)

To this end, we introduce the filtration Gε = (G εt )t≥0, with the general condi-

tions, generated by (Xεt , X

ε,βt )t≥0 and notice that both ΘC (see (4.2)) and ΘβC are

stopping times relative to Gε. Hence,

τβC = ΘC ∧ ΘβC (6.9)

is a stopping time as well relative to Gε. Obviously,

limC→∞

limε→0

ε2 log P(τβC ≤ T

)= −∞.

However, the proof of (6.8) requires a stronger property:

limC→∞

limε→0

ε2 log supβ∈(0,1]

P

(τβC ≤ T

)= −∞. (6.10)



It is clear, that (6.10) is valid if it is valid with τβC replaced by ΘβC . The latter

is verified along the lines of Lemma 4.1 proof:


P(ΘβC ≤ T

)≤ − inf

‖x‖≥CV (x) + V (x0)

+Tε2

2sup

β∈(0,1]

sup‖x‖≤C

∣∣ trace(Ψ(x)[a(x) + βI]

)∣∣+ T supβ∈(0,1]

sup‖x‖≤L

∣∣DβV (x)∣∣

−−−→ε→0

− inf‖x‖≥C

V (x) + V (x0) + T supβ∈(0,1]

sup‖x‖≤L

∣∣DβV (x)∣∣ −−−−→C→∞

−∞,

where DβV (x) = (∇V (x), b(x)) + 12 (∇V (x), aβ(x)∇V (x)).

We are now in the position to prove (6.8). With δ ≤ β1/4, write

sup

t≤τβC∧T


=

supt≤τβ

C∧T‖Xε,β

t − ut‖ ≤ δ⋂

supt≤τβ

C∧T‖Xε

t −Xε,βt ‖ ≤ β1/4

⋃sup

t≤τβC∧T


⋂sup

t≤τβC∧T

‖Xεt −Xε,β

t ‖ > β1/4

⊆

supt≤τβ

C∧T‖Xε,β

t − ut‖ ≤ β1/4⋂

supt≤τβ

C∧T‖Xε

t −Xε,βt ‖ ≤ β1/4

⋃sup

t≤τβC∧T

‖Xεt −Xε,β

t ‖ > β1/4

⊆

supt≤τβ

C∧T‖Xε

t − ut‖ ≤ 2β1/4⋃

supt≤τβ

C∧T‖Xε

t −Xε,βt ‖ > β1/4

⊆

supt≤T

‖Xεt − ut‖ ≤ 2β1/4

⋃sup

t≤τβC∧T

‖Xεt −Xε,β

t ‖ > β1/4

⋃τβC ≤ T

Hence,

P

(sup

t≤τβC∧T


)≤ 3

P

(supt≤T

‖Xεt − ut‖ ≤ 2β1/4

)

∨P

(sup

t≤τβC∧T

‖Xεt −Xε,β

t ‖ > β1/4)∨

P(τβC ≤ T

).



Clearly, ΘβC can be replaced by τβC , and so

− 1

2

∫ T

0

‖us − b(us)‖2(a(us)+βI)−1ds ≤ lim

ε→0ε2 log P

(supt≤T

‖Xεt − ut‖ ≤ 2β1/4

)

∨limε→0

ε2 log P

(sup

t≤τβC∧T

‖Xεt −Xε,β

t ‖ > β1/4)

∨limε→0


P

(τβC ≤ T

). (6.11)

Next we use the following facts:

(1) by Lemma A.2.1 and (6.1),

limβ→0

∫ T

0

‖us − b(us)‖2(a(us)+βI)−1ds =

∫ T

0

‖us − b(us)‖2a⊕(us)ds;

(2) by Lemma A.3.2,

limβ→0

limε→0

ε2 log P

(sup

t≤τβC∧T

‖Xεt −Xε,β

t ‖ > β1/4)

= −∞;

(3) by (6.10), limC→∞ limε→0 ε2 log supβ∈(0,1] P

(τβC ≤ T

)= −∞.

Hence, passing to the limit β→0 and then C→∞ in (6.11) and taking into account

(1)-(3), one gets the required lower bound

limβ→0

limε→0

ε2 log P

(supt≤T

‖Xεt − ut‖ ≤ 2β1/4

)≥ −1

2

∫ T

0

‖us − b(us)‖2a⊕(us)ds.

A.1. Exponential estimates for martingales

Proposition A.1.1. (Lemma A.1 in Ref. [23]) Let M = (Mt)t≥0, Mt ∈ R, be a

continuous local martingale with M0 = 0 and the predictable variation process 〈M〉tdefined on some stochastic basis with general conditions. Let τ be a stopping time,

α and B positive constants and A some measurable set.

(PA-1) if Mτ − 12 〈M〉τ ≥ α on A, then P(A) ≤ e−α;

(PA-2) if Mτ ≥ α and 〈M〉τ ≤ B on A, then P(A) ≤ e−α2

2B ;

(PA-3) P(supt≤T |Mt| ≥ α, 〈M〉T ≤ B) ≤ 2e−α2

2B ;

(PA-4) P(supt≤T |Mt| ≥ α) ≤ 2e−α2

2B

∨P(〈M〉T > B).



A.2. Pseudoinverse of nonnegative definite matrices

Let A⊕ be the Moore-Penrose pseudoinverse matrix of A (see Ref. [17]).

Lemma A.2.1. For d× d nonnegative definite matrix A and x ∈ Rd,

limβ→0

(x, (A+ βI)−1x) =

‖x‖2

A⊕ , AA⊕x = x

∞, otherwise.

Proof. Let S be an orthogonal matrix, S∗S = I, such that D := S∗AS is a

diagonal matrix. Then, due to S∗(A+ βI)S = D + βI, we have S∗(A + βI)−1S =

(D + βI)−1 and S(D + βI)−1S∗ = (A+ βI)−1. Write (y := S∗x)

(x, (A+ βI)−1x) = (x, S(D + βI)−1S∗x) = (S∗x, (D + βI)−1S∗x)

= (y, (D + βI)−1y) = (y, (D + βI)−1DD⊕y)

+ (y, (D + βI)−1(I −DD⊕)y).

Since limβ→0(D + βI)−1DD⊕ = D⊕, one gets

limβ→0

(y, (D + βI)−1DD⊕y) = ‖y‖2D⊕ = ‖x‖2

A⊕

while limβ→0(y, (D + βI)−1(I −DD⊕)y) 6= ∞ only if (I −DD⊕)y = 0. Since the

latter condition is nothing but (I −AA⊕)x = 0, the desired statement holds.

A.3. Exponential negligibility of Xε,βt − Xε

t

We start with an auxiliary result.

Proposition A.3.1. Let Yt be a nonnegative continuous semimartingale defined on

a stochastic basis (with general conditions):

Yt =

∫ t

0

h1(s)Ysds+ ε

∫ t

0

h2(s)YsdM′s

+ ε√β

∫ t

0

h3(s)√YsdM

′′s + ε2β

∫ t

0

h4(s)ds, (A.1)

where hi(s), i = 1, . . . , 4, are bounded predictable processes and M ′t, M

′′t are con-

tinuous martingales, d〈M ′〉t = m′(t)dt, d〈M ′′〉t = m′′(t)dt, 〈M ′,M ′′〉t ≡ 0 with

bounded m′(t) and m′′(t). Assume that for any T > 0 and β > 0,

limL→∞

limε→0

ε2 log P

(supt≤T

√Yt > L

)= −∞. (A.2)

Then, for any T > 0,

limβ→0

limε→0

ε2 log P

(supt≤T

∣∣Yt∣∣ > β1/4

)= −∞.



Proof. Obviously Yt solves the integral equation

Yt = Et∫ t

0

E−1s

[ε√βh3(s)

√YsdM

′′s + ε2βh4(s)ds

],

where Et = exp( ∫ t

0 [h1(s)−ε20.5h22(s)]ds+

∫ t0 εh2(s)dM ′

s

). Let for definiteness |hi| ≤

r, where r is a constant. Then, with ε ≤ 1,

supt≤T

| log Et| ≤ T (r + 0.5r2) + supt≤T

∣∣∣ε∫ t

0

h2(s)dM ′s

∣∣∣.

Hence the random variable supt≤T | log Et| is bounded on the set

supt≤T

∣∣ε∫ t

0

h2(s)dM ′s

∣∣ ≤ C.

Moreover, it is exponentially tight in the sense that

limC→∞

limε→0

ε2 log P

(supt≤T

| log Et| > C)

= −∞. (A.3)

The latter is implied by

limC→∞

limε→0

ε2 log P

(supt≤T

∣∣ε∫ t

0

h2(s)dM ′s

∣∣ > C)

= −∞ (A.4)

since the martingale Nt = ε∫ t0 h2(s)dM ′

s has the quadratic variation process 〈N〉t =

ε2∫ t0 h

2(s)m′(s)ds and, with some positive number r1, we have ε2h2(s)m′(s) ≤ ε2r1.

Then, by taking into account that P(〈N〉T > ε2r1T

)= 0 and applying the state-

ment (PA-4) of Proposition A.1.1, we obtain P(

supt≤T |Nt| > C)≤ 2e−C

2/(2ε2r1T )

providing (A.4).

Now we estimate supt≤T |Yt| on the set

supt≤T∣∣ log Et

∣∣ ≤ C

. Write

supt≤T

|Yt| ≤ eCTrε2β + eC supt≤T

∣∣∣∫ t

0

E−1s ε

√βh3(s)

√YsdM

′′s

∣∣∣.

This upper bound and (A.2), (A.3) reduce the proof of Proposition A.3.1 to:

limβ→0

limε→0

ε2 log P

(supt≤T

∣∣∣∫ t

0

E−1s ε

√βh3(s)

√YsdM

′′s

∣∣∣ > β1/4,

supt≤T

√Yt ≤ L, sup

t≤T| log Et| ≤ C

)= ∞

for any C > 0 and L > 0. Introduce the martingale

N ′′t =

∫ t

0

E−1s ε

√βh3(s)

√YsdM

′′s with 〈N ′′〉t =

∫ t

0

E−2s ε2βh2

3(s)Ysm′′(s)ds

and denote C =

supt≤T√Yt ≤ L, supt≤T | log Et| ≤ C

. With r2 ≥ h2

3(s)Lm′′(s),we find that

〈N ′′〉T ≤ e2Cr2Tε2β.



Hence,

P

(supt≤T

|N ′′t | > β1/4,C

)= P

(supt≤T

|N ′′t | > β1/4, 〈N ′′〉T ≤ e2Cr2Tε

2β,C)

≤ P

(supt≤T

|N ′′t | > β1/4, 〈N ′′〉T ≤ e2Cr2Tε

2β).

By (PA-3) from Proposition A.1.1 the latter term is upper bounded by

2 exp( β1/2

2e2Cr2Tε2β

).

Then we obtain

limε→∞

ε2 log P

(supt≤T

|N ′′t | > β1/4,C

)≤ − 1

2e2Cr2Tβ1/2−−−→β→0

−∞.

We apply Proposition A.3.1 in order to prove

Lemma A.3.2. For any T > 0 and C > 0,

limβ→0

limε→0

ε2 log P

(sup

t≤τβC∧T

∥∥Xε,βt −Xε

t

∥∥ > β1/4)

= −∞.

Proof.

Recall that Xεt and Xε,β

t solve (1.1) and (6.6) respectively and τβC is given in

(6.9). Set 4ε,βt = Xε,β

τβC∧t −Xε

τβC∧t. By (1.1) and (6.6),

4ε,βt =

∫ τβC∧t

0

(b(Xε,β

τβC∧s) − b(Xε

τβC∧s)

)ds+

+ ε

∫ τβC∧t

0

(σ(Xε,β

τβC∧s) − σ(Xε

τβC∧s)

)dBs + ε

√βWτβ

C∧t.

Due to the local Lipschitz continuity of b and σ and with 0/0 = 0, the vector-valued

and matrix-valued functions:

f(s) =b(Xε,β

τβC∧s

)− b(Xετβ

C∧t)

‖4ε,βs ‖

and g(s) =σ(Xε,β

τβC∧s

)− σ

(Xετβ

C∧s)

‖4ε,βs ‖

are well defined and their entries are bounded by a constant depending on C. Hence

4ε,βt =

∫ τβC∧t

0

‖4ε,βs ‖f(s)ds+ ε

∫ τβC∧t

0

‖4ε,βs ‖g(s)dBs + ε

√βWτβ

C∧t.



Since ‖4ε,βt ‖2 = (4ε,β

t ,4ε,βt ), by the Ito formula, we find that

‖4ε,βt ‖2 =

∫ t

0

2‖4ε,βs ‖(4ε,β

s , f(s))ds

+ ε

∫ τβC∧t

0

2‖4ε,βs ‖(4ε,β

s , g(s)dBs)

+ ε√β

∫ τβC∧t

0

2(4ε,βs , dWs)

+ ε2∫ τβ

C∧t

0

‖4ε,βs ‖2 trace

[g(s)g∗(s)

]ds

+ ε2β(τβC ∧ t)d.

(A.5)

Now, by letting ϕ(s) =2(4ε,β

s ,f(s))

‖4ε,βs ‖ and dBs =

2(4ε,βs ,g(s)dBs)

‖4ε,βs ‖ , we rewrite (A.5) as:

‖4ε,βt ‖2 =

∫ τβC∧t

0

‖4ε,βs ‖2

(ϕ(s) + ε2 trace[g(s)g∗(s)]

)ds

+ ε

∫ τβC∧t

0

‖4ε,βs ‖2dBs + ε

√β

∫ τβC∧t

0

‖4ε,βs ‖2(4ε,β

s , dWs)

‖4ε,βs ‖

+ ε2β(τβC ∧ t)d. (A.6)

With the notations

- Yt = ‖4ε,βt ‖2

- h1(s) = IτβC≤s

ϕ(s) + ε2 trace[g(s)g∗(s)]

- h2(s) ≡ 1

- h4(s) = IτβC≤sd

- M ′t = Bt, m

′(s) =4(4ε,β

s , g(s)g∗(s)4ε,βs )

‖4ε,βs ‖2

- M ′′t =

∫ τβC∧t

0 2(4ε,β

s , dWs)

‖4ε,βs ‖

, m′′(s) ≡ 4,

the equation (A.6) is in the form of (A.1). Since hi(s), i = 1, . . . , 4 are bounded

and√Y t ≡ ‖Xε,β

τβC∧t−Xε

τβC∧t‖ ≤ ‖Xε,β

τβC∧t‖+ ‖Xε

τβC∧t‖ ≤ 2C, i.e., (A.2) holds too, the

statement of the lemma follows from Proposition A.3.1.

Acknowledgement

Research of P. Chigansky is supported by a grant from the Israel Science Foundation.

References

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Mathematical Sciences], (Springer-Verlag, New York, 1984). ISBN 0-387-90858-7.Translated from the Russian by Joseph Szucs.

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[3] P. Dupuis and R. S. Ellis, A weak convergence approach to the theory of large devi-ations. Wiley Series in Probability and Statistics: Probability and Statistics, (JohnWiley & Sons Inc., New York, 1997). ISBN 0-471-07672-4. , A Wiley-IntersciencePublication.

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Chapter 8

On the Convergence Rates of a General Class of Weak

Approximations of SDEs

Dan Crisan and Saadia Ghazali∗

Department of Mathematics, Imperial College London180 Queen’s Gate, London SW7 2BZ

[email protected]

In this paper, the convergence analysis of a class of weak approximations of so-lutions of stochastic differential equations is presented. This class includes re-cent approximations such as Kusuoka’s moment similar families method and theLyons-Victoir cubature of Wiener Space approach. We show that the rate of con-vergence depends intrinsically on the smoothness of the chosen test function. Forsmooth functions (the required degree of smoothness depends on the order of theapproximation), an equidistant partition of the time interval on which the ap-proximation is sought is optimal. For functions that are less smooth (for exampleLipschitz functions), the rate of convergence decays and the optimal partition isno longer equidistant. Our analysis rests upon Kusuoka-Stroock’s results on thesmoothness of the distribution of the solution of a stochastic differential equation.Finally the results are applied to the numerical solution of the filtering problem.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

3 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

4 An Application to Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

5 Some Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

1. Introduction

Stochastic differential equations (SDEs) constitute an ideal mathematical model

for a multitude of phenomena and processes encountered in areas such as filtering,

optimal stopping, stochastic control, signal processes and mathematical finance,

most notably in option pricing (see for example Oksendal [1] and Kloeden & Platen

[2]). Unlike their deterministic counterparts, SDEs do not have explicit solutions,

apart from in a few exceptional cases, hence the necessity for a sound theory of

their numerical approximation.∗[email protected]

221


222 D. Crisan and S. Ghazali

In this paper we will be concerned with SDEs written in Stratonovich form, in

other words, we will look at equations of the form,

Xt = X0 +

∫ t

0

V0(Xs)ds+

k∑

j=1

∫ t

0

Vj(Xs) dW js , (1.1)

where the last term is a stochastic integral of Stratonovich type. There are two

classes of numerical methods for approximating SDEs. The objective of the first

is to produce a pathwise approximation of the solution (strong approximation).

The second method involves approximating the distribution of the solution at a

particular instance in time (weak approximation). For example when one is only

interested in the expectation E[ϕ(Xt)] for some function ϕ, it is sufficient to have

a good approximation of the distribution of the random variable Xt rather than of

its sample paths. This observation was first made by Milstein [3] who showed that

pathwise schemes and L2 estimates of the corresponding errors are irrelevant in this

context since the objective is to approximate the law of Xt. This paper contains

approximations that belong to this second class of algorithms.

Classical results in this area concentrate on solving numerically SDEs for which

the so-called ‘ellipticity condition’, or more generally the ‘Uniform Hormander con-

dition’ (UH), is satisfied. For a survey of such schemes see, for example, Kloeden &

Platen [2] or Burrage, Burrage & Tian [4]. Under this condition, for any bounded

measurable function ϕ, the semigroup of operators Ptt∈[0,∞)defined,

(Ptϕ)(x) = E[ϕ(Xt(x))], (1.2)

where X (x) = Xt (x)t∈[0,∞) solves (1.1) with initial condition X0 = x, is smooth

for any t > 0. It is this property upon which the majority of these schemes rely.

For example, the classical Euler-Maruyama scheme requires Ptϕ to be four times

differentiable in order to obtain the optimal rate of convergence. Talay( [5], [6]) and,

independently, Milstein [7] introduced the appropriate methodology to analyse this

scheme. They express the error as a difference including a sum of terms involving

the solution of a parabolic PDE. Their analysis also shows the relationship between

the smoothness of ϕ and the corresponding error. Talay & Tubaro [8] prove an even

more precise result showing that, under the same conditions, the errors correspond-

ing to the Euler-Maruyama and many other schemes can be expanded in terms of

powers of the discretisation step. Furthermore, Bally & Talay [9] show the existence

of such an expansion under a much weaker hypothesis on ϕ: that ϕ need only be

measurable and bounded (even the boundedness condition can be relaxed). Higher

order schemes require additional smoothness properties of Ptϕ (see for example,

Platen & Wagner [10]).

In the eighties, Kusuoka & Stroock( [11], [12], [13]) studied the properties of

Ptϕ under a weaker condition, the so-called UFG condition (see (2.3) in Section

2). Essentially, this condition states that the Lie algebra generated by the vector

fields Viki=1 ∈ C∞b (Rd; Rd) is finite dimensional as a C∞

b (Rd)-module. Kusuoka &


Convergence Rate of Weak Approximations 223

Stroock conclude Malliavin’s undertaking to recover Hormander’s hypo-ellipticity

theory for degenerate second order elliptic operators. They show that under

the UFG condition, Ptϕ retains certain regularity properties, in particular, that

Vi1Vi2 ...VinPtϕ is well defined for any vector fields Vir , where ir ∈ 1, ..., k.

They also give upper bounds for the supremum norm of Vi1Vi2 ...VinPtϕ. Be-

sides the UFG condition, our analysis rests upon a second assumption which

we call the V0 condition. It states that V0 can be expressed in terms of

V1, ...Vk∪[Vi, Vj ] , 1 ≤ i < j ≤ k. This premise is weaker than the ellipticity as-

sumption and has been used, for example, by Jerison and Sanchez-Calle ( [14], [15])

to obtain estimates for the heat kernel. This second condition enables one to con-

trol the supremum norm of Vi1Vi2 ...VinPtϕ for ir ∈ 0, 1, ..., k (see Corollary 5.4 in

Section 5).

A number of schemes have recently been developed to work under these weaker

conditions rather than the ellipticity condition, their convergence depending intrin-

sically on the above estimates of Vi1Vi2 ...VinPtϕ. A further advantage of this new

generation of schemes is a consequence of the classical result stating that the sup-

port of X (x) is the closure of the set S =xϕ : [0,∞) → Rd

where xϕ solves the

ODE,

xϕt = x+

∫ t

0

V0(xϕs )ds+

k∑

j=1

∫ t

0

Vj(xϕs )ϕ (s) ds,

and ϕ : [0,∞) → Rd is an arbitrary smooth function (see Stroock & Varadhan [16],

[17], [18], Millet & Sanz-Sole [19]). These schemes attempt to keep the support

of the approximating process on the set S. In this way, stability problems that

are known to affect classical schemes can be avoided. For example, Ninomiya &

Victoir [20] give an explicit example where the Euler-Maruyama approximation

fails whilst their algorithm succeeds (see Example 3.4 below for the algorithm).

Their example involves an SDE related to the Heston stochastic volatility model in

finance.

In this paper we give a general criterion for the convergence of a class of weak

approximations incorporating this new category of schemes. This criterion is based

upon the stochastic Stratonovich-Taylor expansion of Ptϕ and demonstrates how

the rate of convergence depends on the smoothness of the test function selected.

Our plan is as follows. In section 2 we set out some essential terminol-

ogy, adopted partly from Kusuoka [21], which is required to index the stochastic

Stratonovich-Taylor expansions that follow and to state the UFG and V0 conditions

imposed on the given vector fields (see (2.3) & (2.4) below). Section 3 contains our

main results on the convergence analysis of a class of weak approximations of so-

lutions of SDEs, characterised by introducing the concept of an m-perfect family

(Definition 3.1). The Lyons-Victoir and Ninomiya-Victoir approximations are both

members of this class. Although the Kusuoka approximation is not within this

family, it can effectively be categorised in the same way (see Example 3.6 and the



subsequent comment). In our main Theorem 3.7, we show that the rate of con-

vergence depends intrinsically on the smoothness of the chosen test function; the

higher the order of the approximation, the smoother the test function required.

For a smooth function, an equidistant partition of the time interval on which

the approximation is sought is optimal. For less smooth functions, this is no longer

true. We emphasise that the UFG+V0 conditions are not required for a smooth

test function. Furthermore, the Kusuoka approximation does not require the V0

condition. Finally, in Section 4 we present an application of this theory to the

numerical solution of the filtering problem.

2. Preliminaries

Let (Ω,F ,P) be a probability space satisfying the usual conditions and W =

Wtt∈[0,∞) be a k-dimensional Brownian motion defined on it. We also set W 0t = t

for t ∈ [0,∞) . Let C∞b (Rd,Rd) denote the space of smooth functions ϕ : Rd → Rd

with bounded derivatives, that is, bounded partial derivatives (of all orders) of

the d component functions ϕi : Rd → Rdi=1 exist. We have a dual interpreta-

tion of this space, in that we also regard its elements as vector fields. In other

words, a smooth function V : C∞b (Rd) → C∞

b (Rd) is equivalent to an operator on

C∞b (Rd) ≡ C∞

b (Rd,R) defined by,

V ϕ =

d∑

i=1

V i∂ϕ

∂xi

where V i ∈ C∞b (Rd) is the i-th component of L for i = 1, . . . , d.

We next consider the Stratonovich SDE with drift vector V i0 (x)di=1 and dis-

persion matrix V ij (x)d,ki,j=1 for x ∈ Rd, for some V0, . . . , Vk ∈ C∞b (Rd,Rd). This is

written componentwise as,

X it = X i

0 +

∫ t

0

V i0 (Xs)ds+

k∑

j=1

∫ t

0

V ij (Xs) dW js (2.1)

for i = 1, . . . , d.

The following essential terminology has been adopted from Kusuoka [21]. Let

A be the set of multi-indices,

A = ∅ ∪ ∞∪m=1

0, 1, . . . , km

where k is the dimension of the Brownian Motion introduced above. This set will

be used to index the Stratonovich-Taylor expansions that follow. Let |·| and ‖·‖ be

the following two norms defined on A by

|∅| = 0, |α| = r if α = (i1, . . . , ir) ∈ 0, 1, . . . , kr for r ∈ N

and ‖α‖ = |α| + card 1 ≤ j ≤ |α| : ij = 0 . Furthermore, let A0 = A\∅, A1 =

A\∅, (0) and correspondingly, A(m) = α ∈ A : ‖α‖ ≤ m, A0(m) = α ∈ A0 :



‖α‖ ≤ m and A1(m) = α ∈ A1 : ‖α‖ ≤ m where the integer m ∈ N above

corresponds to the level of the truncation in the Stratonovich-Taylor expansions

that follow (see (3.4)).

We also require the following concatenation on A,

(i1, . . . , ir) ∗ (j1, . . . , js) = (i1, . . . , ir, j1, . . . , js)

and need to define a further operation on the vector fields Vj ∈ C∞b (Rd,Rd)kj=0

introduced above.

Definition 2.1. For α ∈ A , the vector field V[α] is defined inductively by,

V[ϕ] = 0; V[(j)] = Vj ; V[α∗(j)] = [Vα, Vj ]

for j = 0, 1, . . . , k where [Vi, Vj ] := ViVj − VjVi for Vi, Vj ∈ C∞b (Rd,Rd).

In the following we will make use of the semi-norm,

‖ϕ‖V,i =i∑

u=1

∑

α1,...,αu∈A0

‖α1∗...∗αu‖=i

∥∥V[α1] · · ·V[αu]ϕ∥∥∞ .

for i ∈ N. Furthermore, we introduce the semi-norm,

‖ϕ‖p =

p∑

i=1

∥∥∇iϕ∥∥∞ (2.2)

for p ∈ N, ϕ ∈ Cpb (Rd) where,

∥∥∇iϕ∥∥∞ = max

j1,...,ji∈1,...,d

∥∥∥∥∂iϕ

∂xj1 . . . ∂xji

∥∥∥∥∞

and note that it can easily be deduced from the chain rule, for α ∈ A0, ϕ ∈ C|α|b (Rd),

that ‖Vαϕ‖∞ ≤ C ‖ϕ‖|α| and hence ‖ϕ‖V,i ≤ C ‖ϕ‖i for i ∈ N. Also let ‖·‖p,∞ be

the norm ‖ϕ‖p,∞ := ‖ϕ‖∞ + ‖ϕ‖p for ϕ ∈ Cpb (Rd). We define the space,

CV,ib (Rd) =ϕ : ‖ϕ‖V,i <∞

which appears in the definition of an m-perfect family below (see Definition 3.1).

We now introduce the two conditions on the vector fields Vj ∈ C∞b (Rd,Rd)kj=0

required to treat the case when the test function ϕ is not smooth. We emphasise

that the smooth case requires neither the UFG nor the V0 condition. Furthermore,

the Kusuoka approximation (Example 3.6) does not require the V0 condition as the

corresponding error is controlled in terms of V[β] : β ∈ A1 alone.



The UFG condition [Kusuoka & Stroock [13, 21]]. There exists some

l ∈ N such that for any α ∈ A1 we have,

V[α] =∑

β∈A1(l)

ϕα,βV[β]. (2.3)

where ϕα,β ∈ C∞b (Rd) for all β ∈ A1(l).

The V0 condition. There exist ϕβ ∈ C∞b (Rd), β ∈ A1(2) such that,

V0 =∑

β∈A1(2)

ϕβV[β]. (2.4)

As mentioned in the introduction, the UFG condition states that the Lie algebra

generated by the vector fields Viki=1 ∈ C∞b (Rd; Rd) is finite dimensional as a

C∞b (Rd)-module. It is implied by the Uniform Hormander Condition which states

that,

∃l ∈ N and c > 0 s.t.∑

α∈A1(l)

⟨V[α](x), ξ

⟩2 ≥ c |ξ|2 (2.5)

for all x, ξ ∈ Rd, where 〈V, ξ〉 =∑d

i=1 Viξi for V ∈ C∞

b (Rd,Rd).

Under conditions (2.3)+(2.4), one can show that for any r ∈ Z+, αi ∈ A0ri=1

and p = 1, ..., ‖α1 ∗ . . . ∗ αr‖, there exists a constant CTp > 0 such that,∥∥V[α1] . . . V[αr ]Ptϕ

∥∥∞ ≤ CTp t

(p−‖α1∗...∗αr‖)/2 ‖ϕ‖p , t ∈ [0, T ] . (2.6)

where Ptt∈[0,∞) is the semigroup associated with X defined in (1.2). We remark

that condition (2.3) will only give us (2.6) for αi ∈ A1, i = 1, ...r. Under both

(2.3)+(2.4) we have,

‖V0Ptϕ‖∞ ≤ Ct−1 ‖ϕ‖∞ .

However, under condition (2.3) alone, ‖V0Ptϕ‖∞ may be of higher order. Kusuoka

has given an explicit example in which,

ct−l2 ‖ϕ‖∞ ≤ ‖V0Ptϕ‖∞ ≤ Ct−

l2 ‖ϕ‖∞

for some constants c, C > 0 and where l is the constant appearing in (2.3) (see

Proposition 14 and Proposition 16 in Ref. [22]). This coarser bound on ‖V0Ptϕ‖∞results in lower rates of convergence. The authors believe that (2.4) is the most

general condition required to preserve the same rates of convergence as those ob-

tained when Viki=1 satisfy the ellipticity condition (for non-smooth test functions

ϕ).

Result (2.6) is a corollary of a certain representation theorem proved in Kusuoka-

Stroock [13]. For completeness, we state the representation theorem (Theorem 5.2)

in Section 5 where we also sketch a proof of inequality (2.6) in Corollary 5.4. We

note that the case p = 1 has been proved in Ref. [13] for αi ∈ A1ri=1. Inequality

(2.6) proves to be crucial in obtaining upper bounds on the error of of the class

algorithms that we study below (see Theorem 3.7).



3. The Main Theorem

In this section we introduce the concept of an m-perfect family. Such families cor-

respond to various weak approximations of SDEs, including the Lyons-Victoir and

Ninomiya-Victoir schemes. The main result appears in Theorem 3.7 and Corollary

3.8.

For α = (i1, . . . , ir) ∈ A0 and ϕ ∈ Crb (Rd) let fα,ϕ be defined as f(i1,...,ir),ϕ :=

Vi1 ...Virϕ. We also need to define the iterated Stratonovich integral

Ifα,ϕ (t) :=

∫ t

0

∫ s0

0

· · ·(∫ sr−2

0

fα,ϕ(Xsr−1) dW i1sr−1

) · · · dW ir−1

s1 dW irs0 ,

for t ≥ 0. If i1 = 0 then Ifα,ϕ (t) is well defined for ϕ ∈ Crb (Rd). However, if

i1 6= 0 then Ifα,ϕ (t) is well defined provided ϕ ∈ Cr+2b (Rd), since the semimartin-

gale property of fα,ϕ(X) is required in the definition of the first Stratonovich inte-

gral∫ sr−2

0 fα,ϕ(Xsr−1) dW i1sr−1

. Note that the Stratonovich integrals are evaluated

innermost first. Finally let

Iα(t) :=

∫ t

0

∫ s0

0

· · ·(∫ sr−2

0

1 dW i1sr−1

) · · · dW ir−1

s1 dW irs0 .

Let α = (i1, . . . , ir) ∈ A0 be an arbitrary multi-index such that ‖α‖ = m ∈ N (and

|α| = r ∈ N). If m is odd, then E[Iα(t)] = 0 and if m is even then

E[Iα(t)] =

t

m2

2r− m2 ( m

2 )!if α ∈ Am,r

0

0 otherwise, (3.1)

where Am,r0 is the set of multi-indices α = α1 ∗ · · · ∗ αm

2∈ A0 (m) such that each

αi = (0) or (j, j) for some j ∈ 1, . . . , k. Note that r− m2 is equal to the number of

pairs of indices (j, j) occurring in α. A proof of this result can be found in Ref. [23].

The set of iterated Stratonovich integrals plays a central role in the theory of

approximation of solutions of SDEs and there are numerous papers that study its

structure. Here, we adopt the hierarchical set approach introduced by Kloeden

& Platen [24]. An alternative method can be found in Gaines [25] where it is

shown how Lyndon words provide a basis for iterated Stratonovich integrals and

also how shuffle products may be used to obtain moments of stochastic integrals.

Pettersson [26] gives a notationally and computationally convenient Stratonovich-

Taylor expansion. Furthermore, Burrage & Burrage [27] use rooted-tree theory to

describe the aforementioned set and Burrage & Burrage [28] presents an approach

based on B-series.

We state three further results in (3.2), (3.3) and (3.5). The proofs are all el-

ementary and can be found in Ref. [23]. The first two give an upper bound on

the L2 norm of Ifα,ϕ(t) for smooth ϕ. The third provides an explicit form for the

remainder of ϕ(Xt) when expanded in terms of iterated integrals.



For ϕ ∈ C||α||+2b (Rd) and any multi-index α = (i1, . . . , ir) ∈ A0 such that i1 6= 0,

we have

∥∥Ifα,ϕ(t)∥∥

2≤ c1 ‖fα,ϕ‖∞ t

‖α‖2 + c2

k∑

i=1

‖Vifα,ϕ‖∞ t‖α‖+1

2 (3.2)

for some constants c1 ≡ c1(α), c2 ≡ c2(α) ≥ 0. For ϕ ∈ C||α||b (Rd) and any multi-

index α = (i1, . . . , ir) ∈ A0 such that i1 = 0 we have


2≤ c1 ‖fα,ϕ‖∞ t

‖α‖2 . (3.3)

For m ∈ N, ϕ ∈ Cm+3b (Rd) and x ∈ Rd, we define the truncation,

ϕmt (x) := ϕ(x) +∑

α∈A0(m)

fα,ϕ(x)Iα(t). (3.4)

Then for t ≥ 0 the remainder is

Rm,t,ϕ(x) := ϕ(Xt) − ϕmt (x) = (∑

‖α‖=m+1

+∑

‖α‖=m+2,α=0∗β,‖β‖=m)Ifα,ϕ(t). (3.5)

In the following, we define a class of approximations of X expressed in terms of

certain families of stochastic processes, X (x) = Xt (x)t∈[0,∞) for x ∈ Rd, which

are explicitly solvable. In particular, we can explicitly compute the operator,

(Qtϕ)(x) = E[ϕ(Xt(x))]. (3.6)

The semigroup PT will then be approximated by QmhnQmhn−1

. . .Qmh1where hj :=

tj − tj−1nj=1 and πn = tj := ( jn )γTnj=0 for n ∈ N, is a sufficiently fine partition

of the interval [0, T ]. In particular hj ∈ [0, 1) for j = 1, ..., n. The underlying idea is

that Qtϕ will have the same truncation as Ptϕ.

So let X (x) = Xt (x)t∈[0,∞), where x ∈ Rd, be a family of progressively mea-

surable stochastic processes such that, limy→x0 Xt (y) = Xt (x0) P−almost surely,

for any t ≥ 0 and x0 ∈ Rd. As a result, the operatorQt defined in (3.6) has the prop-

erty that Qtϕ ∈ Cb(Rd) for any ϕ ∈ Cb(R

d). In particular, Qt : Cb(Rd) → Cb(R

d) is

a Markov operator.

Definition 3.1. For m ∈ N, the family X (x) = Xt (x)t∈[0,∞) where x ∈ Rd, is

said to be m-perfect for the processX if there exist constants C > 0 andM ≥ m+1

such that for ϕ ∈ CV,Mb (Rd),

supx∈Rd

|Qtϕ(x) − E[ϕmt (x)]| ≤ C

M∑

i=m+1

ti/2 ‖ϕ‖V,i . (3.7)

As we can see from (3.7), the quantity E[ϕmt (x)] plays the same role as the

classical truncation in the standard Taylor expansion of a function. Using (3.1) we



deduce that,

E[ϕ0t (x)] = ϕ(x)

E[ϕ2t (x)] = ϕ(x) + V0ϕ(x)t +

k∑

i=1

V 2i ϕ(x)

t

2= ϕ(x) + Lϕ(x)t

E[ϕ4t (x)] = E[ϕ2

t (x)] +

k∑

i=1

V 20 ϕ(x)

t2

2+

k∑

i=1

V0V2i ϕ(x)

t2

4

+

k∑

i=1

V 2i V0ϕ(x)

t2

4+

k∑

i,j=1

V 2j V

2i ϕ(x)

t2

8

= ϕ(x) + Lϕ(x)t + L2ϕ(x)t2

2,

where L = V0 + 12

∑ki=1 V

2i . Furthermore, since E[Iα(t)] = 0 for odd ‖α‖, it follows

that E[ϕ1t (x)] = E[ϕ0

t (x)],E[ϕ3t (x)] = E[ϕ2

t (x)] and E[ϕ5t (x)] = E[ϕ4

t (x)].

There now follow some examples of m−perfect families corresponding to

Ptt∈[0,∞) as described in (1.2), the Lyons-Victoir method and the Ninomiya-

Victoir algorithm.

Example 3.2. The family of stochastic processes Xt (x)t∈[0,∞), where x ∈ Rd,

is m−perfect. More precisely there exists a constant c3 > 0 such that for ϕ ∈CV,m+2b (Rd),

supx

|Ptϕ(x) − E[ϕmt (x)]| ≤ c3

m+2∑

i=m+1

ti/2 ‖ϕ‖V,i , (3.8)

Proof. For ϕ ∈ CV,m+3b (Rd),

|Ptϕ(x)−E[ϕmt (x)]|=|E[Rm,t,ϕ(x)]|=

∣∣∣∣∣∣E[(

∑

‖α‖=m+1

+∑

‖α‖=m+2,α=0∗β,‖β‖=m)Ifα,ϕ(t)]

∣∣∣∣∣∣

Applying inequality (3.2) to the first sum,

∑

‖α‖=m+1


2≤

∑

‖α‖=m+1

c1(α) ‖fα,ϕ‖∞ tm+1

2 + c2(α)k∑

i=1

‖Vifα,ϕ‖∞ tm+2

2

≤ c4

m+2∑

i=m+1

ti/2 ‖ϕ‖V,i (3.9)

for some constant c4 > 0. Applying result (3.3) to the second sum,∑

‖α‖=m+2,α=0∗β,‖β‖=m


2≤

∑

‖α‖=m+2,α=0∗β,‖β‖=mc1(α) ‖fα,ψ‖∞ t

≤ c5 ‖ϕ‖V,m+2 tm+2

2 (3.10)



for some c5 > 0. The result for ϕ ∈ CV,m+3b (Rd) follows from combining (3.9)

and (3.10). Since none of the terms in (3.8) depend on partial derivatives of order

m+3, the inequality is also valid for any ϕ ∈ CV,m+2b (Rd) (a standard approximation

method can be used).

In the following example, the family of processes X (x) = Xt (x)t∈[0,1], where

x ∈ Rd, corresponds to the Lyons-Victoir approximation (see Ref. [29]). The ex-

ample involves a set of l finite variation paths, ω1, . . . , ωl ∈ C00 ([0, 1],Rk), for some

l ∈ N, together with some weights λ1, . . . , λl ∈ R+ such thatl∑

j=1

λj = 1. These paths

are said to define a cubature formula on Wiener Space of degree m if, for

any α ∈ A0(m),

E[Iα(1)]=l∑

j=1

λjIωjα (1)

where,

Iωj

(i1,...,ir)(1) :=

∫ 1

0

∫ s0

0

· · · (∫ sr−2

0

dωi1j (sr−1)) · · · dωir−1

j (s1)dωirj (s0).

From the scaling properties of the Brownian motion we can deduce, for t ≥ 0,

E[Iα(t)]=

l∑

j=1

λjIωt,jα (t)

where ωt,1, . . . , ωt,l ∈ C00 ([0, t],Rk) is defined by ωt,j (s) =

√tωj(st

), s ∈ [0, t]. In

other words, the expectation of the iterated Stratonovich integrals Iα(t) is the same

under the Wiener measure as it is under the measure,

Qt :=

l∑

j=1

λjδωt,j .

Example 3.3. If we choose X to satisfy the evolution equation (2.1) but with the

driving Brownian motion replaced by the paths ωt,1, . . . , ωt,l defined above then

the family of processes, Xt (x)t∈[0,1], with corresponding operator (Qtϕ)(x) :=

EQt [ϕ(Xt(x))], is m-perfect. More precisely, there exists a constant c6 > 0 such

that for ϕ ∈ CV,m+2b (Rd),

supx

∣∣Qtϕ(x) − E[ϕmt (x)]∣∣ ≤ c6

m+2∑

i=m+1

ti/2 ‖ϕ‖V,i

For example, if (λj , ωt,j) are chosen such that for l = 2k the paths are ωt,j : t 7→t(1, z1

j , .., zkj ) for j = 1, . . . , 2k with points zj ∈ −1, 1k and weights λj = 2−k, we

obtain a cubature formula of degree 3 and a corresponding 3-perfect family.



Proof. Let us first observe that Iωt,jα (t) = t

|α|2 I

ωjα (1) Hence, for ϕ ∈ CV,m+2

b (Rd),∣∣Qtϕ(x) − E[ϕmt (x)]

∣∣ = |EQt [Rm,t,ϕ(x)]|≤ (

∑

‖α‖=m+1

+∑

‖α‖=m+2,α=0∗β,‖β‖=m) ‖fα,ϕ‖∞ ‖EQt [Iα(t)]‖2

≤ (∑

‖α‖=m+1

+∑

‖α‖=m+2,α=0∗β,‖β‖=m) ‖fα,ϕ‖∞

l∑

j=1

λj ‖Iωt,jα (t)]‖2

≤ (∑

‖α‖=m+1

+∑

‖α‖=m+2,α=0∗β,‖β‖=m)kα ‖fα,ϕ‖∞ t

|α|2 .

where kα =l∑

j=1

λj∥∥Iωjα (1)

∥∥2.

Remarks

(i) There has been no change to the underlying measure in the example above.

Merely a representation in terms of the measure Qt has been introduced to ease the

computation of Qt. More precisely, the family of processesXt(x)

t∈[0,1]

where

x ∈ Rd is constructed as follows. We take,

X0(x) = x

and then randomly choose a path ωt,r from the set ωt,1, . . . , ωt,l with correspond-

ing probabilities (λ1, . . . , λl). Each process then follows a deterministic trajectory

driven by the solution of the ordinary differential equation,

dXt = V0(Xt)dt+

k∑

j=1

Vj(Xt)dωjt,k

for some V0, . . . , Vk ∈ C∞b (Rd,Rd) as in (2.1). We can therefore compute the ex-

pected values of a functional of Xt (x) as integrals on the path space with respect

to the Radon measure Qt. Hence the identities,

Qtϕ(x) = E[ϕ(Xt (x))

]= EQt

[ϕ(Xt (x))

]

(ii) The approach adopted by Lyons and Victoir to construct the above approxi-

mation resembles the ideas developed by Clark and Newton in a series of papers

( [30], [31], [32], [33]). Heuristically, Clark and Newton constructed strong approx-

imations of SDEs using flows driven by vector fields which were measurable with

respect to the filtration generated by the driving Wiener process. In a similar vein,

Castell & Gaines [34] provide a method of strongly approximating the solution of

an SDE by means of exponential Lie series.

For the following example, we will denote by exp(V t)f the value at time t of the

solution of the ODE y′ = V (y) , y (0) = f where V ∈ C∞b (Rd,Rd). In particular,



exp(V t) (x) is exp(V t)f for f being the identity function. The family of processes

Y (x) = Yt (x)t∈[0,1] below corresponds to the Ninomiya-Victoir approximation

(see Ref. [20]).

Example 3.4. Let Λ and Z be two independent random variables such that Λ is

Bernoulli distributed P(Λ = 1) = P(Λ = −1) = 12 and Z = (Zi)ki=1 is a standard

normal k− dimensional random variable. Consider the family of processes Y (x) =

Yt (x)t∈[0,1] defined by

Yt(x) =

exp(V0

2 t)k∏i=1

exp(ZiVit1/2) exp(V0

2 t)(x) if Λ = 1

exp(V0

2 t)k∏i=1

exp(Zk+1−iVk+1−it1/2) exp(V0

2 t)(x) if Λ = −1

with the corresponding operator (Qtϕ)(x) := E[ϕ(Yt(x))]. Then there exists a

constant c7 > 0 such that for ϕ ∈ CV,8b (Rd)

supx

∣∣Qtϕ(x) − E[ϕ5t (x)]

∣∣ ≤ c7t3 ‖ϕ‖V,6

Hence Yt (x)t∈[0,1] is 5-perfect.

Proof. We first consider the case Λ = 1. Let Y ik+1i=0 be defined,

dϕ

ds(Y 0s ) = V0ϕ(Y 0

s ) for s ∈ [0,t

2], Y 0

0 = x

dϕ

ds(Y 1s ) = Z1V1ϕ(Y 1

s ) for s ∈ [0,√t], Y 1

0 = Y 0t2

dϕ

ds(Y is ) = ZiViϕ(Y is ) for s ∈ [0,

√t], Y i0 = Y i−1√

t, i = 2, . . . , k

dϕ

ds(Y k+1s ) = V0ϕ(Y k+1

s ) for s ∈ [0,t

2], Y k+1

0 = Y k√t

It follows from the definition of the algorithm and by Ito’s Formula that,

ϕ(Y k+1t2

) = ϕ(Y k√t)+V0ϕ(Y k√

t)t

2+V 2

0 ϕ(Y k√t)t2

8+

∫ t/2

0

∫ s1

0

∫ s2

0

V 30 ϕ(Y k+1

s3 )ds3ds2ds1.

(3.11)

We need to expand the right hand side of (3.11) and divide the resulting expansion

into two parts: the required truncation and a remainder whose expected value

should be bounded by Ct3 ‖ϕ‖V,6. The final term in (3.11) belongs to the remainder

and indeed,

E

[∣∣∣∣∣

∫ t/2

0

∫ s1

0

∫ s2

0

V 30 ϕ(Y k+1

s3 )ds3ds2ds1

∣∣∣∣∣

]≤∥∥V 3

0 ϕ∥∥∞

(t/2)3

6≤ t3

48‖ϕ‖V,6 . (3.12)



Expanding the third term in (3.11),

V 20 ϕ(Y k√

t) = V 2

0 ϕ(Y k−1√t

) +

∫ √t

0

ZkVkV20 ϕ(Y ks )ds

= V 20 ϕ(Y k−1√

t) + ZkVkV

20 ϕ(Y k−1√

t)√t+

∫ √t

0

∫ s1

0

(Zk)2V 2k V

20 ϕ(Y ks2 )ds2ds1

= ...

= V 20 ϕ(Y 0

t2

) + Z1V1V20 ϕ(Y 0

t2

)√t+

k∑

i=2

ZiViV20 ϕ(Y i−1√

t)√t

+

k∑

i=1

∫ √t

0

∫ s1

0

(Zi)2V 2i V

20 ϕ(Y is2)ds2ds1

So

V 20 ϕ(Y k√

t) = V 2

0 ϕ(x) +

∫ t2

0

V 30 ϕ(Y 0

s )ds+ Z1V1V20 ϕ(Y 0

t2

)√t+

k∑

i=2

ZiViV20 ϕ(Y i√

t)√t

+

k∑

i=1

∫ √t

0

∫ s1

0

(Zi)2V 2i V

20 ϕ(Y is2 )ds2ds1.

Now the last four terms are all O(t) since E

[Z1V1V

20 ϕ(Y 0

t2

)√t]

=

E

[ZiViV

20 ϕ(Y i−1√

t)√t]

= 0 because Z is normal, E

[∣∣∣∫ t

2

0V 3

0 ϕ(Y 0s )ds

∣∣∣]≤∥∥V 3

0 ϕ∥∥∞

t2

and finally E

[∣∣∣∫√

t

0

∫ s10

(Zi)2V 2i V

20 ϕ(Y is2)ds2ds1

∣∣∣]≤∥∥V 2

i V20 ϕ∥∥∞

t2 .

So for the third term in (3.11) we have established,∣∣∣∣E[V 2

0 ϕ(Y k√t)t2

8

]− V 2

0 ϕ(x)t2

8

∣∣∣∣ ≤t3

16

(∥∥V 3

0 ϕ∥∥∞ +

k∑

i=1

∥∥V 2i V

20 ϕ∥∥∞

)

≤ t3

16‖ϕ‖V,6 . (3.13)

Similarly, for the second term on the RHS of (3.11),∣∣∣∣∣E[V0ϕ(Y k√

t)t

2

]−(V0ϕ(x) + V 2

0 ϕ(x)t

2+

k∑

i=1

V 2i V0ϕ(x)

t

2

)t

2

∣∣∣∣∣ ≤t3

8‖ϕ‖V,6

(3.14)

Finally, for the first term on the RHS of (3.11),∣∣∣∣∣E[ϕ(Y k√

t)]−(

ϕ(x) + V0ϕ(x) t2 + V 20 ϕ(x) t

2

8 +∑k

i=1 V4i ϕ(x) t

2

4!

+∑ki=1

(V 2i ϕ(x) t2 + V0V

2i ϕ(x) t

2

4

)+∑ki=2

∑i−1j=1 V

2j V

2i ϕ(x) t

2

4

)∣∣∣∣∣

≤ t3

16‖ϕ‖V,6 (3.15)

Substituting (3.12), (3.13), (3.14) and (3.15) in (3.11) gives the bound for the case

Λ = 1. An analogous bound is then established for the case Λ = −1. The final

result is obtained by taking the average of the two cases. See [23] for details.



The following Lemma is required to prove the main theorem below.

Lemma 3.5. For 0 < s ≤ t ≤ 1 and any m−perfect family X t (x)t∈(0,1] with

corresponding operator Q = Qtt∈(0,1] we have,

‖Pt(Psϕ) −Qt(Psϕ)‖∞ ≤ c8 ‖ϕ‖pM∑

j=m+1

tj/2

sj−p2

, (3.16)

where ϕ ∈ Cpb (Rd) for 0 ≤ p < ∞ and some constant c8 > 0. In particular, for

ϕ ∈ CMb (Rd),

‖Pt(Psϕ) −Qt(Psϕ)‖∞ ≤ c8 ‖ϕ‖p tm+1

2 . (3.17)

Proof. Since C∞b (Rd) is dense in Cpb (Rd) in the topology generated by the norm

||·||p,∞ it suffices to prove (3.16) and (3.17) only for a function ϕ ∈ C∞b (Rd). By

Corollary 5.4 in Section 5,

‖Ptϕ‖V,j =

j∑

i=1

∑

α1,...,αi∈A0

‖α1∗...∗αi‖=j

∥∥V[α1] · · ·V[αi]Ptϕ∥∥∞

≤j∑

i=1

∑

α1,...,αi∈A0

‖α1∗...∗αi‖=j

CTp ‖ϕ‖pt(‖α1∗...∗αi‖−p)/2 ≤

c9 ‖ϕ‖pt

j−p2

for some c9 ≡ c9(j, p) ≥ 0. Then (3.16) and (3.17) follow from the definition of an

m−perfect family.

The family of processes X (x) = Xt (x)t∈[0,∞) below corresponds to the

Kusuoka approximation. We recall that Kusuoka’s result requires only the UFG

condition.

Example 3.6. A family of random variables Zα : α ∈ A0 is said to be m-

moment similar if E[ |Zα|r] <∞ for any r ∈ N, α ∈ A0 and Z(0) = 1 with,

E[Zα1 . . . Zαj ] = E[Iα1 . . . Iαj ]

for any j = 1, . . . ,m and α1, . . . , αj ∈ A0 such that ‖α1‖ + · · · + ‖αj‖ ≤ m where

Iα is defined as above.

Let Zα : α ∈ A0 be a family of m−moment similar random variables and let

X (x) = Xt (x)t∈[0,∞) be the family of processes,

Xt (x) =

m∑

j=0

1

j!

∑

α1,...,αj∈A0,‖α1‖+···+‖αj‖≤m

t‖α1‖+···+‖αj‖

2 (P 0α1. . . P 0

αj)(V[α1] . . . V[αj ]H)(x)

(3.18)



where H : Rd → Rd is defined H(x) = x and

P 0α := |α|−1

|α|∑

j=0

(−1)j+1

j

∑

β1∗...∗βj=α

Zβ1 . . . Zβj

with the corresponding operator Q = Qtt∈(0,1] in Cb(Rd) defined by,

Qtϕ(x) = E[ϕ(Xt (x))]

for ϕ ∈ C∞b (Rd) Then,

‖Pt+sϕ−QtPsϕ‖∞ ≤ c10 ‖∇ϕ‖∞mm+1∑

j=m+1

tj/2

sj−12

(3.19)

for some constant c10 > 0.

Proof. See Definition 1, Theorem 3 and Lemma 18 in Kusuoka [35] for (3.19).

The family X (x) , x ∈ Rd as defined in (3.18) is not m-perfect. However,

inequality (3.19) is a particular case of (3.16) where p = 1 and M = mm+1. Since

(3.16) is the only result required to obtain (3.20), we deduce from the proof of

Theorem 3.7 that (3.20), with p = 1, holds for Kusuoka’s method as well. Similarly

part (ii) of Corollary 3.8 holds for Kusuoka’s method.

The set of vector fields appearing in (3.18) belong to the Lie algebra generated by

the original vector fields V0, V1, ..., Vk . Ben Arous [36] and Burrage & Burrage [27]

employ the same set of vector fields to produce strong approximations of solutions of

SDEs. Notably, the same ideas appear much earlier in Magnus [37], in the context

of approximations of the solution of linear (deterministic) differential equations.

Castell [38] also gives an explicit formula for the solution of an SDE in terms of Lie

brackets and iterated Stratonovich integrals.

We now prove our main result on m-perfect families, the gist of which can be

conveyed by the concept of local and global order of an approximation. Local order

measures how close an approximation is to the exact solution on a sub-interval of

the integration, given an exact initial condition at the start of that subinterval.

The global order of an approximation looks at the build up of errors over the entire

integration range. The theorem below states that, in the best possible case, the

global order of an approximation obtained using an m-perfect family is one less

than the local order. More precisely, for a suitable partition, the global error is of

order m−12 whilst the local error is of order m+1

2 .

Let us define the function,

Υp (n) =

n− 1

2 min(γp,(m−1)) if γp 6= m− 1

n−(m−1)/2 lnn for γp = m− 1

In the following,

Eγ,n (ϕ) :=∥∥∥PTϕ−Qmhn

Qmhn−1. . . Qmh1

ϕ∥∥∥∞



for γ ∈ R, n ∈ N.

Theorem 3.7. Let T, γ > 0 and πn = tj = ( jn )γTnj=0 be a partition of the

interval [0, T ] where n ∈ N is such that hj = tj − tj−1nj=1 ⊆ (0, 1]. Then for

any m-perfect family Xt (x)t∈[0,T ] with corresponding operator Q = Qtt∈(0,1]

we have, for ϕ ∈ Cpb (Rd) where p = 1, ...,m,

Eγ,n (ϕ) ≤ c11Υp (n) ‖ϕ‖p +∥∥Ph1ϕ−Qmh1

ϕ∥∥∞ (3.20)

for some constant c11 ≡ c11(γ,M, T ) > 0 where M ≥ m+ 1, as in Definition 3.1.

In particular, if γ ≥ m−1p then,

Eγ,n (ϕ) ≤ c11

nm−1

2

‖ϕ‖p +∥∥Ph1ϕ−Qmh1

ϕ∥∥∞

Proof. We have,

Eγ,n (ϕ) = Phn(PT−hnϕ) −Qmhn(PT−hnϕ)

+

n−1∑

j=1

Qmhn. . . Qmhj+1

(PT−hj+1−···−hnϕ−QmhjPT−hj−···−hnϕ)

= Phn(Ptn−1ϕ) −Qmhn(Ptn−1ϕ)

+

n−1∑

j=1

Qmhn. . . Qmhj+1

(Phj

(Ptj−1ϕ

)−Qmhj

(Ptj−1ϕ

)).

By Lemma 3.5, there exists a constant c8 > 0 such that,

∥∥Phn(Ptn−1ϕ) −Qmhn(Ptn−1ϕ)

∥∥∞ ≤ c8 ‖ϕ‖p

M∑

l=m+1

hl/2n

tl−p2

n−1

.

Since P is a semigroup and Qmhjis a Markov operator for j = 2, . . . , n− 1,

∥∥∥Qmhn. . . Qmhj+1

(Phj

(Ptj−1ϕ

)−Qmhj

(Ptj−1ϕ

))∥∥∥∞

≤∥∥∥Phj

(Ptj−1ϕ

)−Qmhj

(Ptj−1ϕ

)∥∥∥∞

≤ c12 ‖ϕ‖pM∑

l=m+1

hl/2j

tl−p2

j−1

for some c12 > 0. Finally, since Qmhjis a Markov operator, it follows from (3.24)

that,

∥∥Qmhn. . . Qmh2

(Ph1ϕ−Qmh1ϕ)∥∥∞ ≤

∥∥Ph1ϕ−Qmh1ϕ∥∥∞ .

Combining these last four results gives,

Eγ,n (ϕ) =∥∥PTϕ− Qmhn

. . . Qmh1ϕ∥∥∞ ≤

∥∥Ph1ϕ−Qmh1ϕ∥∥∞+c12 ||ϕ||p

n∑

j=2

M∑

l=m+1

hl/2j

tl−p2

j−1

.



It follows, almost immediately from the definition of hj that,

hj =γT (j − 1)γ−1

nγ

∫ j

j−1

(u

j − 1

)γ−1

du

but for j ∈ 2, . . . , n, ( uj−1 )γ−1 ≤ max[( j

j−1 )γ−1, 1] ≤ max[2γ−1, 1]. Hence for

l = m+ 1, . . . ,M ,

hl/2j

t(l−p)/2j−1

≤ (γT (j−1)γ−1

nγ max[2γ−1, 1])l/2

((j−1n

)γT)(l−p)/2

≤ c13(T

nγ)

l2−

(l−p)2 (j − 1)

(γ−1)l2 − γ(l−p)

2 = c13(T

nγ)

p2 (j − 1)

γp−l2

where c13 ≡ c13(γ,M) = max[1, (γmax[2γ−1, 1])M/2]. It follows that,

M∑

l=m+1

hl/2j

t(l−p)/2j−1

≤ c14(1

n)

γp2

M∑

l=m+1

(j − 1)γp−l

2

where c14 ≡ c14(γ,M, T ) = T p/2c13 and since∑M

l=m+1(j−1)γp−l

2 = (j−1)γp−(m+1)

2

×∑M−(m+1)l=0 (j − 1)−

l2 ≤ (j − 1)

γp−(m+1)2 M we have,

M∑

l=m+1

hl/2j

t(l−p)/2j−1

≤ c14M(1

n)

γp2 (j − 1)

γp−(m+1)2 . (3.21)

We now consider (3.21) for three different ranges of γ.

For γ ∈(

0, m−1p

),∑nj=2(j − 1)

γp−(m+1)2 ≤ ∑∞

j=2(j − 1)γp−(m+1)

2 and since the

series on the RHS is convergent, we have,

n−γp2

n∑

j=2

(j − 1)γp−(m+1)

2 ≤ c15n− γp

2

for some constant c15 ≡ c15(γ,M) > 0.

For γ = m−1p ,

∑nj=2(j − 1)−1 ≤ c16 lnn for some constant c16 ≡ c16(γ,M) > 0

so we have,

n− γp2

n∑

j=2

(j − 1)γp−(m+1)

2 ≤ c16n− (m−1)

2 lnn

For γ > m−1p ,

∑nj=2( j−1

n )γp−(m+1)

21n ≤ c17

∫ 1

0 xγp−(m+1)

2 dx = c17∫ 1

0 x−1+ γp−(m−1)

2 dx

≤ c18 (like a Riemann integral) for some constants c17 ≡ c17(γ,M), c18 ≡ c18(γ,M)

so,

n−γp2

n∑

j=2

(j − 1)γp−(m+1)

2 = n−m−12

n∑

j=2

(j − 1

n

) γp−(m+1)2 1

n≤ c18n

−m−12



We observe that the rate of convergence is the controlled by the maximum

between Υ (n) and the rate at which∥∥Ph1ϕ−Qmh1

ϕ∥∥∞ converges to 0. We define

Υk1,k2 (n) = Υk1 (n) + n− γk22 . Hence we have the following corollary:

Corollary 3.8.

(i) For any ϕ ∈ CMb (Rd),

Eγ,n (ϕ) ≤ c19Υm+1,m+1 (n) ‖ϕ‖M .

for some constant c19 > 0. In particular, if γ ≥ 1, then Eγ,n (ϕ) ≤ c19

nm−1

2

‖ϕ‖M .

(ii) For any ϕ ∈ C1b (Rd),

Eγ,n (ϕ) ≤ c21Υ1,1 (n) ‖ϕ‖1

for some constant c21 > 0, if there exists a constant c20 > 0 independent of t such

that,

supx∈Rd

∣∣Xt (x) − x∣∣ ≤ c20

√t. (3.22)

In particular, if γ ≥ m− 1, then Eγ,n (ϕ) ≤ c21

nm−1

2

‖ϕ‖1.

(iii) For any ϕ ∈ C lb(Rd) where 1 < l < M ,

Eγ,n (ϕ) ≤ c24Υl,c23 (n) ‖ϕ‖lfor some constant c24 > 0, if there exist two constants c22, c23 > 0 independent of

t such that,

‖Ptϕ−Qmt ϕ‖∞ ≤ c22tc232 ‖ϕ‖l . (3.23)

In particular, if γ ≥ m− 1, then Eγ,n (ϕ) ≤ c24

nm−1

2

‖ϕ‖l .

Proof.

(i) The result follows from the theorem and the definition of an m−perfect family.

(ii) If ϕ ∈ Cb(Rd) is Lipschitz then,

|Qtϕ(x) − ϕ (x)| ≤ c20 ||∇ϕ||∞√t (3.24)

hence,∥∥Ph1ϕ−Qmh1

ϕ∥∥∞ ≤ c20 ||ϕ||1

√t.

(iii) The result follows from the theorem and (3.23).

Finally we define µt to be the law of Xt :

µt (ϕ) = E [ϕ (Xt)] for ϕ ∈ Cb(Rd).

We also define µNt to be the probability measure defined by,

µNt (ϕ) = E

[Qmhn

Qmhn−1. . .Qmh1

ϕ (X0)]

=

∫

Rd

QmhnQmhn−1

. . . Qmh1ϕ (x) µ0 (dx)



for ϕ ∈ Cb(Rd) and need to introduce the family of norms on the set of signed

measures:

|µ|l = sup|µ (ϕ)| , ϕ ∈ Clb(R

d), ‖ϕ‖l,∞ ≤ 1, l ≥ 1.

Obviously, |µ|l ≤ |µ|l′ if l ≤ l′. In other words, the higher the value of l, the coarser

the norm. We have the following.

Corollary 3.9.

(i) For l ≥M , we have∣∣µt − µNt

∣∣l≤ c19Υm+1,m+1 (n). In particular, if γ ≥ 1, then∣∣µt − µNt

∣∣l≤ c19

nm−1

2

.

(ii) If (3.22) is satisfied then∣∣µt − µNt

∣∣l≤ c21Υ1,1 (n). In particular, if γ ≥ m− 1,

then∣∣µt − µNt

∣∣l≤ c21

nm−1

2

.

(iii) If (3.23) is satisfied then∣∣µt − µNt

∣∣l≤ c24Υl,c23 (n). In particular, if γ ≥ m−1,

then∣∣µt − µNt

∣∣l≤ c24

nm−1

2

.

Throughout, the constants c19, c21, c23, c24 > 0 correspond to those found in Corol-

lary 3.8.

Remark We deduce that there is a payoff between the rate of convergence and

the coarseness of the norm employed: the finer the norm the slower the rate of

convergence. Hence intermediate results such as part (iii) of Corollaries 3.8 and 3.9

may prove useful in subsequent applications. The additional constraint (3.23) holds,

for example, for the Lyons-Victoir method, as a cubature formula of degree m is

also a cubature formula of degree m′ for m′ ≤ m. Similarly, it holds for Kusuoka’s

approximation since an m−similar family is also m′−similar for any m′ ≤ m.

4. An Application to Filtering

We begin with a short description of the filtering problem. Let (X,Y ) be a system

of partially observed random processes. The process X satisfies the stochastic

differential equation (1.1) and is the unobserved component. The process Y is the

observable component and satisfies the evolution equation,

Yt =

∫ t

0

h(Xs)ds+Bt,

where Btt∈[0,∞) is an l-dimensional Brownian motion independent of X and

h =(hi)li=1

∈ C∞b

(Rd,Rl

). Let (Yt)t≥0 be the filtration generated by Y ,

Yt = σ (Ys, 0 ≤ s ≤ t). The problem of stochastic filtering for the partially ob-

served system (X,Y ) involves the construction of πt (ϕ), where π = πt, t ≥ 0 is

the conditional distribution of Xt given Yt and ϕ belongs to a suitably large class

of functions. If ϕ is square integrable with respect to the law of Xt then,

πt (ϕ) = E [ϕ (Xt) |Yt] , P − almost surely.



Using Girsanov’s theorem, one can find a new probability measure P absolutely

continuous with respect to P (and vice versa), so that Y is a Brownian motion

under P, independent of X and, almost surely,

πt(ϕ) =ρt(ϕ)

ρt(1), (4.1)

where,

ρt(ϕ) = E

[ϕ(Xt) exp

(l∑

i=1

∫ t

0

hi(Xs)dYis − 1

2

l∑

i=1

∫ t

0

hi(Xs)2ds

)∣∣∣∣∣Yt]

(4.2)

and E is the expectation with respect to P. The measure ρt is called the un-

normalised conditional distribution of the signal. The identity (4.1) is called the

Kallianpur-Striebel Formula. In the following, we will denote by ||·||p, the Lp-

norm with respect to the probability measure P, |ξ|p = E [|ξ|p]1p , for any ran-

dom variable ξ. The laws of the families X (x) = Xt (x)t∈[0,∞), x ∈ Rd and

X (x) = Xt (x)t∈[0,∞), x ∈ Rd are not affected by the change of measure, hence,

to avoid working with both P and P we can write,

(Ptϕ)(x) = E[ϕ(Xt(x))], (Qtϕ)(x) = E[ϕ(Xt(x))].

In the following, we will only consider equidistant partitions and smooth func-

tions. The method of approximation and the results closely follow the application

of the classical Euler method as described in Picard [39] and Talay [5].

Let yr, r = 1, ..., n be the observation process increments yr = Y (r+1)tn

−Y rtn

and

hr ∈ C∞b

(Rd), r = 0, ..., n − 1, be the (observation dependent) functions defined

by hr =∑li=1(hiyir − t

2n

(hi)2

). Let Rrs, Rrs : C∞

b

(Rd)→ C∞

b

(Rd), r = 0, 1, ..., n

be the following operators,

Rnsϕ (x) = Psϕ (x) , Rnsϕ (x) = Qsϕ (x) for ϕ ∈ C∞b

(Rd), x ∈ Rd

and, for r = 0, 1, ..., n− 1, and for ϕ ∈ C∞b

(Rd), x ∈ Rd,

Rrsϕ (x) = E [ϕ (Xs (x)) exp (hr (Xs (x)))| Ys] = Psϕr (x)

Rrsϕ (x) = E[ϕ(Xs (x)

)exp

(hr(Xs (x)

))∣∣Ys]

= Qsϕr (x) ,

where ϕr = ϕ exp (hr) and s ∈ [0, 1].

Firstly, one approximates ρ by replacing the (continuous) observation path with

a discrete version. We choose the equidistant partitionitn , i = 0, 1, ...n

of the

interval [0, t] and consider only the observation data yr, r = 0, 1, ..., n. We define

the measure,

ρnt (ϕ) = E

[ϕ(Xt) exp

(n−1∑

i=0

hi

(X it

n

))∣∣∣∣∣Yt]

(4.3)

= E

[R0

tnR1

tn...Rnt

nϕ (X0)

∣∣∣Yt]

for ϕ ∈ Cb(Rd).



Following Theorem 1 from Picard [39], for any ϕ ∈ C∞b (Rd) there is a constant

c ≡ c (t, ϕ) such that,

||ρt (ϕ) − ρnt (ϕ)||2 ≤ c

n

The next step is to approximate∏ni=0R

itn

with∏ni=0 R

itn

. For this we need to

adapt the definition of an m−perfect family so that we may use functions which are

parametrized by the observation path Y . Let CY,∞b (Rd) be the set of measurable

functions, f : Rd × C([0, T ],Rl

)→ R with the following properties:

i. for any y ∈ C([0, T ],Rl

)the function x→ f (x, y) belongs to C∞

b (Rd).

ii. for any multi-index α ∈ A, any x ∈ Rd and p ≥ 1, |Dαf (x, Y )|p <∞.

iii. for any multi-index α ∈ A and p ≥ 1, |||Dαf |||p,∞ =

supx∈Rd |Dαf (x, Y )|p <∞.

For f ∈ CY,∞b (Rd) we define the norm |||f |||mp =∑

α∈A(m)

|||Dαf |||p,∞. We note that

if f : Rd × C([0, T ],Rl

)→ R is constant in the y variable, then ||Dαf (x, Y )||p =

|Dαf (x, Y )| and |||f |||mp = ||f ||∞ +∥∥∇1f

∥∥∞ + ... ‖∇mf‖∞.

We now consider an m-perfect family X (x) that satisfies the equivalent of (3.7)

extended to functions in CY,∞b (Rd); more precisely we will assume that for any

f ∈ CY,∞b (Rd),

∣∣∣∣∣∣∣∣∣Qtf − E[fmt |Y ]

∣∣∣∣∣∣∣∣∣p,∞

≤ C

M∑

i=m+1

ti/2 |||f |||ip , (4.4)

for some constants C > 0 and M ≥ m + 1, where fmt is the truncation defined

in (3.4). Note that the original definition (3.7) implies (4.4) due to the inequality

‖ϕ‖V,i ≤ C ‖ϕ‖i for i ∈ N first established in Section 2. Indeed, if f ∈ C∞b (Rd),

in other words it is constant in the y variable, then (3.7) and (4.4) actually coin-

cide. The original Markov family X (x), the family generated by the Lyons-Victoir

method and the one generated by the Ninomiya-Victoir algorithm satisfy (4.4).

The following two lemmas are required to prove the main theorem below.

Lemma 4.1. Let X (x) be an m-perfect family X (x) that satisfies (4.4). Then there

is a constant c25 = c25 (t,m, p) > 0 such that for any ϕ ∈ C∞b (Rd) and r = 1, ..., n,

we have,

∣∣∣∣∣∣∣∣∣Rr−1

tn

RrtnRr+1

tn

...Rntnϕ−Rr−1

tn

RrtnRr+1

tn

...Rntnϕ∣∣∣∣∣∣∣∣∣p,∞

≤ c25n−(m+1)/2 ||ϕ||M .

Proof. Again, using the variational argument in Friedman [40] p.122 (5.17) one

can check that for any ϕ ∈ C∞b (Rd) and r = 1, ..., n the functions,

exp (hr−1)RrtnRr+1

tn

...Rntnϕ ∈ CY,∞b (Rd).



Moreover there is a constant c26 ≡ c26 (M,p) such that for any ϕ ∈ C∞b (Rd) and

r = 1, ..., n,∣∣∣∣∣∣∣∣∣exp (hr−1)Rrt

nRr+1

tn

...Rntnϕ∣∣∣∣∣∣∣∣∣M

p≤ c26 ‖ϕ‖M (4.5)

Let X (x) be an m-perfect family that satisfies (4.4). Then since X (x) also satisfies

(4.4), it follows by the triangle inequality that for s ∈ [0, 1] ,

|||Qsf − Psf |||p,∞ ≤ Cs(m+1)/2 |||f |||MpHence,

∣∣∣∣∣∣∣∣∣Q t

nexp (hr−1)Rrt

nRr+1

tn

...Rntnϕ− P t

nexp (hr−1)Rrt

nRr+1

tn

...Rntnϕ∣∣∣∣∣∣∣∣∣p,∞

≤ C

(t

n

)(m+1)/2 ∣∣∣∣∣∣∣∣∣exp (hr−1)Rrt

nRr+1

tn

...Rntnϕ∣∣∣∣∣∣∣∣∣M

p(4.6)

The result now follows from (4.5) and (4.6) and the fact that Rr−1tn

RrtnRr+1

tn

...Rntnϕ =

Q tn

exp (hr−1)RrtnRr+1

tn

...Rntnϕ and Rr−1

tn

RrtnRr+1

tn

...Rntnϕ = P t

nexp (hr−1)Rrt

nRr+1

tn

...Rntnϕ.

Lemma 4.2. Let X (x) be an m-perfect family that satisfies (4.4). Then there is a

constant c27 ≡ c27 (t,m, p) > 0 such that for any ϕ ∈ C∞b (Rd) we have,

∣∣∣∣∣∣∣∣∣R0

tnR1

tn...Rnt

nϕ− R0

tnR1

tn...Rnt

nϕ∣∣∣∣∣∣∣∣∣p,∞

≤ c27n−(m−1)/2 ||ϕ||M .

Proof. Let us observe that,

R0tnR1

tn...Rnt

nϕ− R0

tnR1

tn...Rnt

nϕ = R0

tnR1

tn...Rnt

nϕ− R0

tnR1

tn...Rnt

nϕ

+

n−1∑

j=1

R0tn...Rj−1

tn

(Rjt

n

Rj+1tn

...Rntnϕ−Rjt

n

Rj+1tn

...Rntnϕ)

+R0tn...Rn−1

tn

(Rnt

nϕ− Rnt

nϕ)

Also note that for p ≥ 1 and r = 1, ..., n,∣∣∣∣∣∣∣∣∣R0

tn...Rj−1

tn

(Rjt

n

Rj+1tn

...Rntnϕ−Rjt

n

Rj+1tn

...Rntnϕ)∣∣∣∣∣∣∣∣∣p

p,∞

≤ c28E

[∣∣∣∣∣∣∣∣∣Rr−1

tn

RrtnRr+1

tn

...Rntnϕ−Rr−1

tn

RrtnRr+1

tn

...Rntnϕ∣∣∣∣∣∣∣∣∣p

p,∞

]

≤ c28n−(m+1)/2 ||ϕ||M .

where c28 ≡ c28 (t,m, p) > 0 is a constant independent of ϕ and r. The claim

follows.

Finally let us define now the measures,

ρnt (ϕ) = E

[R0

tnR1

tn...Rnt

nϕ (X0)

∣∣∣Yt]

for ϕ ∈ Cb(Rd)

and let πnt = ρnt /ρnt (1) be its normalized version.



Theorem 4.3. Let X (x) be an m-perfect family that satisfies (4.4). Then there is

a constant c29 ≡ c29 (t,m, p) > 0 such that for any ϕ ∈ C∞b (Rd) we have,

||ρnt (ϕ) − ρnt (ϕ)||p ≤ c29n−(m−1)/2 ||ϕ||M .

Proof. We have for p ≥ 1,

||ρnt (ϕ) − ρnt (ϕ)||p ≤ E

[E

[(R0

tnR1

tn...Rnt

nϕ− R0

tnR1

tn...Rnt

nϕ)

(X0)∣∣∣Yt]p] 1

p

≤∣∣∣∣∣∣∣∣∣R0

tnR1

tn...Rnt

nϕ− R0

tnR1

tn...Rnt

nϕ∣∣∣∣∣∣∣∣∣p,∞

,

hence the result.

Corollary 4.4. Let X (x) be an m-perfect family that satisfies (4.4) with m ≥ 3

and assume that X0 has all moments finite. Then for any ϕ ∈ C∞b (Rd) there is a

constant c30 ≡ c30 (t,m, p, ϕ) > 0 such that,

||πnt (ϕ) − πnt (ϕ)||p ≤c30(ϕ)

n.

Proof. Since,

|πt(ϕ) − πnt (ϕ)| ≤ 1

ρt(1)|ρt(ϕ) − ρnt (ϕ)| +

||ϕ||ρt(1)

|ρt(1) − ρnt (1)|

the Corollary follows by applying Theorem 4.3 and the fact that∣∣∣∣ρt(1)−1

∣∣∣∣p

is finite

for any p ≥ 0.

5. Some Auxiliary Results

The main aim of this section is to deduce inequality (5.3). In the following we

will adopt the framework of Kusuoka-Stroock [13]. Let (Θ,B,W) be the standard

Wiener space with continuous paths θ : [0,∞) → Rd satisfying θ(0) = 0. Then Θ

with the topology of uniform convergence on compact intervals is a Polish space.

Also let H ⊂ Θ be the Hilbert space of absolutely continuous functions h ∈ Θ such

that ‖h‖H = (∫∞0 |h′(t)|2 dt)1/2 <∞.

Let W 1(R) denote the space of measurable Φ : Θ → R with the following two

properties:

(i) For all h ∈ H , there exists a measurable function Φh : Θ → R such that

Φh = Φ, W − a.s. and t ∈ R → Φh(θ + th) ∈ R is strictly absolutely continuous

for all θ.

(ii) There exist a measurable map, DΦ : Θ → L(H ; R) such that, for all h ∈ H and

ε > 0,

lim|t|↓0

W(θ :

∣∣∣∣Φ(θ + th) − Φ(t)

t−DΦ(θ)(h)

∣∣∣∣ ≥ ε

)= 0.



On W 1(R) the norm ‖·‖(n)q;R is defined as follows,

‖Φ‖(n)q;R :=

∥∥∥‖Φ‖(n)R

∥∥∥Lq(W)

,

where ‖Φ‖(n)R =

n∑m=0

‖DmΦ‖Hm(R) for q ∈ [2,∞) . Note that D0Φ = Φ with

∥∥D0Φ∥∥H0(R)

= |Φ| so E[|Φ|] ≤ ‖Φ‖(n)q;R . Finally let G(L) be the set of all Φ ∈

W 1(R) to which D and the Ornstein-Uhlenbeck operator L as defined in Ref. [11]

can be applied infinitely often.

Definition 5.1. We say that f ∈ ηr(Rd; R) for r ∈ Z if f is a measurable map from

(0,∞) × Rd × Ω into R such that,

(i) f(t, ·, ω) : Rd → R is smooth for each t ∈ (0,∞) and W − a.e ω ∈ Ω.

(ii) f(·, x, ·) : (0,∞) × Ω → R is progressively measurable for each x ∈ Rd.

(iii) ∂α

∂xα f(t, x, ·) ∈ G(L) and is continuous in t ∈ (0,∞) for any α ∈ A1, x ∈ Rd

(iv) sup0<t≤T

supx∈Rd

1tr/2

∥∥ ∂α

∂xα f(t, x, ω)∥∥(k)

q;R< ∞ for any α ∈ A1, k ∈ N, T > 0 and

2 ≤ q <∞.

For a proof of the following Theorem see Kusuoka-Stroock [13], Theorem 2.15 (p.

405).

Theorem 5.2. (Kusuoka-Stroock) Under the UFG condition, for any Φ ∈ηr(R

d; R), r ∈ Z and α ∈ A1 there exists Φα ∈ ηr−‖α‖(Rd; R) such that,

V[α]E[Φ(t, x)f(Xt(x))] = E[Φα(t, x)f(Xt(x))] (5.1)

for any f ∈ C∞b (Rd), t > 0, x ∈ Rd.

The following immediate extension of Theorem 5.2 holds under the additional

condition V0.

Lemma 5.3. Under the UFG+V0 condition, for any Φ ∈ ηr(Rd; R) there exists

Φα ∈ ηr−2(Rd; R) such that,

V0E[Φ(t, x)f(Xt(x))] = E[Φ0(t, x)f(Xt(x))] (5.2)

for any f ∈ C∞b (Rd), t > 0, x ∈ Rd.

Proof. From Theorem 5.2 we get that for any Φ ∈ ηr(Rd; R), r ∈ Z and β ∈ A1 (2)

there exists Φβ ∈ ηr−‖β‖(Rd; R) such that,

V[β]E[Φ(t, x)f(Xt(x))] = E[Φβ(t, x)f(Xt(x))]

for any f ∈ C∞b (Rd), t > 0, x ∈ Rd. Hence,

V0E[Φ(t, x)f(Xt(x))] =∑

β∈A1(2)

ϕβV[β]E[Φ(t, x)f(Xt(x))]

= E[Φ0(t, x)f(Xt(x))]



where Φ0(t, x) =∑

β∈A1(2)ϕβΦβ(t, x) ∈ ηr−2(Rd; R).

The following Corollary is a generalization of Corollary 2.19 (p.407) in Kusuoka-

Stroock [13].

Corollary 5.4. For any r ∈ Z+, αi ∈ A0ri=1and ‖α1 ∗ . . . ∗ αr‖, there exists a

constant CTp > 0 such that,∥∥V[α1] . . . V[αr ]Ptϕ

∥∥∞ ≤ CTp t

(p−‖α1∗...∗αr‖)/2 ‖ϕ‖p , t ∈ [0, T ] . (5.3)

for ϕ ∈ C∞b (Rd) where ‖·‖p for p ∈ N is defined in (2.2).

Proof.

For the case p = 1, let Y (x) be the matrix valued process Y i,jt (x) :=

∂∂xj

(X i(x)) where i, j = 1, ..., d. (see Ikeda & Watanabe [41], Chapter V, for de-

tails). Then for any i, j = 1, ..., d we have Y i,j(t, x) ∈ η0(Rd; R), in particular,

supt∈[0,T ]

supx∈Rd

E

[d∑

i=1

Y i,j(x)

]≤ c(T ). (5.4)

Differentiating under the integral sign (see Friedman [40], p.122 (5.17)) gives,

∂

∂xjE[ϕ(Xt(x))] =

d∑

i=1

E

[∂ϕ

∂xi(Xt(x))Y i,jt (x)

]. (5.5)

Hence,

V[αr ]Ptϕ (x) =

d∑

i=1

E

[∂ϕ

∂xi(Xt(x))Φi(t, x)

]

with Φi(t, x) =∑d

j=1 Vj[αr ] (x)Y i,jt (x) ∈ η0(Rd; R) for i = 1, ..., d. Then by Theorem

5.2 & Lemma 5.3 there exists Φiα ∈ η−(‖α1∗···∗αr−1‖)(Rd; R), i = 1, ..., d such that,

for any ϕ ∈ C1b (Rd),

V[α1] . . . V[αr−1]V[αr]Ptϕ (x) =

d∑

i=1

V[α1] . . . V[αr−1]E

[∂ϕ

∂xi(Xt(x))Φi(t, x)

]

=

d∑

i=1

E

[Φiα(t, x)

∂ϕ

∂xi(Xt(x))

]

for any ϕ ∈ C∞b (Rd). Hence,

∣∣V[α1] . . . V[αr]Ptϕ (x)∣∣ ≤ ||ϕ||1

d∑

i=1

E[|Φiα(t, x)|]

and (5.3) follows from the uniform bound (in (t, x)) on the L1-norm of Φiα(t, x), i =

1, ..., d.



For p > 1 one can obtain an analogue of (5.5) with higher derivatives of Ptϕ in

terms of derivatives of ϕ and a set of processes analogous to Y . The proof follows

in a similar manner.

Acknowledgments

The authors wish to thank the anonymous referee for his comments. We would

also like to thank Alan Bain for carefully reading the paper and his many pertinent

remarks.

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Chapter 9

Flow Properties of Differential Equations Driven by Fractional

Brownian Motion

Laurent Decreusefond and David Nualart∗

GET, Ecole Nationale Superieure des TelecommunicationsLTCI-UMR 5141, CNRS, 46, rue Barrault, 75634 Paris, France

[email protected]

We prove that solutions of stochastic differential equations driven by fractionalBrownian motion for H > 1/2 define flows of homeomorphisms on Rd.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

3 Stochastic differential equations driven by an fBm . . . . . . . . . . . . . . . . . . . . 253

4 Flow of homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

1. Introduction

Suppose that BH = BHt , t ≥ 0 is an m-dimensional fractional Brownian motion

with Hurst parameter H ∈ (0, 1), defined in a complete probability space (Ω,F , P ).

That is, the components BH,j , j = 1, . . . ,m, are independent zero mean Gaussian

processes with the covariance function

RH(t, s) =1

2

(t2H + s2H − |t− s|2H

). (1.1)

For H = 12 , the process BH is an m-dimensional ordinary Brownian motion. On

the other hand, from (1.1) it follows that

E(∣∣∣BH,jt −BH,js

∣∣∣2

) = |t− s|2H .

As a consequence, the processesBH,j have stationary increments, and for any α < H

we can select versions with Holder continuous trajectories of order α on a compact

interval [0, T ].

∗Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, Kansas 66045-7523, [email protected]

249


250 L. Decreusefond and D. Nualart

This process was first studied by Kolmogorov [1] and later by Mandelbrot and

Van Ness [2], where a stochastic integral representation in terms of an ordinary

Brownian motion was established.

The fractional Brownian motion has the following self-similar property: For

any constant a > 0, the processesa−HBHat , t ≥ 0

and

BHt , t ≥ 0

have the same

distribution. For H = 12 the process BH has independent increments, but for

H 6= 12 , this property is no longer true. In particular, if H > 1

2 , the fractional

Brownian motion has the long range dependence property, which means that for

each j = 1, . . . ,m

∞∑

n=1

Corr(BH,jn+1 −BH,jn , BH,j1 ) = ∞.

The self-similar and long range dependence properties make the persistent frac-

tional Brownian motion a convenient model for some input noises in a variety of

topics from finance to telecommunication networks, where the Markov property is

not required. This fact has motivated the recent development of the stochastic

calculus with respect to the fractional Brownian motion. See Refs. [3–5], and the

references therein.

In this paper we are interested in stochastic differential equations on Rd driven

by a multi-dimensional fractional Brownian motion with Hurst parameter H > 12 ,

that is, equations of the form

X it = X i

0 +

m∑

j=1

∫ t

0

σi,j(s,Xs)dBH,js +

∫ t

0

bi(s,Xs)ds, (1.2)

i = 1, . . . , d. The stochastic integral appearing in (1.2) is a path-wise Riemann-

Stieltjes integral. In fact, under suitable conditions on σ, the processes σ(s,Xs)

and BHs have trajectories which are Holder continuous of order strictly larger than12 and we can use the approach introduced by Young [6]. A first result on the

existence and uniqueness of a solution for this kind of equations was obtained by

Lyons [7], using the notion of p-variation. On the other hand, the theory of rough

path analysis introduced by Lyons [7] (see also the monograph by Lyons and

Qian [8]), has allowed Coutin and Qian [9] to establish the existence of strong

solutions and a Wong-Zakai type approximation limit for the stochastic differential

equations of the form (1.2) driven by a fractional Brownian motion with parameter

H > 14 . In Ref. [9] sufficient conditions on the vector fields b and σ are given to

ensure existence and uniqueness of the solution of (1.2) even when vector fields do

not commute.

Zahle [10] has introduced a generalized Stieltjes integral using the techniques

of fractional calculus. This integral is expressed in terms of fractional derivative

operators and it coincides with the Riemann-Stieltjes integral∫ bafdg, when the

functions f and g are Holder continuous of orders λ and µ, respectively and λ+µ > 1.

Using this formula for the Riemann-Stieltjes integral, Nualart and Rascanu [11] have


Flows Properties of SDEs Driven by Fractional Brownian Motion 251

obtained the existence of a unique solution for the stochastic differential equations

(1.2) under general conditions on the coefficients.

Later on, Nualart and Saussereau [12] have studied the regularity in the sense of

Malliavin Calculus of the solution of Equation (1.2), and they have established the

absolute continuity of the law of the random variableXt under some non-degeneracy

conditions on the coefficient σ.

The main result of this paper is the flow and homeomorphic properties of X as

a function of the initial condition x. Since the solution of (1.2) is defined path-wise,

ordinary (i.e., deterministic) methods are in use here. Namely, we use the estimates

found in Ref. [11] and approximate fractional Brownian motion by a sequence of

regular processes to prove that the solution Xrt(x), 0 ≤ r ≤ t ≤ T, x ∈ Rd of

Xrt(x) = x+

∫ t

r

σ (s,Xrs(x)) dBH(s) +

∫ t

r

b(s,Xrs(x))ds,

defines a flow of Rd-homeomorphisms.

The paper is organized as follows. Section 2 contains some preliminaries on

fractional calculus. In Section 3 we review some results on the properties of the

solution of Equation (1.2) and we establish some continuity estimates as a function

of the initial condition and the driven input, which are needed later. Finally in

Section 4 we show that Equation (1.2) defines a flow of homeomorphisms.

2. Preliminaries

Let a, b ∈ R, a < b. Let f ∈ L1 (a, b) and α > 0. The left-sided and right-sided

fractional Riemann-Liouville integrals of f of order α are defined for almost all

x ∈ (a, b) by

Iαa+f (x) =1

Γ (α)

∫ x

a

(x− y)α−1

f (y) dy

and

Iαb−f (x) =(−1)

−α

Γ (α)

∫ b

x

(y − x)α−1

f (y) dy,

respectively, where (−1)−α

= e−iπα and Γ (α) =∫∞0 rα−1e−rdr is the Euler func-

tion. Let Iαa+(Lp) (resp. Iαb−(Lp)) be the image of Lp(a, b) under the action of

operator Iαa+ (resp. Iαb−). If f ∈ Iαa+ (Lp) (resp. f ∈ Iαb− (Lp)) and 0 < α < 1 then

the Weyl derivative

Dαa+f (x) =

1

Γ (1 − α)

(f (x)

(x− a)α + α

∫ x

a

f (x) − f (y)

(x− y)α+1 dy

)1(a,b)(x) (2.1)

(resp. Dα

b−f (x) =(−1)α

Γ (1 − α)

(f (x)

(b− x)α+ α

∫ b

x

f (x) − f (y)

(y − x)α+1 dy

)1(a,b)(x)

)

(2.2)



is defined for almost all x ∈ (a, b) .

For any 0 < λ ≤ 1, denote by Cλ(0, T ) the space of λ-Holder continuous func-

tions f : [0, T ] → R, equipped with the norm ‖f‖∞ + ‖f‖λ, where

‖f‖λ = sup0≤s<t≤T

|f (t) − f (s)|(t− s)λ

.

Recall from Ref. [13] that we have:

• If α <1

pand q =

p

1 − αpthen Iαa+ (Lp) = Iαb− (Lp) ⊂ Lq (a, b) .

• If α >1

pthen Iαa+ (Lp) ∪ Iαb− (Lp) ⊂ Cα−

1p (a, b) .

The linear spaces Iαa+ (Lp) are Banach spaces with respect to the norms

‖f‖Iαa+(Lp) = ‖f‖Lp +

∥∥Dαa+f

∥∥Lp ∼

∥∥Dαa+f

∥∥Lp ,

and the same is true for Iαb− (Lp).

Suppose that f ∈ Cλ(a, b) and g ∈ Cµ(a, b) with λ + µ > 1. Then, from

the classical paper by Young [6], the Riemann-Stieltjes integral∫ ba fdg exists. The

following proposition can be regarded as a fractional integration by parts formula,

and provides an explicit expression for the integral∫ ba fdg in terms of fractional

derivatives (see Ref. [10]).

Proposition 2.1. Suppose that f ∈ Cλ(a, b) and g ∈ Cµ(a, b) with λ+ µ > 1. Let

λ > α and µ > 1 − α. Then the Riemann-Stieltjes integral∫ bafdg exists and it can

be expressed as

∫ b

a

fdg = (−1)α∫ b

a

Dαa+f (t)D1−α

b− gb− (t) dt, (2.3)

where gb− (t) = g (t) − g (b).

Zahle [10] introduced a generalized Stieltjes integral of f with respect to g defined

by the right-hand side of (2.3), assuming that f and g are functions such that g(b−)

exists, f ∈ Iαa+ (Lp) and gb− ∈ I1−αb− (Lq) for some p, q ≥ 1, 1/p+1/q ≤ 1, 0 < α < 1.

Let α < 12 and d ∈ N∗. Denote by Wα,∞

0 (0, T ; Rd) the space of measurable

functions f : [0, T ] → Rd such that

‖f‖α,∞ := supt∈[0,T ]

(|f(t)| +

∫ t

0

|f (t) − f (s)|(t− s)

α+1 ds

)<∞.

We have, for all 0 < ε < α

Cα+ε(0, T ; Rd) ⊂Wα,∞0 (0, T ; Rd) ⊂ Cα−ε(0, T ; Rd).



Denote by W 1−α,∞T (0, T ; Rm) the space of measurable functions g : [0, T ] → Rm

such that

‖g‖1−α,∞,T := sup0<s<t<T

( |g(t) − g(s)|(t− s)1−α

+

∫ t

s

|g(y) − g(s)|(y − s)2−α

dy

)<∞.

Clearly, for all ε > 0 we have

C1−α+ε (0, T ; Rm) ⊂W 1−α,∞T (0, T ; Rm) ⊂ C1−α (0, T ; Rm) . (2.4)

Moreover, if g belongs to W 1−α,∞T (0, T ; Rm), its restriction to (0, t) belongs to

I1−αt− (L∞(0, t; Rm)) for all t and

Λα(g) :=1

Γ(1 − α)sup

0<s<t<T

∣∣(D1−αt− gt−

)(s)∣∣

≤ 1

Γ(1 − α)Γ(α)‖g‖1−α,∞,T <∞. (2.5)

The integral∫ t0fdg can be defined for all t ∈ [0, T ] if g belongs to W 1−α,∞

T (0, T )

and f satisfies

‖f‖α,1 :=

∫ T

0

|f(s)|sα

ds+

∫ T

0

∫ s

0

|f(s) − f(y)|(s− y)α+1

dy ds <∞.

Furthermore the following estimate holds

∣∣∣∣∣

∫ T

0

fdg

∣∣∣∣∣ ≤ Λα(g) ‖f‖α,1 .

3. Stochastic differential equations driven by an fBm

We are going to consider first the case of a deterministic equation. Let 0 < α < 12 be

fixed. Let g ∈ W 1−α,∞T (0, T ; Rm). Consider the deterministic differential equation

on Rd

ξit = xi0 +

∫ t

0

bi(s, ξs)ds+

m∑

j=1

∫ t

0

σi,j (s, ξs) dgjs , t ∈ [0, T ] , (3.1)

i = 1, ..., d, where x0 ∈ Rd, and the coefficients σi,j , bi : [0, T ]×Rd→ R are measur-

able functions. Set σ =(σi,j)d×m , b =

(bi)d×1

and for a matrix A =(ai,j)d×m

and a vector y =(yi)d×1

denote |A|2 =∑

i,j

∣∣ai,j∣∣2 and |y|2 =

∑i

∣∣yi∣∣2 .

Let us consider the following assumptions on the coefficients.

(H1) σ(t, x) is differentiable in x, and there exist some constants 0 < β, δ ≤ 1,



M1,M2,M3 > 0 such that the following properties hold:

i) Lipschitz continuity

|σ(t, x) − σ(t, y)| ≤M1|x− y|, ∀x ∈ Rd, ∀ t ∈ [0, T ]

ii) Holder continuity

|∂xiσ(t, x) − ∂xiσ(t, y)| ≤M2|x− y|δ,∀ |x| , |y| ∈ Rd, ∀t ∈ [0, T ] , i = 1, . . . , d,

iii) Holder continuity in time

|σ(t, x) − σ(s, x)| + |∂xiσ(t, x) − ∂xiσ(s, x)| ≤M3|t− s|β∀x ∈ Rd, ∀ t, s ∈ [0, T ] .

(H2) There exist constants L1, L2 > 0 such that the following properties hold:

i) Local Lipschitz continuity

|b(t, x) − b(t, y)| ≤ L1|x− y|, ∀ |x| , |y| ∈ Rd, ∀t ∈ [0, T ] ,

ii) Linear growth

|b(t, x)| ≤ L2(1 + |x|), ∀x ∈ Rd, ∀t ∈ [0, T ] .

Set

α0 = min

1

2, β,

δ

1 + δ

.

The following existence and uniqueness result has been proved in Ref. [11].

Theorem 3.1. Suppose that the coefficients σ(t, x) and b(t, x) satisfy assumptions

(H1) and (H2). Then, if α < α0 there exists a unique solution of Equation (3.1) in

the space C1−α (0, T ; Rd).

Actually, these conditions can be slightly relaxed. For instance, the Holder

continuity of the partial derivatives of σ and the Holder continuity of the coefficient

b may hold only locally (see Ref. [11] for the details).

We now state two theorems which are consequences of the estimates found in

Ref. [11].

For any λ ≥ 0 we introduce the equivalent norm in the space W α,∞0

(0, T ; Rd

)

defined by

‖f‖α,λ = supt∈[0,T ]

e−λt(|f(t)| +

∫ t

0

|f(t) − f(s)|(t− s)α+1

ds

).

Theorem 3.2. Let ξt(x0) denote the solution of (3.1) at time t with initial condition

x0. Fix R > 1. Then there exists a constant C such that for any x0 and x1 in the ball

B(0, R) = x ∈ Rd, |x| ≤ R and for any λ >[R exp

(C (1 + Λα(g))

11−2α

)] 11−2α

we have

‖ξ(x0) − ξ(x1)‖α, λ≤(

1 −R exp(C (1 + Λα(g))

11−2α

)λ2α−1

)−1

|x0 − x1|.



Proof. It is proved in Ref. [11] that there exists a constant C1 such that if λ0 =

C1 (1 + Λα(g))1

1−2α then for any initial condition x0 in the ball of radius R we have

‖ξ(x0)‖α,λ0≤ 2 (1 + |x0|) ≤ 4R. (3.2)

Given a function f ∈ Wα,∞0 (0, T ; Rd) we define as in Ref. [11]

F(b)t (f) =

∫ t

0

b(s, f(s))ds,

G(σ)t (g, f) =

∫ t

0

σ(s, f(s))dg(s),

∆(f) = supr∈[0,T ]

∫ r

0

|f(r) − f(s)|δ(r − s)α+1

ds.

If f, h ∈Wα,∞0 (0, T ; Rd) (see Ref. [11]) there exist constants C2 and C3 such that

∥∥∥F (b)(f) − F (b)(h)∥∥∥α,λ

≤ C2λα−1 ‖f − h‖α,λ , (3.3)

∥∥∥G(σ) (g, f) −G(σ) (g, h)∥∥∥α,λ

≤ C3Λα(g)λ2α−1

× (1 + ∆ (f) + ∆ (h)) ‖f − h‖α,λ , (3.4)∥∥∥G(σ) (g, f)

∥∥∥α,λ

≤ C4Λα(g)λ2α−1(

1 + ‖f‖α,λ)

(3.5)

Also, if f ∈Wα,∞0 (0, T ; Rd) and h is a bounded measurable function, then

∥∥∥F (b)(h)∥∥∥

1−α≤ C5(1 + ‖h‖∞), (3.6)

∥∥∥G(σ) (g, f)∥∥∥

1−α≤ C6Λα(g)

(1 + ‖f‖α,∞

). (3.7)

We have the following estimate

∆(ξ(x0)) = supr∈[0,T ]

∫ r

0

|ξr(x0) − ξs(x0)|δ(r − s)α+1

ds

≤ T δ−α(1+δ)

δ − α (1 + δ)‖ξ(x0)‖1−α , (3.8)

and using (3.6), (3.7), and (3.2) we obtain

‖ξ(x0)‖1−α ≤ |x0| +∥∥∥F (b) (ξ(x0))

∥∥∥1−α

+∥∥∥G(σ) (ξ(x0))

∥∥∥1−α

≤ |x0| + C5 (1 + ‖ξ(x0)‖∞) + Λα(g)C6

(1 + ‖ξ(x0)‖α,∞

)

≤ C7eλ0T (1 + |x0|)(1 + Λα(g))

≤ 2C7eλ0TR(1 + Λα(g))

= 2C7 exp(TC1 (1 + Λα(g))

11−2α

)R(1 + Λα(g)).



Hence, from (3.8) we get

∆(ξ(x0)) ≤ C8 exp(TC1 (1 + Λα(g))

11−2α

)R(1 + Λα(g)). (3.9)

Using (3.3), (3.4), and (3.9) we obtain for x0 and x1 in the ball of radius R

‖ξ(x0) − ξ(x1)‖α, λ ≤ |x0 − x1| + ‖F (b)(ξ(x0)) − F (b)(ξ(x1))‖α, λ+∥∥∥G(σ)(g, ξ(x0)) −G(σ)(g, ξ(x1))

∥∥∥α,λ

≤ |x0 − x1| + C1λα−1 ‖ξ(x0) − ξ(x1)‖α,λ

+ C2Λα(g)λ2α−1 (1 + ∆ (ξ(x0)) + ∆ (ξ(x1))) ‖ξ(x0) − ξ(x1)‖α,λ≤ |x0 − x1| + C1λ

α−1 ‖ξ(x0) − ξ(x1)‖α,λ+ C2Λα(g)λ2α−1

(1 + 2C8 exp

(TC1 (1 + Λα(g))

11−2α

)

×R(1 + Λα(g)))‖ξ(x0) − ξ(x1)‖α,λ .

As a consequence,

‖ξ(x0) − ξ(x1)‖α, λ≤ |x0 − x1| + λ2α−1

(exp

(C (1 + Λα(g))

11−2α

)R)‖ξ(x0) − ξ(x1)‖α,λ

for some constant C, which implies the result.

Theorem 3.3. The map

ξ : W 1−α,∞T (0, T ; Rm) −→Wα,∞

0 (0, T ; Rd)

g 7−→ ξ,

where ξ is the solution of (3.1) with x0 ∈ B(0, R), is continuous. Namely, for λ > 0

large enough we have

‖ξ(g) − ξ(h)‖α, λ ≤ C1λ2α−1‖ξ(g)‖α, λ

1 − C2λ2α−1 (1 + Λα(h))Λα(g − h).

Proof. With the previous notations, we have

ξ(g) = x0 + F (b)(ξ(g)) +G(σ)(g, ξ(g))

and

ξ(h) = x0 + F (b)(ξ(h)) +G(σ)(h, ξ(h)).

Thus,

ξ(g) − ξ(h) = F (b)(ξ(g)) − F (b)(ξ(h))

+G(σ)(g − h, ξ(g)) +(G(σ)(h, ξ(g)) −G(σ)(h, ξ(h))

).



According to (3.5), for λ sufficiently large,

‖ξ(g) − ξ(h)‖α, λ ≤ C2λα−1 ‖ξ(g) − ξ(h)‖α,λ

+ C4λ2α−1(1 + Λα(g − h))‖ξ(g)‖α, λ

+ C3λ2α−1Λα(h)‖ξ(g) − ξ(h)‖α, λ.

Hence the result.

Consider Equation (1.2) on Rd, for t ∈ [0, T ], where X0 is a d-dimensional

random variable, and the coefficients σi,j , bi : Ω × [0, T ] × Rd→R are measurable

functions.

Theorem 3.4. Suppose that X0 is an Rd-valued random variable, the coefficients

σ(t, x) and b(t, x) satisfy assumptions (H1) and (H2), where the constants might

depend on ω, with β > 1−H, δ > 1/H−1. Then if α ∈ (1 −H,α0), then there exists

a unique stochastic process, whose trajectories belong to the space W α,∞0 (0, T ; Rd),

solution of the stochastic equation (1.2) and, moreover, for P-almost all ω ∈ Ω

X (ω, ·) ∈ C1−α (0, T ; Rd).

Consider the particular case where b = 0 and σ is time independent, that is,

Xt = X0 +

∫ t

0

σ(Xs)dBHs . (3.10)

By the above theorem this equation has a unique solution provided σ is continuously

differentiable, and σ′ is bounded and Holder continuous of order δ > 1H − 1.

Hu and Nualart [14] have established the following estimates. Choose θ ∈(12 , H

). Then, the solution to Equation (3.10) satisfies

sup0≤t≤T

|Xt| ≤ 21+kT(‖σ′‖∞∨|σ(0)|)1/θ‖BH‖1/θθ (|X0| + 1) , (3.11)

where k is a constant depending only on θ. Moreover, if σ is bounded and ‖σ′‖∞ 6= 0

this estimate can be improved in the following way

sup0≤t≤T

|Xt| ≤ |X0| + k‖σ‖∞(T θ‖BH‖θ ∨ T‖σ′‖

1−θθ∞ ‖BH‖

1θ

θ

), (3.12)

where again k is a constant depending only on θ.

These estimates improve those obtained by Nualart and Rascanu [11] based on

a suitable version of Gronwall’s lemma. The estimates (3.11) and (3.12) lead to the

following integrability properties for the solution of Equation (3.10).

Theorem 3.5. Consider the stochastic differential equation (3.10), and assume

that E(|X0|p) < ∞ for all p ≥ 2. If σ′ is bounded and Holder continuous of order

δ > 1H − 1, then

E

(sup

0≤t≤T|Xt|p

)<∞ (3.13)



for all p ≥ 2. If furthermore σ is bounded and E (exp(λ|X0|γ)) <∞ for any λ > 0

and γ < 2H, then

E

(expλ

(sup

0≤t≤T|Xt|γ

))<∞ (3.14)

for any λ > 0 and γ < 2H.

Nualart and Saussereau [12] have proved that the random variable Xt belongs

locally to the space D∞ if the function σ is infinitely differentiable and bounded

together with all its partial derivatives. As a consequence, they have derived the

absolute continuity of the law of Xt for any t > 0 assuming that the initial condition

is constant and the vector space spanned by (σi,j(x0))1≤i≤d, 1 ≤ j ≤ m is Rd.

Applying Theorem 3.5 Hu and Nualart have proved [14] that if the function σ is

infinitely differentiable and bounded together with all its partial derivatives, then for

any t ∈ [0, T ] the random variable Xt belongs to the space D∞. As a consequence,

if the matrix a(x) = σσT (x) is uniformly elliptic, then, for any t > 0 the probability

law of Xt has a C∞ density. In a recent paper, Baudoin and Coutin [15] have

extended this result and derived the regularity of the density under Hormander

hypoellipticity conditions.

4. Flow of homeomorphisms

Let π = 0 = t0 < t1 < · · · < tn = T be the uniform partition of the interval [0, T ].

That is tk = kTn , k = 0, . . . , n. We denote by Bn,H the polygonal approximation of

the fractional Brownian motion defined by

Bn,Ht =n−1∑

k=0

(BHtk +

n

T(t− tk)

(BHtk+1

−BHtk

))1(tk,tk+1](t).

In order to get a precise rate for these approximations we will make use of the

following exact modulus of continuity of the fractional Brownian motion (see Ref.

[16]). There exists a random variable G such that almost surely for any s, t ∈ [0, T ]

we have∣∣BHt −BHs

∣∣ ≤ G|t− s|H√

log (|t− s|−1). (4.1)

Fix θ < H . We have the following result, which provides the rate of convergence of

these approximations in Holder norm.

Lemma 4.1. There exist a random variable CT,β such that

‖BH −Bn,H‖Cθ(0,T ;Rm) ≤ CT,βnθ−H√logn. (4.2)

Proof. To simplify the notation we will assume that m = 1. Fix 0 < s < t < T

and assume that s ∈ [tl, tl+1] and t ∈ [tk, tk+1]. Let us first estimate

h1(s, t) =1

(t− s)θ|Bn,Ht −BHt − (Bn,Hs −BHs )| .



If t− s ≥ Tn , then using (4.1) we obtain

|h1(s, t)| ≤ T−βnβ[∣∣∣BHtk −BHt +

n

T(t− tk)

(BHtk+1

−BHtk

)∣∣∣

+∣∣∣BHtl −BHs +

n

T(s− tl)

(BHtl+1

−BHtl

)∣∣∣]

≤ 4GT−θ+Hn−H+θ√

log (n/T ).

If t − s < Tn , then there are two cases. Suppose first that s, t ∈ [tk, tk+1]. In this

case, if n is large enough we obtain using (4.1)

|h1(s, t)| ≤ |BHt −BHs |(t− s)θ

+n

T

|BHtk+1−BHtk |

(t− s)θ(t− s)

≤ G|t− s|H−θ√log |t− s|−1 +GT−1+H√

log(n/T ) n1−H(t− s)1−θ


log (n/T ).

On the other hand, if s ∈ [tk−1, tk] and t ∈ [tk, tk+1] we have, again if n is large

enough

|h1(s, t)| ≤ 1

(t− s)θ

∣∣∣BHtk −BHt +n

T(t− tk)

(BHtk+1

−BHtk

)

−BHtk −BHs − n

T(tk − s)

(BHtk −BHtk−1

)∣∣∣

≤ 1

(t− s)θ

[|BHt −BHs | +

n

T(t− s)

(|BHtk −BHtk−1

| + |BHtk+1−BHtk |

)]

≤ G

(t− s)θ

[|t− s|H

√log |t− s|−1 + 2(t− s)

( nT

)H√log (n/T )

]


log (n/T ).

This proves (4.2).

Corollary 4.2. For any α ∈ (1 −H, 1/2), we have:

supn

Λα(Bn,H) < +∞ and limn→+∞

Λα(Bn,H −BH) = 0.

Proof. Choose η > 0 in such a way that 1 − α+ η < H . According to (2.4) and

(2.5), we have

Λα(Bn,H) ≤ cη‖Bn,H‖C1−α+η(0,T ;Rm)

and

Λα(Bn,H −BH) ≤ cη‖Bn,H −BH‖C1−α+η(0,T ;Rm).

Then, Lemma 4.1 implies that the sequence Bn,H converges to BH in the norm of

C1−α+η(0, T ; Rm) which yields the results.



Consider for any 0 ≤ r ≤ t ≤ T and any natural number n ≥ 1 the following

equations

Xnrt(x) = x+

∫ t

r

σ (s,Xnrs(x)) dBn,H(s) +

∫ t

r

b(s,Xnrs(x))ds, (4.3)

and

Y nrt(x) = x+

∫ t

r

σ (s, Y nst(x)) dBn,H(s) +

∫ t

r

b(s, Y nst(x))ds. (4.4)

We know from standard results on ordinary differential equations that for any n ≥ 1,

(1) Equations (4.3) and (4.4) have a unique solution.

(2) For any x ∈ Rd, for any 0 ≤ r ≤ τ ≤ t ≤ T , Xnτt(X

nrτ (x)) = Xn

rt(x).

(3) For any x ∈ Rd, for any 0 ≤ r ≤ τ ≤ t ≤ T , Y nrτ (Y nτt(x)) = Y nrt(x).

(4) The maps (x 7→ Xnrt(x)) and (x 7→ Y nrt(x)) are Rd-homeomorphisms inverse of

each other:

Xnrt(Y

nrt(x)) = x and Y nrt(X

nrt(x)) = x.

We are then in position to prove our main theorem:

Theorem 4.3. Assume that Hypotheses (H1) and (H2) hold. Then, claims 1, 2, 3

and 4 also hold for the equations

Xrt(x) = x+

∫ t

r

σ (s,Xrs(x)) dBH(s) +

∫ t

r

b(s,Xrs(x))ds, (4.5)

and

Yrt(x) = x+

∫ t

r

σ (s, Yst(x)) dBH (s) +

∫ t

r

b(s, Yst(x))ds. (4.6)

Proof. Point 1 is proved in Ref. [11]. As to the second claim, proceed as follows:

Xnτt(X

nrτ (x)) −Xτt(Xrτ (x)) = Xn

τt(Xnrτ (x)) −Xn

τt(Xrτ (x))

+ (Xnτt −Xτt)(Xrτ (x)).

Fix ε > 0 and α such that 1−H < α < 12 . Fix a trajectory ω ∈ Ω. Choose n0 so that

Λα(Bn,H−BH ) ≤ ε for all n ≥ n0 and choose λ such that λ2α−1C2 supn Λα(Bn,H) ≤12 . Then, according to Theorem 3.3, for any n ≥ n0,

‖Xnr· −Xr·‖α,λ ≤ C1λ

2α−1‖Xr·‖α, λ1 − C2λ2α−1 (1 + Λα(Bn,H))

Λα(Bn,H −BH)

≤ 2C1λ2α−1‖Xr·‖α, λε.

Hence, for n ≥ n0,

|(Xnτt −Xτt)(Xrτ (x))| ≤ cε.



The convergence of Xnr· implies that there exists R such that for any τ ∈ [r, t] and

for any n ≥ n0, Xnrτ (x) ∈ B(0, R). Then, Theorem 3.2 implies that for λ large

enough

|Xnτt(X

nrτ (x)) −Xn

τt(Xrτ (x))|

≤(

1 −R exp

(C

(1 + sup

nΛα(Bn,H)

) 11−2α

)λ2α−1

)−1

× |Xnrτ (x) −Xrτ (x)|

≤ c|Xnrτ (x) −Xrτ (x)|.

We have thus proved that

0 = limn→+∞

Xnτt(X

nrτ (x)) −Xτt(Xrτ (x))

= limn→+∞

Xnrt(x) −Xτt(Xrτ (x))

= Xrt(x) −Xτt(Xrτ (x)).

Other points are handled similarly.

Acknowledgement

This work was carried out during a stay of Laurent Decreusefond at Kansas Univer-

sity, Lawrence KS. He would like to thank KU for warm hospitality and generous

support.

References

[1] A. N. Kolmogorov, Wienershe Spiralen und einige andere interessante Kurven imHilbertschen Raum, C. R. (Doklady) Acd. Sci. URSS (N. S.). 26, 115–118, (1940).

[2] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noisesand applications, SIAM Rev. 10, 422–437, (1968). ISSN 1095-7200.

[3] L. Decreusefond and A. S. Ustunel, Stochastic analysis of the fractional Brownianmotion, Potential Anal. 10(2), 177–214, (1999). ISSN 0926-2601.

[4] D. Feyel and A. de La Pradelle, On fractional Brownian processes, Potential Anal.10(3), 273–288, (1999). ISSN 0926-2601.

[5] D. Nualart. Stochastic integration with respect to fractional Brownian motion andapplications. In Stochastic models (Mexico City, 2002), vol. 336, Contemp. Math.,pp. 3–39. Amer. Math. Soc., Providence, RI, (2003).

[6] L. C. Young, An inequality of the Holder type connected with Stieltjes integration,Acta Math. 67, 251–282, (1936).

[7] T. Lyons, Differential equations driven by rough signals. I. An extension of an in-equality of L. C. Young, Math. Res. Lett. 1(4), 451–464, (1994). ISSN 1073-2780.

[8] T. Lyons and Z. Qian, System control and rough paths. Oxford Mathematical Mono-graphs, (Oxford University Press, Oxford, 2002). ISBN 0-19-850648-1. Oxford SciencePublications.



[9] L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brown-ian motions, Probab. Theory Related Fields. 122(1), 108–140, (2002). ISSN 0178-8051.

[10] M. Zahle, Integration with respect to fractal functions and stochastic calculus. I,Probab. Theory Related Fields. 111(3), 333–374, (1998). ISSN 0178-8051.

[11] D. Nualart and A. Rascanu, Differential equations driven by fractional Brownianmotion, Collect. Math. 53(1), 55–81, (2002). ISSN 0010-0757.

[12] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equationsdriven by a fractional Brownian motion, (2005). Preprint.

[13] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives.(Gordon and Breach Science Publishers, Yverdon, 1993). ISBN 2-88124-864-0. Theoryand applications, Edited and with a foreword by S. M. Nikol′skiı, Translated fromthe 1987 Russian original, Revised by the authors.

[14] Y. Hu and D. Nualart. Differential equations driven by Holder continuous functionsof order greater than 1/2, (2006). Preprint.

[15] F. Baudoin and M. Hairer. A version of hormander’s theorem for the fractional brow-nian motion, (2006). Preprint.

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Chapter 10

Regularity of Transition Semigroups Associated to a 3D

Stochastic Navier-Stokes Equation

Franco Flandoli and Marco Romito∗

Dipartimento di Matematica Applicata, Universit di Pisavia Bonanno Pisano, 25/b, 56126 Pisa, Italy

[email protected]

A 3D stochastic Navier-Stokes equation with a suitable non degenerate additivenoise is considered. The regularity in the initial conditions of every Markovtransition kernel associated to the equation is studied by a simple direct approach.A by-product of the technique is the equivalence of all transition probabilitiesassociated to every Markov transition kernel.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

2.2 Definitions, assumptions and known results . . . . . . . . . . . . . . . . . . . . 266

3 The Log-Lipschitz estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

3.1 Probability of blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

3.2 Derivative of the regularised problem . . . . . . . . . . . . . . . . . . . . . . . . 271

4 Equivalence of all transition probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 272

5 Conclusion and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

A.1 An exponential tail estimate for the Stokes problem . . . . . . . . . . . . . . . . . . . 275

A.2 The deterministic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

1. Introduction

An old dream in stochastic fluid dynamics is to prove the well posedness of a stochas-

tic version of the 3D Navier-Stokes equations, taking advantage of the noise, as

one can do for finite dimensional stochastic equations with non regular drift (see

for instance Stroock & Varadhan [1]). The problem is still open, although some

intriguing results have been recently proved, see for instance Da Prato & Debuss-

che [2], Mikulevicius & Rozovski [3], Flandoli & Romito [4, 5]. We recall here the

framework constructed in Ref. [4] and prove some additional results.

∗Dipartimento di Matematica, Universit di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy,[email protected]

263


264 F. Flandoli and M. Romito

We consider a viscous, incompressible, homogeneous, Newtonian fluid described

by the stochastic Navier-Stokes equations on the torus T = [0, L]3, L > 0,

∂u

∂t+ (u · ∇) u+ ∇p = ν4u+

∞∑

i=1

σihi (x)·βi (t) (1.1)

with div u = 0 and periodic boundary conditions, with suitable fields hi (x) and

independent Brownian motions βi (t). The 3D random vector field u = u (t, x) is

the velocity of the fluid and the random scalar field p = p (t, x) is the pressure. To

simplify the exposition, we avoid generality and focus on one of the simplest set of

assumptions:

σ2i = λ−3

i

where λi are the eigenvalues of the Stokes operator (see the next section). This

assumption also allows us to compare more closely the results in Da Prato & De-

bussche [2] and Flandoli [6]. However, following Flandoli & Romito [4], we could

treat any power law for σi. Under this assumption, one can associate a transi-

tion probability kernel P (t, x, ·) to equation (2.2), which is the abstract version of

(1.1), in D(A) (see the definitions in Section 2.1 below), satisfying the Chapman-

Kolmogorov equation. In other words, there exists a Markov selection in D(A)

for equation (2.2). To avoid misunderstandings, this does not mean that equation

(2.2) has been solved in D(A) with continuous trajectories: this would imply well

posedness. What has been proved is that the law of weak martingale solutions is

supported on D(A) for all times, with a number of related additional properties,

but a priori the typical trajectory may sometimes blow-up in the topology of D(A).

The transition probabilities P (t, x, ·) are irreducible and strong Feller, hence

equivalent, in D(A). These results and the existence of P (t, x, ·) have been proved

first in Da Prato & Debussche [2] and Debussche & Odasso [7] by a careful selection

from the Galerkin scheme. Then another proof by an abstract selection principle and

the local-in-time regularity of equation (2.2) has been given in Flandoli & Romito [4].

More precisely, first one proves the existence of a Markov kernel P (t, x, ·) by means

of a general and abstract method, then one proves that any such kernel is irreducible

and strong Feller, hence equivalent, in D(A).

We complement here the approach of Ref. [4] with two results. First, the simple

idea used in Ref. [4] to prove the strong Feller property is here developed further, to

show a weak form of Lipschitz continuity of P (t, x, ·) in x ∈ D(A). More precisely,

we prove the estimate

|P (t, x0 + h,Γ) − P (t, x0,Γ)| ≤ CTt ∧ 1

(1 + |Ax0|6)|Ah| log(|Ah|−1) (1.2)

for t ∈ (0, T ], x0, h ∈ D(A), with |Ah| ≤ 1. This result has been proved in a

stronger version in Da Prato & Debussche [2] for the transition kernel constructed

from the Galerkin scheme, and also in Flandoli [6] for any Markov kernel associated

to equation (2.2). In both cases the proof is based on the very powerful approach


Regularity for Stochastic Navier-Stokes Equation 265

introduced in Ref. [2] which however requires a considerable amount of technical

work. Here we give a rather elementary proof along the lines of Flandoli & Romito

[4], based on the following simple idea: given x0, h ∈ D(A), for a short random

time the solution is regular, unique and differentiable in the initial conditions; then

the propagation of regularity in x from small time to arbitrary time is due to the

Markov property. Unfortunately we cannot prove in this way the stronger estimate

obtained in Ref. [2] (where the right-hand-side of (1.2) has the form t−1+ε(1 +

|Ax0|2)|Ah|), so our first result here has mostly a pedagogical character, since the

proof is conceptually very easy.

The second result, which follows from the same main estimates used to prove

(1.2), is the equivalence

P (1) (t, x, ·) ∼ P (2) (t′, x′, ·)

for any t, t′ > 0 and x, x′ ∈ D(A), when P (i) (t, x, ·), i = 1, 2, are any two Markov

transition kernels associated to equation (2.2) in D(A). We have not proved yet

the existence of invariant measures associated to such kernelsa, but if we assume to

have such invariant measures, it also follows that they are equivalent. This result

and the gradient estimates discussed above could be steps to understand better the

open question of well posedness for equation (2.2). In particular, it seems to be not

so easy to produce examples of stochastic differential equations without uniqueness

but where all Markov solutions are equivalent.

Among the open problems related to this research we mention the relation be-

tween the regularity results for P (t, x, ·) in the initial condition discussed above and

the properties of Malliavin derivatives, investigated for stochastic 3D Navier-Stokes

equations by Mikulevicius and Rozovsky [3, 8].

2. Preliminaries

2.1. Notations

Denote by T = [0, 1]3 the three-dimensional torus, and let L2 (T ) be the space of

vector fields u : T → R3 with L2 (T )-components. For every α > 0, let Hα (T )

be the space of fields u ∈ L2 (T ) with components in the Sobolev space Hα (T ) =

Wα,2 (T ).

Let D∞ be the space of infinitely differentiable divergence free periodic fields u

on T , with zero mean. Let H be the closure of D∞ in the topology of L2 (T ): it

is the space of all zero mean fields u ∈ L2 (T ) such that div u = 0 and u · n on the

boundary is periodic. We denote by 〈., .〉H and |.|H (or simply by 〈., .〉 and |.|) the

usual L2-inner product and norm in H . Let V (resp. D(A)) be the closure of D∞

in the topology of H1 (T ) (in the topology of H2 (T ), respectively): it is the space

of divergence free, zero mean, periodic elements of H1 (T ) (respectively of H2 (T )).

aThis is apparently due to technical reasons and it is the subject of a work in progress.



The spaces V and D(A) are dense and compactly embedded in H . From Poincare

inequality we may endow V with the norm ‖u‖2V :=

∫T |Du (x)|2 dx.

Let A : D(A) ⊂ H → H be the operator Au = −4u (component wise). Since

A is a selfadjoint positive operator in H , there is a complete orthonormal system

(hi)i∈N ⊂ H of eigenfunctions of A, with eigenvalues 0 < λ1 ≤ λ2 ≤ . . . (that is,

Ahi = λihi). The fields hi in equation (2.2) will be these eigenfunctions. We have

〈Au, u〉H = ‖u‖2V

for every u ∈ D(A).

Let V ′ be the dual of V ; with proper identifications we have V ⊂ H ⊂ V ′ with

continuous injections, and the scalar product 〈·, ·〉H extends to the dual pairing

〈·, ·〉V,V ′ between V and V ′. We may enlarge this scheme to D(A) ⊂ V ⊂ H ⊂V ′ ⊂ D(A)′. Let B (·, ·) : V × V → V ′ be the bi linear operator defined as

〈w,B (u, v)〉V,V ′ =

3∑

i,j=1

∫

Tui∂vj∂xi

wj dx

for every u, v, w ∈ V . We shall repeatedly use the following inequality:∣∣∣A1/2B (u, v)

∣∣∣H

≤ C0 |Au| |Av| (2.1)

for u, v ∈ D(A). The proof is elementary (see Flandoli [9]).

2.2. Definitions, assumptions and known results

We (formally) rewrite equations (1.1) as an abstract stochastic evolution equation

in H ,

du(t) + [νAu(t) +B (u(t), u(t))] dt =

∞∑

i=1

σihi dβi (t) . (2.2)

Let us set

Ω = C ([0,∞);D(A)′)

and denote by (ξt)t≥0 the canonical process on Ω, defined as ξt (ω) = ω (t), by Fthe Borel σ-algebra in Ω and by Ft the σ-algebra generated by the events ξs ∈ Γwith s ∈ [0, t] and Γ a Borel set of D(A)′. Finally, denote by B(D(A)) the Borel

σ-algebra of D(A) and by Bb(D(A)) the set of all real valued bounded measurable

functions on D(A)).

Definition 2.1. Given a probability measure µ0 on H , we say that a probability

measure P on (Ω,F) is a solution to the martingale problem associated to equation

(2.2) with initial law µ0 if

(MP1) P [ξ ∈ L∞loc([0,∞);H) ∩ L2

loc([0,∞);V )] = 1,



(MP2) for each ϕ ∈ D∞ the process (Mϕt ,Ft, P )t≥0, defined P -a. s. on (Ω,F) as

Mϕt := 〈ξt − ξ0, ϕ〉H +

∫ t

0

ν〈ξs, Aϕ〉H ds−∫ t

0

〈B(ξs, ϕ), ξs〉H ds

is a continuous square integrable martingale with quadratic variation

[Mϕ]t = t∑

i∈N

σ2i |〈ϕ, hi〉|2,

(MP3) the marginal of P at time 0 is µ0.

Remark 2.2. Among all test functions in property [MP2], we can choose ϕ = hi.

Set for all i, βi(t) = 1σiMhit (and 0 if σi = 0). The (βi)i∈N are a sequence of

independent standard Brownian motions. Under the assumption∑

i σ2i < ∞, the

series∑∞i=1 σihiβi (t) defines an H-valued Brownian motion on (Ω,F ,Ft, P ), that

we shall denote by W (t). The canonical process (ξt) is a weak martingale solution

of (2.2), in the sense that it satisfies (2.2) in the following weak form: there exists

a Borel set Ω0 ⊂ Ω with P (Ω0) = 1 such that on Ω0 for every ϕ ∈ D∞ and t ≥ 0

we have

〈ξt − ξ0, ϕ〉H +

∫ t

0

ν 〈ξs, Aϕ〉H ds−∫ t

0

〈B (ξs, ϕ) , ξs〉H ds = 〈W (t) , ϕ〉H . (2.3)

The following theorem is well known, see for instance the survey paper of Flan-

doli [9] and the reference therein.

Theorem 2.3. Assume∑i σ

2i < ∞. Let µ be a probability measure on H such

that∫H|x|2H µ (dx) < ∞. Then there exists at least one solution to the martingale

problem with initial condition µ.

Definition 2.4. We say that P (·, ·, ·) : [0,∞) × D(A) × B (D(A)) → [0, 1] is a

Markov kernel in D(A) of transition probabilities associated to equation (1.1) if

P (·, ·,Γ) is Borel measurable for every Γ ∈ B (D(A)), P (t, x, ·) is a probability

measure on B (D(A)) for every (t, x) ∈ [0,∞) × D(A), the Chapman-Kolmogorov

equation

P (t+ s, x,Γ) =

∫

D(A)

P (t, x, dy)P (s, y,Γ)

holds for every t, s ≥ 0, x ∈ D(A), Γ ∈ B (D(A)), and for every x ∈ D(A) there is a

solution Px on (Ω, F ) of the martingale problem associated to equation (2.2) with

initial condition x such that

P (t, x,Γ) = Px [ξt ∈ Γ] for all t ≥ 0.

We recall the following result from Da Prato & Debussche [2], Debussche &

Odasso [7] or Flandoli & Romito [4]:

Theorem 2.5. There exists at least one Markov kernel P (t, x,Γ) in D(A) of tran-

sition probabilities associated to equation (1.1).



We recall that a P (t, x,Γ) is called irreducible in D(A) if for every t > 0, x0,

x1 ∈ D(A), ε > 0, we have

P (t, x0, BA (x1, ε)) > 0,

where BA (x1, ε) is the ball in D(A) of centre x1 and radius ε.

We say that P (t, x,Γ) is strong Feller in D(A) if

x 7→∫

D(A)

ϕ (y)P (t, x, dy)

is continuous on D(A) for every bounded measurable function ϕ : D(A) → R and for

every t > 0. It is well known (see for example Da Prato & Zabczyk [10, Proposition

4.1.1]) that irreducibility and strong Feller in D(A) imply that the laws P (t, x, ·) are

all mutually equivalent, as (t, x) varies in (0,∞)×D(A). Because of this equivalence

property, we say that P (t, x,Γ) is regular.

We recall also that P (t, x,Γ) is called stochastically continuous in D(A) if

limt→0 P (t, x, BA (x, ε)) = 1 for every x ∈ D(A) and ε > 0.

In Da Prato & Debussche [2], the transition probability kernel constructed by

Galerkin approximations is proved to be stochastically continuous, irreducible and

strong Feller in D(A), hence regular. More generally (see Flandoli & Romito [4]):

Theorem 2.6. Every Markov kernel P (t, x,Γ) in D(A) of transition probabilities

associated to equation (1.1) is stochastically continuous, irreducible and strong Feller

in D(A), hence regular.

3. The Log-Lipschitz estimate

Theorem 3.1. Let P (t, x,Γ) be a Markov kernel in D(A) of transition probabilities

associated to equation (1.1). Then, given T > 0, there is a constant CT such that

the inequality

|P (t, x0 + h,Γ) − P (t, x0,Γ)| ≤ CTt ∧ 1

(1 + |Ax0|6)|Ah| log(|Ah|−1)

holds for every t ∈ (0, T ], x0, h ∈ D(A), with |Ah| ≤ 1, and Γ ∈ B (D(A)).

We explain here only the logical skeleton of the proof, which is very simple. The

two main technical ingredients will be treated in the next two separate subsections.

The first idea is to decompose:

P (t, x0 + h,Γ) − P (t, x0,Γ)

=

∫

D(A)

[P (ε, x0 + h, dy) − P (ε, x0, dy)]P (t− ε, y,Γ) .

To shorten some notation, let us write

(Ptϕ) (x) =

∫

D(A)

ϕ (y)P (t, x, dy)



so, with the function ϕ (x) = 1x∈Γ the previous identity reads

(Ptϕ) (x0 + h) − (Ptϕ) (x0) = (Pε (Pt−εϕ)) (x0 + h) − (Pε (Pt−εϕ)) (x0) . (3.1)

It is now sufficient to estimate

(Pεψ) (x0 + h) − (Pεψ) (x0)

uniformly in ψ ∈ Bb (D(A)). The value of ε has to be chosen depending on the size

of x0 and h, as we shall see.

The second idea is to use an initial coupling : we introduce the equation with

cut-off χR(|Au|2), where χR (r) : [0,∞) → [0, 1] is a non-increasing smooth function

equal to 1 over [0, R], to 0 over [R+ 2,∞), and with derivative bounded by 1. The

equation is

du+[Au+B(u, u)χR

(|Au|2

)]dt =

∑∞i=1 σihi dβi (t) ,

u (0) = x.(3.2)

The definition of martingale problem for this equation is the same (with obvious

adaptations) as the definition given above for equation (1.1). Let τR : Ω → [0,∞]

be defined as

τR (ω) = inf t ≥ 0 : |Aω (t)| ≥ R .We recall the following result from Flandoli & Romito [4, Lemma 5.11]:

Lemma 3.2. For every x ∈ D(A) there is a unique solution P(R)x of the martingale

problem associated to equation (3.2), with the additional property

P (R)x [ξ ∈ C ([0,∞) ;D(A))] = 1.

Let Px be any solution on (Ω, F ) of the martingale problem associated to equation

(2.2) with initial condition x. Then

EP(R)x[ϕ (ξt) 1τR≥t

]= EPx

[ϕ (ξt) 1τR≥t

]

for every t ≥ 0 and ϕ ∈ Bb (D(A)).

Introduce the notation

(P(R)t ϕ) (x) = EP

(R)x [ϕ (ξt)] .

The previous lemma implies that for every ψ ∈ Bb (D(A)) we have

|(Pεψ)(x) − (P (R)ε ψ)(x)| ≤ 2Px[τR < ε] ‖ψ‖∞. (3.3)

Summarising:

Corollary 3.3. For every x0, h ∈ D(A) and ψ ∈ Bb (D(A)) we have

|(Pεψ)(x0 + h) − (Pεψ)(x0)| ≤ 2 (Px0+h[τR < ε] + Px0 [τR < ε]) ‖ψ‖∞+∣∣∣(P (R)

ε ψ)(x0 + h) − (P (R)ε ψ)(x0)

∣∣∣ .



Let us give now the proof of Theorem 3.1. Assume t ∈ (0, T ], x0, h ∈ D(A)

be given, with |Ah| ≤ 1. Let K > 0 be such that |Ax0| + 1 ≤ K. We have

|A(x0 + h)| ≤ K, so we may apply Proposition 3.5 below to both x0 and x0 + h.

We thus get, for ε ∈ (0, 15C∗K2 ), where C∗ > 0 is the constant defined by (A.3) in

the Appendix, we have

Px0+h [τ2K < ε] + Px0 [τ2K < ε] ≤ 2C#e−η#K2

4ε .

Given h, K and t as above, let us look for a value ε ∈ (0, 15C∗K2 ) such that ε ≤ t

and the latter exponential quantity is smaller than |Ah|. We impose

η#K2

4ε≥ log(|Ah|−1)

hence it is sufficient to take

ε ≤ η#K2

4 log(|Ah|−1)∧ t

2∧ 1

5C∗K2. (3.4)

We have proved so far the first claim of the following lemma. The second claim is

a simple consequence of (3.1) and the previous corollary.

Lemma 3.4. Given t > 0, x0, h ∈ D(A), with |Ah| ≤ 1, and Γ ∈ B (D(A)), if ε is

chosen as in (3.4), then

Px0+h [τ2K < ε] + Px0 [τ2K < ε] ≤ 2C#|Ah|and for ϕ(x) = 1x∈Γ and ψ = Pt−εϕ,

|Ptϕ(x0 + h) − Ptϕ(x0)| ≤ 4C#|Ah| ‖ϕ‖∞ + |P (2K)ε ψ(x0 + h) − P (2K)

ε ψ(x0)|.

Finally, from Proposition 3.6 below, renaming the constant C, with ϕ (x) =

1x∈Γ and ψ = Pt−εϕ,∣∣∣(P (2K)

ε ψ) (x0 + h) − (P (2K)ε ψ) (x0)

∣∣∣ ≤ C

ε|Ah| eCK6ε.

Thus, for ε as in (3.4), we get

|Ptϕ (x0 + h) − Ptϕ (x0)| ≤ 4C#|Ah| +C

ε|Ah|eCK6ε.

Let us further restrict ourselves to

ε ≤ η#K2

4 log(|Ah|−1)∧ t

2∧ 1

5C∗K2∧ 1

K6,

so that we have

|Ptϕ(x0 + h) − Ptϕ(x0)| ≤ 4C#|Ah| +C

ε|Ah|.

The choice

ε = Ct ∧ 1

K6 log(|Ah|−1)

is admissible for a suitable constant C > 0, and we finally get (1.2). The proof of

Theorem 3.1 is complete.



3.1. Probability of blow-up

Proposition 3.5. Let K ≥ 1 and assume that x0 ∈ D(A) and ε > 0 are given such

that |Ax0| ≤ K and ε ≤ 15C∗K2 , where C∗ is the constant defined in (A.3). Then

Px0 [τ2K < ε] ≤ C#e−η#K2

4ε ,

for suitable universal constants η# > 0 and C# > 0.

Proof. From Corollary A.2.2 we know that if ε ≤ 15C∗K2 and |Ax0| ≤ K, then

one has

θ2ε ≤ 1

4K2 ⇒ |Au(s)| < 2K for s ∈ [0, ε] ⇒ τ2K ≥ ε,

where θε is defined in (A.2) in the Appendix. Therefore, with the constraints

|Ax0| ≤ K and ε ≤ 15C∗K2 , by Proposition A.1.1 one gets

Px0 [τ2K < ε] ≤ Px0

[Θ2ε >

1

4K2

]≤ C#e−η#

K2

4ε .

3.2. Derivative of the regularised problem

Here we show the regularity of the transition semigroup associated to the regularised

problem (3.2).

Proposition 3.6. For every R ≥ 1 and x0, h ∈ D(A),

∣∣∣(P (R)ε ψ

)(x0 + h) −

(P (R)ε ψ

)(x0)

∣∣∣ ≤ C ‖ψ‖∞ε

|Ah| eCR6ε,

where C is a universal constant.

Proof. We write the following computations for the limit problem but the under-

standing is that we do it on the Galerkin approximations. For every ψ ∈ Bb (H),

ε > 0, from the Bismut-Elworthy-Li formula (see Da Prato & Zabczyk [10]),

∣∣∣(P (R)ε ψ

)(x0 + h) −

(P (R)ε ψ

)(x0)

∣∣∣

≤ C ‖ψ‖∞ε

supη∈[0,1]

E

[(∫ ε

0

∣∣∣A 32Dhu

(R)x0+ηh

(s)∣∣∣2

ds

) 12

],



where, for each R ≥ 1 and x ∈ D(A), u(R)x is the solution, starting at x, of problem

(3.2). From the regularised equation we have

1

2

d

dt

∣∣∣ADhu(R)x (t)

∣∣∣2

+∣∣∣A 3

2Dhu(R)x (t)

∣∣∣2

≤ χR(|Au(R)x (t)|2)

∣∣∣⟨ADhu

(R)x , AB(Dhu

(R)x , u(R)

x ) +AB(u(R)x , Dhu

(R)x )

⟩∣∣∣

+ 2χ′R(|Au(R)

x (t)|2)〈Au(R)x , ADhu

(R)x 〉|〈ADhu

(R)x , AB(u(R)

x , u(R)x )〉|

≤ CχR

(|Au(R)

x (t)|2) ∣∣∣A 3

2Dhu(R)x (t)

∣∣∣∣∣∣ADhu

(R)x (t)

∣∣∣∣∣∣Au(R)

x (t)∣∣∣

+ Cχ′R

(∣∣∣Au(R)x (t)

∣∣∣2) ∣∣∣Au(R)

x (t)∣∣∣3 ∣∣∣ADhu

(R)x (t)

∣∣∣∣∣∣A 3

2Dhu(R)x (t)

∣∣∣

≤ 1

2

∣∣∣A 32Dhu

(R)x (t)

∣∣∣2

+ Cχ2R

(∣∣∣Au(R)x (t)

∣∣∣2)∣∣∣ADhu

(R)x (t)

∣∣∣2 ∣∣∣Au(R)

x (t)∣∣∣2

+ Cχ′R

(|Au(R)

x (t)|2)2 ∣∣∣ADhu

(R)x (t)

∣∣∣2 ∣∣∣Au(R)

x (t)∣∣∣6

≤ 1

2

∣∣∣A 32Dhu

(R)x (t)

∣∣∣2

+ CR6∣∣∣ADhu

(R)x (t)

∣∣∣2

.

Thus

1

2

d

dt

∣∣∣ADhu(R)x (t)

∣∣∣2

+1

2

∣∣∣A 32Dhu

(R)x (t)

∣∣∣2

≤ CR6∣∣∣ADhu

(R)x (t)

∣∣∣2

.

This implies∣∣∣ADhu

(R)x (t)

∣∣∣2

≤ eCR6t |Ah|2

and ∫ ε

0

∣∣∣A 32Dhu

(R)x0+ηh

(s)∣∣∣2

ds ≤ |Ah|2(

1 +

∫ ε

0

CR6eCR6s ds

)= |Ah|2 eCR

6ε.

Thus∣∣∣(P (R)ε ψ

)(x0 + h) −

(P (R)ε ψ

)(x0)

∣∣∣ ≤ C ‖ψ‖∞ε

|Ah| eCR6ε.

The proposition is proved.

4. Equivalence of all transition probabilities

To make the following statement independent of previous results, we shall assume

stochastic continuity, irreducibility and the strong Feller property in the theorem

below, but we recall that these properties have been proved for every Markov kernel

in D(A) associated to equation (1.1), under the assumptions of the introduction.

Theorem 4.1. Let P (i) (t, x,Γ) be two Markov kernels in D(A) of transition prob-

abilities associated to equation (1.1). Assume they are stochastically continuous,

irreducible and strong Feller in D(A). Then the probability measures P (1) (t, x, ·)and P (2) (t′, x′, ·) are equivalent, for any t, t′ > 0 and x, x′ ∈ D(A).



Proof. Step 1. Let us recall the following fact (see Ref. [10]): if a Markov kernel

of transition probabilities P (t, x,Γ) in D(A) (or any Polish space) is stochastically

continuous, irreducible and strong Feller, then the laws P (t1, x1, .) and P (t2, x2, .)

are equivalent, for every t1, t2 > 0, x1, x2 ∈ D(A). Therefore, under our assump-

tions, it is sufficient to prove that, given t0 > 0, x0 ∈ D(A), the laws P (1) (t0, x0, .)

and P (2) (t0, x0, .) are equivalent. Since the argument is symmetric between the two

kernels, it is sufficient to prove the following statement: given t0 > 0, x0 ∈ D(A), if

Γ is a Borel set in D(A) such that P (2) (t0, x0,Γ) = 0 then P (1) (t0, x0,Γ) = 0. In

the following steps we prove such claim.

Let us remark that, again by the equivalence property for P (2), the assumption

that P (2) (t0, x0,Γ) = 0 implies that P (2) (t, x,Γ) = 0 for every t > 0, x ∈ D(A).

Therefore what we are going to prove is the following claim: given t0 > 0, x0 ∈D(A), let Γ be a Borel set in D(A) such that P (2) (t, x,Γ) = 0 for every t > 0,

x ∈ D(A); then P (1) (t0, x0,Γ) = 0.

Step 2. Since both P (1)(·, ·, ·) and P (2)(·, ·, ·) satisfy (3.3),

P (1)(t, x,Γ) = |P (1)(t, x,Γ) − P (2)(t, x,Γ)| ≤ 2(P (1)x [τR < t] + P (2)

x [τR < t]).

Now, for every pair (ε, x), with ε > 0 and x ∈ D(A), such that 5C∗(1 + |Ax|)2ε ≤ 1

(the constant C∗ is defined in (A.3), in the appendix), Proposition 3.5 implies that

P (1)(ε, x,Γ) ≤ 2C#e−η#(1+|Ax|)2

4ε ≤ 2C#e−14εη# .

Step 3. For every ε < 15C∗ , set Aε = x ∈ D(A) : 5C∗(1 + |Ax|)2ε ≤ 1, then by the

Markov property and the previous step,

P (1)(t0 + ε, x0,Γ) =

∫

Acε

P (1)(ε, x,Γ)P (1)(t0, x0, dx)

+

∫

Aε

P (1)(ε, x,Γ)P (1)(t0, x0, dx)

≤ 2C#e−14εη# + P (1)(t0, x0, A

cε)

Since P (1) (s, x0, D(A)) = 1, we have P (1)(t0, x0, Acε) −→ 0, as ε→ 0, and thus

limε→0

P (1) (t0 + ε, x0,Γ) = 0.

Step 4. Again by the Markov property, for every neighborhood G of x0 in D(A),

P (1)(t0 + ε, x0,Γ) =

∫P (1)(t0, y,Γ)P (1)(ε, x0, dy)

≥ P (1)(ε, x0, G) infy∈G

P (1)(t0, y,Γ).

Since the kernel P (1) is stochastically continuous, it follows that P (1)(ε, x0, G) → 1,

as ε→ 0, and so, by the previous step, infy∈G P (1)(t0, y,Γ) −→ 0 as ε→ 0. By the

strong Feller property, the map y 7→ P (1)(t0, y,Γ) is continuous, hence in conclusion

P (1)(t0, x0,Γ) = 0. The proof is complete.



5. Conclusion and remarks

We have proved that the transition probabilities associated to any Markov selection

are all equivalent to each other. However, the problem of uniqueness of Markov

selections remains open. We stress that it would imply uniqueness of solutions

to the martingale problem, by the argument of Stroock & Varadhan [1, Theorem

12.2.4].

The estimates proved in this work allows us at least to state a sufficient condition

for uniqueness of Markov selections. The proof is inspired by a well known proof

in semigroup theory as well as by the proof of uniqueness given by Bressan and

co-authors (see for instance Ref. [11]).

Proposition 5.1. Assume that a Markov selection (Px)x∈D(A) has the following

property: for every t > 0 and x ∈ D(A),

limn→∞

n∑

k=1

P

(t− k

nt, x,BA

(0,

√n

t

)c)= 0

where BA (0, n) is the ball in D(A) of radius n. Then (Px)x∈D(A) coincides with

any other Markov selection.

Proof. Let (Qx)x∈D(A) be another Markov selection. Let us rewrite, for ϕ ∈Cb (D(A)):

Ptϕ−Qtϕ = Pt− tnP t

nϕ− Pt− t

nQ t

nϕ

+Pt− tnQ t

nϕ−Qt− t

nQ t

nϕ

and so on iteratively until we have

Ptϕ−Qtϕ =n∑

k=1

Pt− ktn

(P t

nψ (k−1)t

n

−Q tnψ (k−1)t

n

)

where ψs = Qsϕ. We have, by using (3.3) and Proposition 3.5,∣∣∣Pt− kt

n

(P t

nψ (k−1)t

n

−Q tnψ (k−1)t

n

)(x)∣∣∣

=∣∣∣EPx

[(P t

nψ (k−1)t

n

−Q tnψ (k−1)t

n

)(ξt− kt

n

)]∣∣∣

≤ EPx

[∣∣∣(P t

nψ (k−1)t

n−Q t

nψ (k−1)t

n

)(ξt− kt

n

)∣∣∣1ξt− k

nt∈A t

n]

+EPx

[∣∣∣(P t

nψ (k−1)t

n

−Q tnψ (k−1)t

n

)(ξt− kt

n

)∣∣∣1ξt− k

nt∈Ac

tn

]

≤ 4C#e−nt η# + 2Px[ξt− k

n t∈ Act

n]

≤ 4C#e−nt η# + 2P

(t− k

nt, x,BA

(0,

√n

t

)c),



where we have set At = 5C∗t(1 + |Ax|)2 ≤ 1 and, roughly, A tn≈ BA

(0,√

nt

).

Hence

|Ptϕ(x) −Qtϕ(x)| ≤ 4nC#e−nt η# + 2

n∑

k=1

P

(t− k

nt, x,BA

(0,

√n

t

)c)

which completes the proof of the proposition.

The criterion of this proposition is apparently not really useful at the present

stage of our understanding. Indeed, if we apply Chebichev inequality we get the

sufficient condition

limn→∞

n∑

k=1

(t

n

)1+ε

EPx

[∣∣∣Aξt− ktn

∣∣∣2(1+ε)

]= 0

with is implied by the condition

EPx

[∫ t

0

|Aξs|2(1+ε) ds]<∞

which however would easily imply the well posedness of the 3D Navier-Stokes equa-

tion by direct estimates of the difference of two solutions.

A.1. An exponential tail estimate for the Stokes problem

Consider the following Stokes problem

dZ +AZ dt = A− 32 dW,

Z(0) = 0,

and set Θt = sups∈[0,t] |AZ(s)|. The next result is well known, but we give a proof

to keep track of the dependence on the constants of interest in this paper.

Proposition A.1.1. There exist η# > 0 and C# > 0 such that for every K ≥ 12

and ε > 0,

P[Θε ≥ K] ≤ C#e−η#

K2

ε .

Proof. Step 1. Set y(t) = ε−12Z(εt), then it is easy to see that y solves the equa-

tion dy + εAy dt = Q12 dW . Next, fix a value α ∈ ( 1

6 ,14 ), then by the factorisation

method (see Da Prato & Zabczyk [12, Chapter 5]),

y(t) =

∫ t

0

e−ε(t−s)A dWs = Cα

∫ t

0

e−ε(t−s)A(t− s)α−1Y (s) ds,

where Y (s) =∫ s0 e−ε(s−r)A(s−r)−α dWr and Cα denotes a generic constant depend-

ing only on α (it will keep changing value along the proof). For every t ∈ (0, 1],

since α > 16 , it follows from Holder’s inequality that

|Ay(t)|H ≤ Cα

∫ t

0

(t− s)α−1|AY (s)|H ds ≤ Cα

(∫ 1

0

|AY (s)|6H ds) 1

6

.



In conclusion, since ε−1Θ2ε = supt∈[0,1] |Ay(t)|2H , it follows by the above inequality

and standard arguments that

P[Θε ≥ K] ≤ e−

aεK

2

E

[exp

(a

(∫ 1

0

|AY (s)|6 ds) 1

3

)], (A.1)

with a constant a that will be specified later (and a = aCα).

Step 2. In order to estimate the expectation in (A.1), notice that

exp

(a

(∫ 1

0

|AY (s)|6 ds) 1

3

)=

∞∑

n=0

an

n!

[∫ 1

0

|AY (s)|6H ds]n

3

(A.2)

≤2∑

n=0

an

n!

[∫ 1

0

|AY (s)|6H]n

3

+

∫ 1

0

∞∑

n=3

an

n!|AY (s)|2nH

≤ a

[∫ 1

0

|AY (s)|10 ds] 1

5

+a2

2

[∫ 1

0

|AY (s)|8 ds] 1

2

+

∫ 1

0

ea|AY (s)|2H ds

Step 3. Now, AY (s) is a centered Gaussian process with covariance (cfr. proof of

Theorem 5.9 in Da Prato & Zabczyk [12])

Qs =

∫ s

0

(s− r)−2αA−1e−2ε(s−r)A dr,

so that, by Proposition 2.16 in Ref. [12],

E[ea|AY (s)|2H ] = e−12Tr[log(1−2aQs)],

provided that a ≤ infλ∈σ(Qs)12λ , where σ(Qs) is the spectrum of Qs. Similarly,

E|AY (s)|2p = Cp(Tr(Qs))p, for all integers p.

In order to choose a suitable value of a, let µ ∈ σ(Qs), then there is a eigenvalue

λ of A such that µ = µ(λ) is given by

µ = λ−1

∫ s

0

r−2αe−2rλε dr = λ−2+2α(2ε)−(1−2α)

∫ 2λεs

0

r−2αe−r dr ≤ Cαλ−10 ,

where λ0 is the smallest eigenvalue of A. Hence a can be chosen as Cαλ0, for a

suitable Cα.

Step 4. We conclude the proof: since a is small enough, we have that −Tr[log(1 −2aQs)] ≤ CαTr[Qs] and, as in step 3,

Tr[Qs] =∑

λ∈σ(A)

λ−2+2α(2ε)−(1−2α)

∫ 2λεs

0

r−2αe−r dr ≤ Cαε−(1−2α),



where the sum in λ converges since α < 14 and λn ≈ n

23 . Hence, by (A.1) and (A.2),

P[Θε ≥ K] ≤ e−

aK2

ε E

[a

[∫ 1

0

|AY (s)|10] 1

5

+a2

2

[∫ 1

0

|AY (s)|8] 1

2

+

∫ 1

0

ea|AY (s)|2H

]

≤ Cαe−aK2

ε

(eCαε

−(1−2α)

+ ε−(1−2α) + ε−2(1−2α))

≤ C#e−η#K2

ε ,

where η# and C# can be easily found, since K ≥ 12 .

A.2. The deterministic equation

The basic ingredient of our approach is the bunch of regular paths that every weak

solution has for a positive local (random) time, when the initial condition is regular.

It was called regular jet in Flandoli [6]. It is based on the solutions of the following

deterministic equation

u(t) +

∫ t

0

(Au (s) +B (u, u)) ds = x+ w (t) . (A.3)

We say that

u ∈ C ([0,∞;Hσ) ∩ L2loc ([0,∞);V )

is a weak solution of (A.3) if

〈u(t), ϕ〉 +

∫ t

0

(〈u (s) , Aϕ〉 − 〈B (u (s) , ϕ) , u (s)〉) ds = 〈x, ϕ〉 + 〈w(t), ϕ〉

for every ϕ ∈ D∞. Notice that all terms in the above definition are meaningful,

included the quadratic one in u due to the estimate

|〈B (u, v) , z〉| ≤ C |Dv|L∞ |u|L2 |z|L2 .

We take w ∈ Ω∗ where

Ω∗ =⋂

β∈(0, 12 )α∈(0, 34 )

Cβ ([0,∞);D(Aα)) .

Consider also the auxiliary Stokes equations

z(t) +

∫ t

0

Az (s) ds = w (t)

having the unique mild solution

z(t) = e−tAw (t) −∫ t

0

Ae−(t−s)A (w (s) − w (t)) ds.



From elementary arguments based on the analytic estimates∣∣Aαe−tA

∣∣ ≤ Cα,T

tα for

t ∈ (0, T ), we have (see for instance Flandoli [13] for details)

z ∈ C ([0,∞);D(A)) .

Let us set

θT = supt∈[0,T ]

|Az (t)| . (A.4)

Let C0 > 0 be the constant of inequality (2.1) and let

C∗ := 4C20 . (A.5)

Lemma A.2.1. Given x ∈ D(A) and w ∈ Ω∗, let K ≥ |Ax| and ε > 0 be such that

(K2 + θ2ε

)( 1

2K2+ C∗ε

)< 1

Then there exists a solution u ∈ C ([0, ε] ;D(A)), which is unique in the class of

weak solutions, and |Au (s)| < 2K for s ∈ [0, ε].

Proof. We show only the quantitative estimate, the other statements being stan-

dard in the theory of Navier-Stokes equations. For simplicity, all computations

will be made on the limit problem, although they should be made on its Galerkin

approximations. The uniqueness of local solution ensures that the procedure is

nevertheless correct.

Set v = u− z, then

dv

dt+Av +B (u, u) = 0

and, by using (2.1),

d

dt|Av|2 + 2 ‖Av‖2

V ≤ 2 |〈Av,AB (u, u)〉| ≤ 2 ‖Av‖V∣∣∣A1/2B (u, u)

∣∣∣

≤ 2C0 ‖Av‖V |Au|2 ≤ ‖Av‖2V + C2

0 |Au|4

≤ ‖Av‖2V + C∗(|Av|2 + |Az|2)2.

Hence on [0, ε] we have that

d

dt|Av|2 ≤ C∗(|Av|2 + θ2ε)

2,

and so, if we set y(t) = |Av (t)|2 + θ2ε , it follows that

dy

dt≤ C∗y2, on [0, ε] .

Consequently, since y > 0 (except for the irrelevant case w ≡ 0), we have

y(s) ≤ y(0)

1 − C∗sy(0),



namely,

|A (u (s) − z (s))|2 + θ2ε ≤ |Ax|2 + θ2ε

1 − C∗s(|Ax|2 + θ2ε

)

for s ∈ [0, ε]. Therefore

|Au (s)|2 ≤ 2(|Ax|2 + θ2ε)

1 − C∗s(|Ax|2 + θ2ε

) ≤ 2(K2 + θ2ε)

1 − C∗s (K2 + θ2ε).

This result is true until 1 −C∗s(K2 + θ2ε

)> 0, namely for s ∈ [0, 1

C∗(K2+θ2ε) ). The

assumption of the lemma ensures that [0, ε] is included in this interval. Thus the

last inequality is true at least on [0, ε]. Moreover, again by the assumption of the

lemma,

2(K2 + θ2ε)

4K2< 1 − C∗s

(K2 + θ2ε

)

that implies

2(K2 + θ2ε)

1 − C∗s (K2 + θ2ε)< 4K2,

and thus |Au(s)|2 < 4K2, for s ∈ [0, ε].

Corollary A.2.2. Assume there are K > 0 and ε > 0 such that

ε ≤ 1

5C∗K2and θ2ε ≤

1

4K2,

then, for every x ∈ D(A) such that |Ax| ≤ K, we have |Au (s)| < 2K for s ∈ [0, ε].

References

[1] D. W. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes. vol.233, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences], (Springer-Verlag, Berlin, 1979). ISBN 3-540-90353-4.

[2] G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equa-tions, J. Math. Pures Appl. (9). 82(8), 877–947, (2003). ISSN 0021-7824.

[3] R. Mikulevicius and B. L. Rozovskii, Global L2-solutions of stochastic Navier-Stokesequations, Ann. Probab. 33(1), 137–176, (2005). ISSN 0091-1798.

[4] F. Flandoli and M. Romito. Markov selections for the 3D stochastic Navier-Stokesequations. http://www.arxiv.org/abs/math.PR/0602612.

[5] F. Flandoli and M. Romito, Markov selections and their regularity for the three-dimensional stochastic Navier-Stokes equations, C. R. Math. Acad. Sci. Paris. Toappear.

[6] F. Flandoli, On the method of Da Prato and Debussche for the 3D stochastic NavierStokes equations, J. Evol. Eq. To appear.

[7] A. Debussche and C. Odasso. Markov solutions for the 3d stochastic Navier-Stokes equations with state dependent noise. http://www.arxiv.org/abs/math.AP/0512361.



[8] R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulentflows, SIAM J. Math. Anal. 35(5), 1250–1310 (electronic), (2004). ISSN 0036-1410.

[9] F. Flandoli. An introduction to 3D stochastic fluid dynamics. In SPDE in hydrody-namics: recent progress and prospects (CIME course), Lecture Notes in Mathematics.Springer. http://www.cime.unifi.it.


[11] A. Bressan, Hyperbolic systems of conservation laws. vol. 20, Oxford Lecture Seriesin Mathematics and its Applications, (Oxford University Press, Oxford, 2000). ISBN0-19-850700-3. The one-dimensional Cauchy problem.

[12] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. vol. 44, En-cyclopedia of Mathematics and its Applications, (Cambridge University Press, Cam-bridge, 1992). ISBN 0-521-38529-6.

[13] F. Flandoli, Stochastic differential equations in fluid dynamics, Rend. Sem. Mat. Fis.Milano. 66, 121–148, (1996). ISSN 0370-7377.


Chapter 11

Rate of Convergence of Implicit Approximations for Stochastic

Evolution Equations

Istvan Gyongy and Annie Millet∗

School of Mathematics and Maxwell Institute for Mathematical Sciences,University of Edinburgh

King’s Buildings, Edinburgh, EH9 3JZ, United Kingdom

[email protected]

Stochastic evolution equations in Banach spaces with unbounded nonlinear driftand diffusion operators are considered. Under some regularity condition assumedfor the solution, the rate of convergence of implicit Euler approximations is es-timated under strong monotonicity and Lipschitz conditions. The results areapplied to a class of quasilinear stochastic PDEs of parabolic type.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

2 Preliminaries and the approximation scheme . . . . . . . . . . . . . . . . . . . . . . . 284

3 Convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

4.1 Quasilinear stochastic PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

4.2 Linear stochastic PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

1. Introduction

Let V → H → V ∗ be a normal triple of spaces with dense and continuous em-

beddings, where V is a separable and reflexive Banach space, H is a Hilbert space,

identified with its dual by means of the inner product in H , and V ∗ is the dual of V .

Thus 〈v, h〉 = (v, h) for all v ∈ V and h ∈ H∗ = H , where 〈v, v∗〉 = 〈v∗, v〉 denotes

the duality product of v ∈ V , v∗ ∈ V ∗, and (h1, h2) denotes the inner product of

h1, h2 ∈ H . Let W = W (t) : t ≥ 0 be a d1-dimensional Brownian motion carried

∗Laboratoire de Probabilites et Modeles Aleatoires (CNRS UMR 7599), Universites Paris 6-Paris 7, Boıte Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, [email protected], and Centred’Economie de la Sorbonne (CNRS UMR 8174), Equipe SAMOS-MATISSE, Universite Paris 1Pantheon Sorbonne, 90 Rue de Tolbiac, 75634 Paris Cedex 13, [email protected]

281


282 I. Gyongy and A. Millet

by a stochastic basis (Ω,F , (Ft)t≥0, P ). Consider the stochastic evolution equation

u(t) = u0 +

∫ t

0

A(s, u(s)) ds+

d1∑

k=1

∫ t

0

Bk(s, u(s)) dW k(s) , (1.1)

where u0 is a V -valued F0-measurable random variable, A and B are (non-linear)

adapted operators defined on [0,∞[×V ×Ω with values in V ∗ andHd1 := H×...×H ,

respectively.

It is well-known, see Refs. [1–3], that this equation admits a unique solution

if the following conditions are met: There exist constants λ > 0, K ≥ 0 and an

(Ft)-adapted non-negative locally integrable stochastic process f = ft : t ≥ 0such that

(i) (Monotonicity) There exists a constant K such that

2〈u− v, A(t, u) −A(t, v)〉 +

d1∑

k=1

|Bk(t, u) −Bk(t, v)|2H ≤ K|u− v|2H , (1.2)

(ii) (Coercivity)

2〈v, A(t, v)〉 +

d1∑

k=1

|Bk(t, v)|2H ≤ −λ|v|2V +K|v|2H + f(t),

(iii) (Linear growth)

|A(t, v)|2V ∗ ≤ K|v|2V ∗ + f(t),

(iv) (Hemicontinuity)

limλ→0

〈w,A(t, v + λu)〉 = 〈w,A(t, v)〉

hold for all for u, v, w ∈ V , t ∈ [0, T ] and ω ∈ Ω.

Under these conditions equation (1.1) has a unique solution u on [0, T ]. (See

Definition 2.3 below for the definition of the solution.) Moreover, if E|u0|2H < ∞and E

∫ T0f(t) dt <∞, then

E supt≤T

|u(t)|2H +E

∫ T

0

|u(t)|2V dt <∞.

In Ref. [4] it is shown that under these conditions, approximations defined by various

implicit and explicit schemes converge to u.

Our aim is to prove rate of convergence estimates for these approximations.

To achieve this aim we require stronger assumptions: a strong monotonicity con-

dition on A,B and a Lipschitz condition on B in v ∈ V . In the present paper

we consider implicit time discretizations. Note that without space discretizations,

in general, explicit time discretizations do not converge. Consider, for example,

the heat equation du(t) = ∆u(t), with initial condition u(0) = u0 ∈ L2(Rd).

Then the explicit time discretization on the grid k/nnk=0 gives the approxima-

tion un(k/n) := (I + ∆/n)ku0 at time t = k/n. Hence clearly, if u0 /∈ ∩∞i=1W

i2(Rd),


Rate of Convergence of Implicit Approximations 283

then u(k/n) does not belong to the Sobolev space W l2(Rd), with any fixed negative

index l, when k is sufficiently large.

The study of various space-time discretization schemes will be done in the con-

tinuation of the present paper.

We require also the following time regularity from the solution u (see condition

(T2)): E|u0|2V < ∞, almost surely ut ∈ V for every t ∈ T , and there exist some

constants C and ν > 0 such that

E|u(t) − u(s)|2V ≤ C |t− s|2ν ,

for all s, t ∈ [0, T ]. Note that unlike the solutions to stochastic differential equa-

tions, the solutions to stochastic PDEs can satisfy this condition with a variety of

exponents ν, different from 1/2, due to the interplay between space and time regu-

larities of the solutions. (See Ref. [5] for space and time regularity of the solutions

to stochastic parabolic PDEs of second order.) Note also that our general setting

allows us to cover a large class of stochastic parabolic PDEs of order 2m for any

m ≥ 1 (see Ref. [1] for the class of stochastic parabolic SPDEs of order 2m and

Ref. [6] for the stochastic Cahn-Hilliard equation).

In the case of time independent operators A and B we obtain the rate of conver-

gence for the implicit approximation uτ corresponding to the mesh size τ = T/m

of the partition of [0, T ]

Emaxi≤m

|u(iτ) − uτ (iτ)|2H +E∑

i≤m|u(iτ) − uτ (iτ)|2V τ ≤ Cτν ,

where C is a constant independent of τ . If in addition to the above assumptions A

is also Lipschitz continuous in v ∈ V then the order of convergence is doubled,

Emaxi≤m

|u(iτ) − uτ (iτ)|2H +E∑

i≤m|u(iτ) − uτ (iτ)|2V τ ≤ Cτ2ν .

In the case of time dependent A and B it is natural to assume that they are Holder

continuous in t in order to control the error due to their discretization in time.

However, it is possible to control this discretization error when the operator A is

not even continuous in t, if we discretize it by taking the average of A(s) over

the intervals [ti, ti+1]. This explains the discretization of A(t) and condition (T1)

below. If both operators A and B are Holder continuous in time then we use also

the obvious discretization: Aτti = A(ti+1, .) and Bτk,ti = B(ti, .).

As examples we present a class of quasi-linear stochastic partial differential equa-

tions (SPDEs) of parabolic type, and show that it satisfies our assumptions. Thus

we obtain rate of convergence results also for implicit approximations of linear

parabolic SPDEs, in particular, for the Zakai equation of nonlinear filtering. We

refer to Refs. [3, 7–9] for basic results for the stochastic PDEs of nonlinear filtering.

We will extend these results to degenerate parabolic SPDEs, and to space-time

explicit and implicit schemes for stochastic evolution equations in the continuation

of this paper.



In Section 2 we give a precise description of the schemes and state the as-

sumptions on the coefficients which ensure the convergence of these schemes to the

solution u of (1.1). In Section 3 estimates for the speed of convergence of time

implicit schemes are stated and proved. Finally, in the last section, we give a class

of examples of quasi-linear stochastic PDEs for which all the assumptions of the

main theorem, Theorem 3.4, are fulfilled.

As usual, we denote by C a constant which can change from line to line.

2. Preliminaries and the approximation scheme

Let (Ω,F , (Ft)t≥0, P ) be a stochastic basis, satisfying the usual conditions, i.e.,

(Ft)t≥0 is an increasing right-continuous family of sub-σ-algebras of F such that

F0 contains every P -null set. Let W = W (t) : t ≥ 0 be a d1-dimensional Wiener

martingale with respect to (Ft)t≥0, i.e., W is an (Ft)-adapted Wiener process with

values in Rd1 such that W (t) −W (s) is independent of Fs for all 0 ≤ s ≤ t.

Let T be a given positive number. Consider the stochastic evolution equation

(1.1) for t ∈ [0, T ] in a triplet of spaces

V → H ≡ H∗ → V ∗,

satisfying the following conditions: V is a separable and reflexive Banach space

over the real numbers, embedded continuously and densely into a Hilbert space H ,

which is identified with its dual H∗ by means of the inner product (·, ·) in H , such

that (v, h) = 〈v, h〉 for all v ∈ V and h ∈ H , where 〈·, ·〉 denotes the duality product

between V and V ∗, the dual of V . Such triplet of spaces is called a normal triplet.

Let us state now our assumptions on the initial value u0 and the operators A,

B in the equation. Let

A : [0, T ] × V × Ω → V ∗ , B : [0, T ] × V × Ω → Hd1

be such that for every v, w ∈ V and 1 ≤ k ≤ d1, 〈A(s, v), w〉 and (Bk(s, v), w) are

adapted processes and the following conditions hold:

(C1) The pair (A,B) satisfies the strong monotonicity condition, i.e., there exist

constants λ > 0 and L > 0 such almost surely

2 〈u− v, A(t, u) −A(t, v)〉 +

d1∑

k=1

|Bk(t, u) −Bk(t, v)|2H

+λ |u− v|2V ≤ L |u− v|2H (2.1)

for all t ∈]0, T ], u and v in V .

(C2) (Lipschitz condition on B) There exists a constant L1 such that almost

surely

d1∑

k=1

|Bk(t, u) −Bk(t, v)|2H ≤ L1 |u− v|2V (2.2)



for all t ∈ [0, T ], u and v in V .

(C3) (Lipschitz condition on A) There exists a constant L2 such that almost

surely

|A(t, u) −A(t, v)|2V ∗ ≤ L2 |u− v|2V (2.3)

for all t ∈ [0, T ], u and v in V .

(C4) u0 : Ω → V is F0-measurable and E|u0|2V < ∞. There exist non-negative

random variables K1 and K2 such that EKi <∞, and

d1∑

k=1

|Bk(t, 0)|2H ≤ K1 , (2.4)

|A(t, 0)|2V ∗ ≤ K2 (2.5)

for all t ∈ [0, T ] and ω ∈ Ω.

Remark 2.1. If λ = 0 in (2.1) then one says that (A,B) satisfies the monotonicity

condition. Notice that this condition together with the Lipschitz condition (2.3) on

A implies the Lipschitz condition (2.2) on B.

Remark 2.2. (1) Clearly, (2.3)–(2.5) and (2.2)–(2.4) imply that A and B satisfy

the growth condition

d1∑

j=1

|Bk(t, v)|2H ≤ 2L1|v|2V + 2K1, (2.6)

and

|A(t, v)|2V ∗ ≤ 2L2 |v|2V + 2K2 (2.7)

respectively, for all t ∈ [0, T ], ω ∈ Ω and v ∈ V .

(2) Condition (2.3) obviously implies that the operator A is hemicontinuous:

limε→0

〈A(t, u+ εv), w〉 = 〈A(t, u), w〉 (2.8)

for all t ∈ [0, T ] and u, v, w ∈ V .

(3) The strong monotonicity condition (C1), (C2), (2.4) and (2.5) yield that the

pair (A,B) satisfies the following coercivity condition: there exists a non-negative

random variable K3 such EK3 <∞ and almost surely

2 〈v, A(t, v)〉 +

d1∑

k=1

|Bk(t, v)|2H + λ2 |v|2V ≤ L|v|2H +K3 (2.9)

for all t ∈]0, T ], ω ∈ Ω and v ∈ V .

Proof. We show only (3). By the strong monotonicity condition (C1)

2 〈v, A(t, v)〉 +

d1∑

k=1

|Bk(t, v)|2H + λ |v|2V ≤ L|v|2H + R1(t) +R2(t) (2.10)



with

R1(t) = 2 〈v, A(t, 0)〉,

R2(t) =

d1∑

k=1

|Bk(t, 0)|2H + 2

d1∑

k=1

(Bk(t, v) −Bk(t, 0) , Bk(t, 0)

).

Using (C2) and (2.5), we have

|R1(t)| ≤ λ4 |v|2V + 4K2

λ ,

|R2(t)| ≤ 2

d1∑

j=1

|Bk(t, v) −Bk(t, 0)|2H

12 ( d1∑

k=1

|Bk(t, 0)|2H

) 12

+K1

≤ λ4 |v|2V + CK1.

Thus, (2.10) concludes the proof of (2.9).

Definition 2.3. An H-valued adapted continuous process u = u(t) : t ∈ [0, T ] is

a solution to equation (1.1) on [0, T ] if almost surely :

u(t) ∈ V for almost every t ∈ [0, T ],∫ T

0

|u(t)|2V dt <∞ , (2.11)

and

(u(t), v) = (u0, v) +

∫ t

0

〈A(s, u(s)), v〉 ds +

d1∑

k=1

∫ t

0

(Bk(s, u(s)), v) dW k(s) (2.12)

holds for all t ∈ [0, T ] and v ∈ V . We say that the solution to (1.1) on [0, T ] is

unique if for any solutions u and v to (1.1) on [0, T ] we have

P(

supt∈[0,T ]

|u(t) − v(t)|H > 0)

= 0.

The following theorem is well-known (see Refs. [1–3]).

Theorem 2.4. Let A and B satisfy the monotonicity, coercivity, linear growth and

hemicontinuity conditions (i)-(iv) formulated in the Introduction. Then for every

H-valued F0-measurable random variable u0, equation (1.1) has a unique solution

u on [0, T ]. Moreover, if E|u0|2H <∞ and E∫ T0 f(t) dt <∞, then

E(

supt∈[0,T ]

|u(t)|2H)

+E

∫ T

0

|u(t)|2V dt <∞ (2.13)

holds.

Hence by the previous remarks we have the following corollary.

Corollary 2.5. Assume that conditions (C1), (C2) hold. Then for every H-valued

random variable u0 equation (1.1) has a unique solution u, and if E|u0|2H < ∞,

then (2.13) holds.



Approximation scheme. For a fixed integer m ≥ 1 and τ := T/m we define the

approximation uτ for the solution u by an implicit time discretization of equation

(1.1) as follows:

uτ (t0) = u0 ,

uτ (ti+1) = uτ (ti) + τ Aτti(uτ (ti+1)

)

+

d1∑

k=1

Bτk,ti(uτ (ti)

) (W k(ti+1) −W k(ti)

)for 0 ≤ i < m, (2.14)

where ti := iτ and

Aτti(v) =1

τ

∫ ti+1

ti

A(s, v) ds , (2.15)

Bτk,0(v) = 0, Bτk,ti+1(v) =

1

τ

∫ ti+1

ti

Bk(s, v) ds (2.16)

for i = 0, 1, 2, ...,m.

A random vector uτ := uτ(ti) : i = 0, 1, 2, ...,m is called a solution to scheme

(2.14) if uτ (ti) is a V -valued Fti-measurable random variable such that E|uτ (ti)|2V <

∞ and (2.14) holds for every i = 0, · · · ,m− 1.

We use the notation

κ1(t) := iτ for t ∈ [iτ, (i+ 1)τ [, and κ2(t) := (i+ 1)τ for t ∈]iτ, (i+ 1)τ ] (2.17)

for integers i ≥ 0, and set

At(v) = Ati(v), Bk,t(v) = Bti(v)

for t ∈ [ti, ti+1[, i = 0, 1, 2, ...m− 1 and v ∈ V .

Another possible choice is

Aτti(u) = A(ti+1, u) and Bτk,ti(u) = Bk(ti, u) for i = 0, 1, · · · ,m− 1. (2.18)

The following theorem establishes the existence and uniqueness of uτ for large

enough m, and provides estimates in V and in H . We remark that in practice

(2.14) should also be solved numerically. This is possible for example by Galerkin’s

approximations and by finite elements methods. In the continuation of this pa-

per we consider explicit and implicit time discretization schemes together with si-

multaneous ‘space discretizations’, and we estimate the error of the corresponding

approximations for (1.1).

Theorem 2.6. Assume that A and B satisfy the monotonicity, coercivity, linear

growth and hemicontinuity conditions (i)–(iv). Assume also that (C4) holds. Let

Aτ and Bτ be defined either by (2.15) and (2.16), or by (2.18). Then there exist an



integer m0 and a constant C, such that for m ≥ m0 equation (2.14) has a unique

solution uτ (ti) : i = 0, 1, ...,m, and

E max0≤i≤m

∣∣uτ (iτ)∣∣2H

+E

m∑

i=1

∣∣uτ (iτ)∣∣2Vτ ≤ C . (2.19)

Proof. For the sake of simplicity, we only give the proof in the case Aτ and Bτ

are defined by (2.15) and (2.16). This theorem with estimate

max0≤i≤m

E∣∣uτ (iτ)

∣∣2H

+E

m∑

i=1

∣∣uτ (iτ)∣∣2Vτ ≤ C (2.20)

in place of (2.19) is proved in Ref. [4] for a slightly different implicit scheme. For the

above implicit scheme the same proof can be repeated without essential changes.

For the convenience of the reader we recall from Ref. [4] that the existence and

uniqueness of the solution uτ(ti) : i = 0, 1, 2, ...,m to (2.14)–(2.16) is based on

the following proposition (Proposition 3.4 from Ref. [4]): Let D : V → V ∗ be a

mapping such that

(a) D is monotone, i.e., for every x, y ∈ V , 〈D(x) −D(y), x− y〉 ≥ 0;

(b)D is hemicontinuous, i.e., limε→0

〈D(x+ εy), z〉 = 〈D(x), z〉 for every x, y, z ∈ V ;

(c) there exist positive constants K, C1 and C2, such that

|D(x)|V ∗ ≤ K (1 + |x|V ), 〈D(x), x〉 ≥ C1 |x|2V − C2 , ∀x ∈ V.

Then for every y ∈ V ∗, there exists x ∈ V such that D(x) = y and

|x|2V ≤ C1 + 2C2

C1+

1

C21

|y|2V ∗ .

If there exists a positive constant C3 such that

〈D(x1) −D(x2), x1 − x2〉 ≥ C3 |x1 − x2|2V ∗ , ∀x1, x2 ∈ V , (2.21)

then for any y ∈ V ∗, the equation D(x) = y has a unique solution x ∈ V .

Note that for each i = 1, 2, ...m−1 equation (2.14) for x := uτ (ti+1) can be rewritten

as Dx = y with

D := I − τAτti , y := uτ (ti) +

d1∑

k=1

Bτk,ti(uτ (ti)

) (W k(ti+1) −W k(ti)

)

where I denotes the identity on V . It is easy to verify that due to conditions (i)–(iv)

and (C4) the operator D satisfies the conditions (a), (b) and (c) for sufficiently large

m. Thus a solution uτ (ti) : i = 0, 1, ...,m can be obtained by recursion on i for

all m greater than some m0. To show the uniqueness we need only verify (2.21).

By (2.15) and by the monotonicity condition (i) we have

〈D(x1) −D(x2), x1 − x2〉 = |x1 − x2|2H −∫ ti+1

ti

〈A(s, x1) −A(s, x2), x1 − x2〉 ds



≥ |x1 − x2|2H −Kτ |x1 − x2|2H = (1 −Kτ)|x1 − x2|2H ,

where the constant K is from (1.2). Hence it is clear that (2.21) holds if m is

sufficiently large.

Now we show (2.20). From the definition of uτ (ti+1) we have

|uτ (tj)|2H = |u0|2H + I(tj) + J (tj) + K(tj) −j∑

i=1

|Aτti(uτ (iτ))|2H τ (2.22)

for tj = jτ , j = 0, 1, 2, ...m, where

I(tj) := 2

∫ tj

0

〈uτ (κ2(s)), A(s, uτ (κ2(s)))〉 ds,

J (tj) :=∑

1≤i<j|∑

k

Bτk,ti(uτ (iτ))(W k(ti+1) −W k(ti))|2H ,

K(tj) := 2∑

k

∫ tj

0

(uτ (κ1(s)), Bτk,s(u

τ (κ1(s))))dW k(s),

and κ1, κ2 are piece-wise constant functions defined by (2.17). By Ito’s formula for

every k, l = 1, 2, ..., d1,

(W k(ti+1) −W k(ti))(Wl(ti+1) −W l(ti)) = δkl(ti+1 − ti) +Mkl(ti+1) −Mkl(ti),

where δkl = 1 for k = l and 0 otherwise, and

Mkl(t) :=

∫ t

0

(W k(s) −W k(κ1(s)

)) dW l(s) +

∫ t

0

(W l(s) −W l(κ1(s))

)dW k(s).

Thus we get

J (tj) = J1(tj) + J2(tj),

with

J1(tj) :=∑

1≤i<j

∑

k

|Bτk,ti(uτ (ti))|2Hτ ,

J2(tj) :=

∫ tj

0

∑

k,l

(Bτk,s(uτ (κ1(s))), Bτl,s(u

τ (κ1(s)))) dMkl(s).



By the Davis inequality we have

Emaxj≤m

|J2(tj)| ≤ 3∑

k,l

E

∫ T

0

|Bτk,s(uτ (κ1(s)))|2H |Bτl,s(uτ (κ1(s)))|2H d〈Mkl〉(s)1/2

≤ C1

∑

k,l

E

∫ T

0

|Bτk,s(uτ (κ1(s)))|4H |W l(s) −W l(κ1(s))∣∣2 ds

1/2

≤ C1

∑

k,l

E[

maxj≤m

∣∣Bτk,tj (uτ (tj))∣∣H

√τ

× 1

τ

∫ T

0

|Bτk,s(uτ (κ1(s)))|2H∣∣W l(s) −W l(κ1(s))

∣∣2ds1/2]

≤ d1C1

∑

k

τE maxj≤m

∣∣Bτk,tj (uτ (tj))∣∣2H

+ C1τ−1∑

k,l

E

∫ T

0

|Bτk,s(uτ (κ1(s)))|2H∣∣W l(s) −W l(κ1(s))

∣∣2ds

≤ C2

(1 +E

∑

j≤m|uτ (jτ)|2V τ

),

where C1 and C2 are constants, independent of τ . Here we use that by Jensen’s

inequality for every k

∑

1≤i<j|Bτk,ti(uτ (iτ))|2Hτ ≤

∫ tj

0

|Bk(s, uτ (κ2(s))|2H ds,

and that the coercivity condition (ii) and the growth condition on (iii) imply the

growth condition (2.6) on B with some constant L1 and random variable K1 satis-

fying EK1 <∞. Hence by taking into account the coercivity condition we obtain

E maxj≤m

[I(tj) + J (tj)

]

≤ Emaxj≤m

∫ tj

0

[2⟨uτ (κ2(s)) , A(s, uτ (κ2(s)))

⟩+∑

k

|Bk(s, uτ (κ2(s))|2H]ds

+E maxj≤m

|J2(tj)|

≤ C(

1 + maxj≤m

E|uτ (jτ)|2H +Em∑

j=1

|uτ (jτ)|2V τ)

(2.23)



with a constant C independent of τ . By using the Davis inequality again we obtain

E maxj≤m

∣∣K(tj)∣∣ ≤ 6 E

∫ T

0

∑

k

∣∣(uτ (κ1(s)), Bτk,s(uτ (κ1(s))

)∣∣2 ds1/2

≤ 6 E

maxj≤m

∣∣uτ (jτ)∣∣H

∫ T

0

∑

k

|Bτk,s(uτ (κ1(s)))|2H ds1/2

≤ 12 Emax

j≤m|uτ (jτ)|2H + 18 E

∫ T

0

∑

k

∣∣Bτk,s(uτ (κ1(s)))∣∣2Hds

≤ 12 E max

j≤m|uτ (jτ)|2H + C

1 +E

∑

j≤m|uτ (jτ)|2V τ

(2.24)

with a constant C independent of τ . From (2.20)–(2.24) we get

E maxj≤m

|uτ (jτ)|2H ≤ E|u0|2H +E maxj≤m

(I(tj) + J (tj)

)+E max

j≤m|K(tj)|

≤ 12 E max

j≤m|uτ (jτ)|2H + C (1 + max

j≤mE|uτ (jτ)|2H +E

∑

j≤m|uτ (jτ)|2V τ)

≤ 12 E max

j≤m|uτ (jτ)|2H + C (1 + L) <∞

by virtue of (2.20), which proves the estimate (2.19).

3. Convergence results

In order to obtain a speed of convergence, we require further properties from B(t, v)

and from the solution u of (1.1).

We assume that there exists a constant ν ∈]0, 1/2] such that:

(T1) The coefficient B satisfies the following time-regularity: There exists a

constant C and a random variable η ≥ 0 with finite first moment, such that almost

surely

d1∑

k=1

|Bk(t, v) −Bk(s, v)|2H ≤ |t− s|2ν(η + C|v|2V ) (3.1)

for all s ∈ [0, T ] and v ∈ V .

(T2) The solution u to equation (1.1) satisfies the following regularity property:

almost surely u(t) ∈ V for all t ∈ [0, T ], and there exists a constant C > 0 such that

E|u(t) − u(s)|2V ≤ C |t− s|2ν (3.2)

for all s, t ∈ [0, T ].

Remark 3.1. Clearly, (3.2) implies

supt∈[0,T ]

E|u(t)|2V <∞. (3.3)



Finally, in order to prove a convergence result in the H norm uniformly in time,

we also have to require the following uniform estimate on the V -norm of u:

(T3) There exists a random variable ξ such that Eξ2 <∞ and

supt≤T

|u(t)|V ≤ ξ (a.s.).

To establish the rate of convergence of the approximations we first suppose that

the coefficients A and B satisfy the Lipschitz property.

Theorem 3.2. Suppose that the conditions (C1)-(C4), (T1) and (T2) hold. Let

Aτ and Bτ be defined by (2.15) and (2.16). Then there exist a constant C and an

integer m0 ≥ 1 such that

sup0≤l≤m

E|u(lτ) − uτ (lτ)|2H +E

m∑

j=0

|u(jτ) − uτ (jτ)|2V τ ≤ C τ2ν (3.4)

for all integers m ≥ m0.

The following proposition plays a key role in the proof.

Proposition 3.3. Assume assumptions (i) through (iv) from the Introduction and

let Aτ and Bτ be defined by (2.15) and (2.16). Suppose, moreover condition (C4).

Then

|u(tl) − uτ (tl)|2H = 2

∫ tl

0

⟨u(κ2(s)) − uτ (κ2(s)), A(s, u(s)) −A(s, uτ (κ2(s)))

⟩ds

+l−1∑

i=0

∣∣∣∣∣

∫ ti+1

ti

d1∑

k=1

[Bk(s, u(s)) −Bτk,s(u

τ (ti))]dW k(s)

∣∣∣∣∣

2

H

+ 2

d1∑

k=1

∫ tl

0

(Bk(s, u(s)) −Bτk,s(u

τ (ti)) , u(κ1(s)) − uτ (κ1(s)))dW k(s)

−l−1∑

i=0

∣∣∣∣∫ ti+1

ti

[A(s, u(s)) −A(s, uτ (ti+1))

]ds

∣∣∣∣2

H

(3.5)

holds for every l = 1, 2, ...,m.



Proof. Using (2.14) we have for any i = 0, · · · ,m− 1

|u(ti+1) − uτ (ti+1)|2H − |u(ti) − uτ (ti)|2H =

2

∫ ti+1

ti

⟨u(ti+1) − uτ (ti+1), A(s, u(s)) −A(s, uτ (ti+1))

⟩ds

+ 2

d1∑

k=1

(∫ ti+1

ti


τ (ti))]dW k(s) , u(ti+1) − uτ (ti+1)

)

−∣∣∣∫ ti+1

ti

[A(s, u(s)) −A(s, uτ (ti+1))

]ds

+

d1∑

k=1

∫ ti+1

ti


τ (ti))]dW k(s)

∣∣∣2

H

=2

∫ ti+1

ti

⟨u(ti+1) − uτ (ti+1), A(s, u(s)) −A(s, uτ (ti+1))

⟩ds

+

∣∣∣∣∣d1∑

k=1

∫ ti+1

ti


τ (ti))]dW k(s)

∣∣∣∣∣

2

H

+ 2

d1∑

k=1

(∫ ti+1

ti


τ (ti))]dW k(s) , u(ti) − uτ (ti)

)

−∣∣∣∣∫ ti+1

ti

[A(s, u(s)) −A(s, uτ (ti+1))

]ds

∣∣∣∣2

H

Summing up for i = 1, · · · , l − 1, we obtain (3.5).

Proof of Theorem 3.2.

Taking expectations in both sided of (3.5) and using the strong monotonicity con-

dition (C1), we deduce that for l = 1, · · · , m,

E|u(tl) − uτ (tl)|2H

≤ E

∫ tl

0

2⟨u(κ2(s)) − uτ (κ2(s)), A(s, u(κ2(s))) −A(s, uτ (κ2(s)))

⟩ds

+

d1∑

k=1

E

∫ tl−1

0

|Bk(s, u(κ2(s))) −Bk(s, uτ (κ2(s)))|2H ds+

3∑

k=1

Rk

≤ −λE∫ tl

0

|u(κ2(s)) − uτ (κ2(s))|2V ds+ LτE|u(tl) − uτ (tl)|2H

+ Lτ

l−1∑

i=1

E|u(ti) − uτ (ti)|2H ds+

3∑

k=1

Rk , (3.6)



where

R1 =E

∫ tl

0

2⟨u(κ2(s)) − uτ (κ2(s)), A(s, u(s)) −A(s, u(κ2(s)))

⟩ds ,

R2 =

d1∑

k=1

E

∫ τ

0

|Bk(s, u(s))|2H ds ,

R3 =

d1∑

k=1

l−1∑

i=1

E[ ∫ ti+1

ti

ds∣∣∣Bk(s, u(s)) − 1

τ

∫ ti

ti−1

Bk(t, uτ (ti)) dt∣∣∣2

H

−∫ ti

ti−1

|Bk(t, u(ti)) −Bk(t, uτ (ti))|2H dt].

The Lipschitz property of A imposed in (2.3), (3.2) and Schwarz’s inequality imply

|R1| ≤ L2E

∫ tl

0

|u(κ2(s)) − uτ (κ2(s))|V |u(s) − u(κ2(s))|V ds ,

≤ L2

(E

∫ tl

0

|u(κ2(s)) − uτ (κ2(s))|2V ds) 1

2(E

∫ tl

0

|u(s) − u(κ2(s))|2V ds) 1

2

≤ λ

3E

∫ tl

0

|u(κ2(s)) − uτ (κ2(s))|2V ds+ Cτ2ν . (3.7)

A similar computation based on (2.2) yields

|R3| ≤d1∑

k=1

l−1∑

i=1

E

∫ ti

ti−1

dt1

τ

∫ ti+1

ti

ds(|Bk(s, u(s)) −Bk(t, uτ (ti))|2H

− |Bk(t, u(ti))) −Bk(t, uτ (ti)))|2H)

≤ λ

3E

∫ tl−1

0

|u(κ2(t)) − uτ (κ2(t))|2V dt+ C R′3

where

R′3 =

d1∑

k=1

E1

τ

∫ tl

t1

ds

∫ κ1(s)

κ1(s)−τdt |Bk(s, u(s)) −Bk(t, u(κ2(t)))|2H .

Hence, using (2.2), (3.1) and (3.2) we have

R′3 ≤

d1∑

k=1

E1

τ

∫ tl

t1

ds

∫ κ1(s)

κ1(s)−τdt[|Bk(s, u(s)) −Bk(t, u(s))|2H

+ |Bk(t, u(s)) −Bk(t, u(t))|2H + |Bk(t, u(t)) −Bk(t, u(κ2(t)))|2H]

≤ E

∫ tl

t1

τ2ν |u(s)|2V ds+ CE1

τ

∫ tl−1

0

dt

∫ κ2(t)+τ

κ2(t)

ds[|u(s) − u(t)|2V

+ |u(t) − u(κ2(t)|2V]≤ C τ2ν .



Hence

|R3| ≤ C τ2ν +λ

3E

∫ tl

0

|u(κ2(s)) − uτ (κ2(s))|2V ds . (3.8)

Furthermore (2.6) and (3.3) imply

|R2| ≤ Cτ (3.9)

with a constant C independent of τ . By inequalities (3.6)–(3.9), for sufficiently

large m,

E|u(tl) − uτ (tl)|2H +λ

3E

∫ tl

0

|u(κ2(s)) − uτ (κ2(s))|2V ds

≤l−1∑

i=1

L τ E|u(ti) − uτ (ti)|2H + Cτ2ν . (3.10)

Since supm∑mi=1 L τ < +∞, a discrete version of Gronwall’s lemma yields that

there exists C > 0 such that for m large enough

sup0≤l≤m

E|u(tl) − uτ (tl)|2H ≤ Cτ2ν .

This in turn with (3.2) implies

E

∫ T

0

|u(s) − uτ (κ2(s))|2V ds ≤ Cτ2ν ,

which completes the proof of the theorem.

Assume now that the solution u of equation (1.1) satisfies also the condition

(T3) Then we can improve the estimate (3.4) in the previous theorem.

Theorem 3.4. Let (C1)-(C4) and (T1)–(T3) hold, and let Aτ and Bτ be defined

by (2.15) and (2.16). Then for all sufficiently large m

E max0≤j≤m

|u(jτ) − uτ (jτ)|2H +E

m∑

j=0

|u(jτ) − uτ (jτ)|2V τ ≤ C τ2ν (3.11)

holds, where C is a constant independent of τ .

Proof. For k = 1, · · · , d1, set

Fk(t) = Bk(t, u(t)) −Bτk,t(uτ (κ1(t)))

m(t) =

d1∑

k=1

∫ t

0

Fk(s) dW k(s) and G(s) = m(s) −m(κ1(s)).



Then by Ito’s formula

|m(ti+1) −m(ti)|2H = 2

∫ ti+1

ti

∑

k

(G(s) , Fk(s)) dW k(s)

+

d1∑

k=1

∫ ti+1

ti

|Fk(s)|2H ds

for i = 0, ...,m− 1. Hence by using (3.5) we deduce that for l = 1, · · · ,m

|u(tl) − uτ (tl)|2H ≤ I1(tl) + I2(tl) + 2M1(tl) + 2M2(tl) (3.12)

with

I1(t) := 2

∫ t

0

〈u(κ2(s)) − uτ (κ2(s)) , A(s, u(s)) −A(s, uτ (κ2(s)))〉 ds,

I2(t) :=

d1∑

k=1

∫ t

0

|Bk(s, u(s)) −Bτk,s(uτ (κ1(s)))|2H ds,

M1(t) :=

d1∑

k=1

∫ t

0

(G(s) , Fk(s)

)dW k(s),

M2(t) :=

d1∑

k=1

∫ t

0

(Fk(s) , u(κ1(s)) − uτ (κ1(s))

)dW k(s).

By (C3)

sup0≤l≤m

|I1(tl)| ≤∫ T

0

|u(κ2(s)) − uτ (κ2(s))|2V ds+ L2

∫ T

0

|u(s) − uτ (κ2(s))|2V ds

≤ (1 + 2L2)

m∑

i=1

|u(ti) − uτ (ti)|2V τ + 2L2

∫ T

0

|u(s) − u(κ2(s))|2V ds.

Hence by Theorem 3.2 and by condition (T2)

E sup0≤l≤m

|I1(tl)| ≤ Cτ2ν , (3.13)

where C is a constant independent of τ .

Using Jensen’s inequality, (2.6) and condition (T1) we have for s ≤ τ

∑

k

|Fk(s)|2H =∑

k

|Bk(s, u(s))|2H ≤ 2L1 |u(s)|2H + 2K1, (3.14)



while for s ∈ [ti, ti+1], 1 ≤ i ≤ m, one has for some constant C independent of τ

∑

k

|Fk(s)|2H ≤ 1

τ

∑

k

∫ ti

ti−1

|Bk(s, u(s)) −Bk(r, uτ (ti))|2H dr

≤ 31

τ

∑

k

∫ ti

ti−1

[|Bk(s, u(s)) −Bk(r, u(s))|2H

+ |Bk(r, u(s)) −Bk(r, u(ti))|2H + |Bk(r, u(ti)) −Bk(uτ (ti))|2H]dr

≤ C[τ2ν

(η + |u(s)|2V

)+ |u(s) − u(ti)|2V + |u(ti) − uτ (ti)|2V

]. (3.15)

Thus, (3.14) and (3.15) yield

sup0≤l≤m

|I2(tl)| ≤ C

∫ τ

0

|u(s)|2V ds+ Cτ + C τ2ν

∫ T

0

(η + C |u(s)|2V

)ds

+ C

∫ T

0

|u(s) − u(κ2(s))|2V ds+ Cm∑

i=1

|u(ti) − uτ (ti)|2V τ.

Hence by Theorem 3.2 and by condition (T2)

E sup0≤l≤m

|I2(tl)| ≤ Cτ2ν , (3.16)

where C is a constant independent of τ . Since sup0≤s≤T∑

k |Fk(s)|2H need not be

measurable, we denote by Γ the set of random variables ζ satisfying

sup0≤s≤T

∑

k

|Fk(s)|2H ≤ ζ (a.s.).

For ζ ∈ Γ, the Davis inequality, and the simple inequality ab ≤ τ2a

2 + 12τ b

2 yield

E sup1≤l≤m

|M1(tl)| ≤ 3E

(∫ T

0

d1∑

k=1

|(Fk(s) , G(s)

)|2 ds

) 12

≤ 3E

ζ1/2

[∫ T

0

|G(s)|2H ds] 1

2

≤ 3

2τ infζ∈Γ

Eζ +3

2τE

∫ T

0

|G(s)|2H ds. (3.17)

By (2.6) and (3.15) we deduce

sup0≤s≤T

∑

k

|Fk(s)|2H ≤ C τ2ν(

sup0≤s≤T

|u(s)|2V + η + 1)

+ C max1≤i≤m

|u(ti) − uτ (ti)|2V

≤ C(

1 + ξ + max1≤i≤m

|u(ti) − uτ (ti)|2V),



where ξ is the random variable from condition (T3) and C is a constant, independent

of τ . Hence Theorem 3.2 yield

τ infζ∈Γ

Eζ ≤ τ C (Eη +Eξ) + C τ

m∑

i=1

E|u(ti) − uτ (ti)|2V ≤ C1 τ2ν , (3.18)

where C1 is a constant, independent of τ . Similarly, due to conditions (T1)-(T2)

and Theorem 3.2

E∑

k

∫ T

0

|Fk(s)|2H ds ≤ C τ2ν(

1 +E

∫ T

0

|u(s)|2V ds)

+ C τ2ν + C τ E

m∑

i=1

|u(ti) − uτ (ti)|2V ≤ C τ2ν (3.19)

with a constant C, independent of τ . Furthermore, the isometry of stochastic

integrals and (3.22) yield

1

τE

∫ T

0

|G(t)|2H dt ≤1

τE

∫ T

0

∣∣∣∣∣

∫ t

κ1(t)

∑

k

Fk(s) dW k(s)

∣∣∣∣∣

2

H

dt

≤ 1

τE

∫ T

0

dt

∫ t

κ1(t)

∑

k

|Fk(s)|2H ds ≤ C τ2ν . (3.20)

Thus from (3.17) by (3.18) and (3.20) we have

E sup1≤l≤m

|M1(tl)| ≤ Cτ2ν (3.21)

Finally, the Davis inequality implies

E sup1≤l≤m

|M2(tl)|H ≤ 3E

(∫ T

0

∑

k

|(Fk(s) , u(κ1(s)) − uτ (κ1(s))

)|2 ds

) 12

≤ 1

4E sup

1≤l≤m|u(κ1(s)) − uτ (κ1(s))

)|2H + 18E

∫ tj

0

|Fk(s)|2H ds. (3.22)

Thus, from (3.12) by inequalities (3.13), (3.16), (3.21) and (3.22) we obtain

1

2E sup

1≤l≤m|u(tl) − uτ (tl)|2H ≤ C τ2ν ,

with a constant C, independent of τ , which with (3.4) completes the proof of the

theorem.

We now prove that if the coefficient A does not satisfy the Lipschitz property

(C3) but only the coercivity and growth conditions (2.7)-(2.9), then the order of

convergence is divided by two.

Theorem 3.5. Let A and B satisfy the conditions (C1), (C2) and (C4). Suppose

that conditions (T1) and (T2) hold, and let Aτ and Bτ be defined by (2.15) and



(2.16). Then there exists a constant C, independent of τ , such that for all sufficiently

large m

sup0≤j≤m

E|u(jτ) − uτ (jτ)|2H +E

m∑

j=1

|u(jτ) − uτ (jτ)|2V τ ≤ C τν . (3.23)

Proof. Using (3.5), taking expectations and using (C1) with u(s) and uτ (κ2(s)),

we obtain for every l = 1 · · · ,m

E|u(tl) − uτ (tl)|2H ≤ −λE∫ tl

0

|u(s) − uτ (κ2(s)|2V ds

+E

∫ tl

0

K1 |u(s) − uτ (κ2(s)|2H ds+

3∑

k=1

Ri , (3.24)

where

R1 =

r∑

j=1

2E

∫ tl

0

⟨u(κ2(s)) − u(s) , A(s, u(s)) −A(s, uτ (κ2(s)))

⟩ds ,

R2 =

d1∑

k=1

E

∫ τ

0

|Bk(s, u(s))|2H ds ,

R3 =

d1∑

k=1

l−1∑

i=1

E1

τ

∫ ti+1

ti

ds

∫ ti

ti−1

dt[|Bk(s, u(s)) −Bk(t, uτ (ti)))|2H

− |Bk(t, u(t)) −Bk(t, uτ (ti))|2H].

Using (2.7), (3.2), (3.3) and Schwarz’s inequality, we deduce

|R1| ≤ C E

∫ tl

0

|u(κ2(s)) − u(s)|V[|u(s)|V + |uτ (κ2(s))|V +K2

]ds

≤ C

(E

∫ tl

0

|u(s) − u(κ2(s))|2V ds) 1

2(E

∫ tl

0

(|u(s)|2V + |u(κ2(s))|2V

)ds

) 12

+ C

(E

∫ tl

0

|u(s) − u(κ2(s))|2V ds) 1

2

≤ Cτν . (3.25)

Furthermore, Schwarz’s inequality, (C2) and computations similar to that proving

(3.8) yield for any δ > 0 small enough

|R3| ≤ δ

d1∑

k=1

E

∫ tl−1

0

|Bk(t, u(t)) −Bk(t, uτ (κ2(t)))|2H dt

+ C

d1∑

k=1

l−1∑

i=1

1

τE

∫ ti+1

ti

ds

∫ ti

ti−1

dt |Bk(s, u(s))) −Bk(t, u(t))|2H

≤ λ

2E

∫ tl−1

0

|u(s) − uτ (κ2(s))|2V ds+ Cτ2ν .



This inequality and (3.25) imply that

E|u(tl) − uτ (tl)|2H +λ

2E

∫ tl

0

|u(s) − uτ (κ2(s))|2V ds

≤ K1

∫ tl

0

E|u(s) − uτ (κ2(s))|2H ds+ C τν .

Hence for any t ∈ [0, T ],

E|u(t) − uτ (κ2(t)|2H ≤ 2E|u(κ2(t)) − uτ (κ2(t))|2H + 2E|u(t) − u(κ2(t))|2H

≤ 2K1

∫ κ2(t)

0

E|u(s) − uτ (κ2(s))|2Hds+ C τν + 2E|u(t) − u(κ2(t))|2H

≤ 2K1

∫ t

0

E|u(s) − uτ (κ2(s))|2H ds+ C τν + 2E|u(t) − u(κ2(t))|2H

+ C τ[

supsE(|u(s)|2H + |uτ (κ2(s))|2H

)].

Ito’s formula and (2.9) imply that for any t ∈ [0, T ],

E|u(t) − u(κ2(t))|2H = E

∫ κ2(t)

t

[2〈A(s, u(s)) , u(s)〉 +

d1∑

k=1

|Bk(s, u(s))|2H]ds

≤ K1E

∫ κ2(t)

t

|u(s)|2H ds ≤ K1 τ sup0≤s≤T

E|u(s)|2H .

Hence (2.13) and (2.19) imply that

E|u(t) − uτ (κ2(t))|2H ≤ 2K1

∫ t

0

E|u(s) − uτ (κ2(s))|2H ds+ C τν

and Gronwall’s lemma yields

sup0≤t≤T

E|u(t) − uτ (κ2(t))|2H ≤ Cτν . (3.26)

Therefore,

E

∫ T

0

|u(t) − uτ (κ2(t)|2V dt < Cτν (3.27)

follows by (3.24). Finally taking into account that by (T2) there exists a constant

C such that

E|u(t) − u(κ2(t)|2V ≤ Cτ2ν for all t ∈ [0, T ],

from (3.26) and (3.27) we obtain (3.23).

Using the above result one can easily obtain the following theorem in the same

way as Theorem 3.2 is obtained from Theorem 3.4.



Theorem 3.6. Let A and B satisfy the conditions (C1), (C2) and (C4). Suppose

that conditions (T1)–(T3) hold and let Aτ and Bτ be defined by (2.15) and (2.16).

Then there exists a constant C such that for m large enough,

E max0≤j≤m

|u(jτ) − uτ (jτ)|2H +E

m∑

j=0

|u(jτ) − uτ (jτ)|2V τ ≤ C τν . (3.28)

Remark 3.7. By analyzing their proof, it is not difficult to see that Theorems

3.2, 3.4, 3.5 and 3.6 remain true, if instead of (2.15) and (2.16), one uses (2.18)

in the definition of the implicit scheme, and requires furthermore that A satisfies

the following time-regularity similar to (T1): there exist a constant C ≥ 0 and a

random variable η ≥ 0 with finite expectation, such that almost surely

|A(t, u) −A(s, u)|2V ∗ ≤ |t− s|2ν(η + C‖u‖2

V

)

for 0 ≤ s ≤ t ≤ T and u ∈ V .

4. Examples

4.1. Quasilinear stochastic PDEs

Let us consider the stochastic partial differential equation

du(t, x) =[Lu(t, x) + F (t, x,∇u(t, x), u(t, x)

]dt

+

d1∑

k=1

[Mku(t, x) +Gk(t, x, u(t, x))

]dW k(t), (4.1)

for t ∈ (0, T ], x ∈ Rd with initial condition

u(0, x) = u0(x), x ∈ Rd, (4.2)

where W is a d1-dimensional Wiener martingale with respect to the filtration

(Ft)t≥0, F and Gk are Borel functions of (ω, t, x, p, r) ∈ Ω × [0,∞) × Rd × Rd × R

and of (ω, t, x, r) ∈ Ω × [0,∞) × Rd × R, respectively, and L, Mk are differential

operators of the form

L(t)v(x) =∑

|α|≤1,|β|≤1

Dα(aαβ(t, x)Dβv(x)), Mk(t)v(x) =∑

|α|≤1

bαk (t, x)Dαv(x),

(4.3)

with functions aαβ and bαk of (ω, t, x) ∈ Ω × [0,∞) × Rd, for all multi-indices α =

(α1, ..., αd), β = (β1, ..., βd) of length |α| =∑

i αi ≤ 1, |β| ≤ 1.

Here, and later on Dα denotes Dα11 ...Dαd

d for any multi-indices α = (α1, ..., αd) ∈0, 1, 2, ...d, where Di = ∂

∂xiand D0

i is the identity operator.

We use the notation ∇p := (∂/∂p1, ..., ∂/∂pd). For r ≥ 0 let W r2 (Rd) denote

the space of Borel functions ϕ : Rd→R whose derivatives up to order r are square



integrable functions. The norm |ϕ|r of ϕ in W r2 is defined by

|ϕ|2r =∑

|γ|≤r

∫

Rd

|Dγϕ(x)|2 dx.

In particular, W 20 (Rd) = L2(Rd) and |ϕ|0 := |ϕ|L2(Rd). Let us use the notation

P for the σ-algebra of predictable subsets of Ω × [0,∞), and B(Rd) for the Borel

σ-algebra on Rd.

We fix an integer l ≥ 0 and assume that the following conditions hold.

Assumption (A1) (Stochastic parabolicity). There exists a constant λ > 0 such

that

∑

|α|=1,|β|=1

(aαβ(t, x) − 1

2

d1∑

k=1

bαk bβk(t, x)

)zα zβ ≥ λ

∑

|α|=1

|zα|2 (4.4)

for all ω ∈ Ω, t ∈ [0, T ], x ∈ Rd and z = (z1, ..., zd) ∈ Rd, where zα := zα11 zα2

2 ...zαd

d

for z ∈ Rd and multi-indices α = (α1, α2, ..., αd).

Assumption (A2) (Smoothness of the linear term). The derivatives of aαβ and

bαk up to order l are P ⊗ B(Rd) -measurable real functions such that for a constant

K

|Dγaαβ(t, x)| ≤ K, |Dγbαk (t, x)| ≤ K, for all |α| ≤ 1, |β| ≤ 1, k = 1, · · · , d1,

(4.5)

for all ω ∈ Ω, t ∈ [0, T ], x ∈ Rd and multi-indices γ with |γ| ≤ l.

Assumption (A3) (Smoothness of the initial condition). Let u0 be a W l2-valued

F0-measurable random variable such that

E|u0|2l <∞. (4.6)

Assumption (A4) (Smoothness of the nonlinear term). The function F and

their first order partial derivatives in p and r are P ⊗ B(Rd) ⊗ B(Rd) ⊗ B(R)-

measurable functions, and gk and its first order derivatives in r are P⊗B(Rd)⊗B(R)

-measurable functions for every k = 1, .., d1. There exists a constant K such that

|∇pF (t, x, p, r)| + | ∂∂rF (t, x, p, r)| +

d1∑

k=1

| ∂∂rGk(t, x, r)| ≤ K (4.7)

for all ω ∈ Ω, t ∈ [0, T ], x ∈ Rd, p ∈ Rd and r ∈ R. There exists a random variable

ξ with finite first moment, such that

|F (t, ·, 0, 0)|20 +

d1∑

k=1

|Gk(t, ·, 0)|20 ≤ ξ (4.8)

for all ω ∈ Ω and t ∈ [0, T ].



Definition 4.1. An L2(Rd)-valued continuous (Ft)-adapted process u = u(t) : t ∈[0, T ] is called a generalized solution to the Cauchy problem (4.1)-(4.2) on [0, T ] if

almost surely u(t) ∈W 12 (Rd) for almost every t,

∫ T

0

|u(t)|21 dt <∞,

and

d(u(t), ϕ) = ∑

|α|≤1,|β|≤1

(−1)|α|(aαβ(t)Dβu(t) , Dαϕ

)+(F (t,∇u(t), u(t)) , ϕ

)dt

+

d1∑

k=1

∑

|α|≤1

(bαk (t)Dαu(t) , ϕ

)+(Gk(t, u(t)) , ϕ

)dW k(t)

holds on [0, T ] for every ϕ ∈ C∞0 (Rd), where (v, ϕ) denotes the inner product of v

and ϕ in L2(Rd).

Set H = L2(Rd), V = W 12 (Rd) and consider the normal triplet V → H → V ∗

based on the inner product in L2(Rd), which determines the duality 〈 , 〉 between

V and V ∗ = W−12 (Rd). By (4.5), (4.7) and (4.8) there exist a constant C and a

random variable ξ with finite first moment, such that∣∣∣

∑

|α|≤1,|β|≤1

(−1)|α|(aαβ(t)Dβv , Dαϕ

)∣∣∣ ≤ C|v|1|ϕ|1,

d1∑

k=1

|(bαk (t)Dαv , ϕ)|2 ≤ C|v|21|ϕ|20,

|(F (t,∇v, v) , ϕ

)|2 ≤ C|v|21|ϕ|21 + ξ|ϕ|20,

d1∑

k=1

|(Gk(t, v(t)) , ϕ)|2 ≤ C|v|21|ϕ|20 + ξ|ϕ|20

for all ω, t ∈ [0, T ] and v, ϕ ∈ V . Therefore the operators A(t), Bk(t) defined by

〈A(t, v), ϕ〉 =∑

|α|≤1,|β|≤1


)+(F (t,∇v, v) , ϕ

),

(Bk(t, v) , ϕ) =(bαk (t)Dαv , ϕ

)+(Gk(t, v) , ϕ

), v, ϕ ∈ V (4.9)

are mappings from V into V ∗ and H , respectively, for each k and ω, t, such that

the growth conditions (2.6) and (2.7) hold. Thus we can cast the Cauchy problem

(4.1)– (4.2) into the evolution equation (1.1), and it is an easy exercise to show that

Assumptions (A1), (A2) with l = 0 and Assumption (A4) ensure that conditions

(C1) and (C2) hold. Hence Corollary 2.5 gives the following result.

Theorem 4.2. Let Assumptions (A1)-(A4) hold with l = 0. Then problem (4.1)-

(4.2) admits a unique generalized solution u on [0, T ]. Moreover,

E(

supt∈[0,T ]

|u(t)|20)

+E

∫ T

0

|u(t)|21 dt <∞. (4.10)



Next we formulate a result on the regularity of the generalized solution. We

need the following assumptions.

Assumption (A5) The first order derivatives of Gk in x are P ⊗ B(Rd) ⊗ B(R)-

measurable functions, and there exist a constant L, a P⊗B(R)-measurable function

K of (ω, t, x) and a random variable ξ with finite first moment, such that

d1∑

k=1

|DαGk(t, x, r)| ≤ L|r| +K(t, x), |K(t)|20 ≤ ξ

for all multi-indices α with |α| = 1, for all ω ∈ Ω, t ∈ [0, T ], x ∈ Rd and r ∈ R.

Assumption (A6) The first order derivatives of F in x are P ⊗B(Rd)⊗B(Rd)⊗B(R)-measurable functions, and there exist a constant L, a P ⊗ B(R)-measurable

function K of (ω, t, x) and a random variable ξ with finite first moment, such that

|∇xF (t, x, p, r)| ≤ L(|p| + |r|) +K(t, x), |K(t)|20 ≤ ξ

for all ω, t, x, p, r.

Assumption (A7) There exist P ⊗ B(R)-measurable functions gk such that

Gk(t, x, r) = gk(t, x) for all k = 1, 2, ..., d1, t, x, r,

and the derivatives in x of gk up to order l are P ⊗B(R)-measurable functions such

thatd1∑

k=1

|gk(t)|2l ≤ ξ,

for all (ω, t), where ξ is a random variable with finite first moment.

Theorem 4.3. Let Assume (A1)-(A4) with l = 1. Then for the generalized solution

u of (4.1)-(4.2) the following statements hold:

(i) Suppose (A5). Then u is a W 12 (Rd)-valued continuous process and

E(

supt≤T

|u(t)|21)

+E

∫ T

0

|u(t)|22 dt <∞ ; (4.11)

(ii) Suppose (A6) and (A7) with l = 2. Then u is a W 22 (Rd)-valued continuous

process and

E(

supt≤T

|u(t)|22)

+E

∫ T

0

|u(t)|23 dt <∞ . (4.12)

Proof. Define

ψ(t, x) = F (t, x,∇u(t, x), u(t, x)), ϕk(t, x) = Gk(t, x, u(t, x))

for t ∈ [0, T ], ω ∈ Ω and x ∈ Rd, where u is the generalized solution of (4.1)-(4.2).

Then due to (4.10)

E

∫ T

0

|ψ(t)|20 dt <∞, E∑

k

∫ T

0

|ϕk(t)|21 dt <∞.



Therefore, the Cauchy problem

dv(t, x) =[Lv(t, x) + ψ(t, x)

]dt

+

d1∑

k=1

[Mkv(t, x) + ϕk(t, x)

]dW k(t), t ∈ (0, T ] , x ∈ Rd , (4.13)

v(0, x) =u0(x), x ∈ Rd (4.14)

has a unique generalized solution v on [0, T ]. Moreover, by Theorem 1.1 in Ref. [10],

v is a W 12 -valued continuous Ft-adapted process and

E(

supt≤T

|v(t)|21)

+E

∫ T

0

|v(t)|22 dt <∞.

Since u is a generalized solution to (4.13)–(4.14), by virtue of the uniqueness of the

generalized solution we have u = v, which proves (i). Assume now (A6) and (A7)

with l = 2. Then obviously (A5) holds, and therefore due to (4.11)

E

∫ T

0

|ψ(t)|21 dt <∞, E∑

k

∫ T

0

|ϕk(t)|22 dt <∞.

Thus by Theorem 1.1 in Ref. [10] the generalized solution v = u of (4.13)–(4.14)

is a W 22 (Rd)-valued continuous process such that (4.12) holds. The proof of the

theorem is complete.

Corollary 4.4. Let (A1)-(A4) hold with l = 2. Assume also (A6) and (A7). Then

there exists a constant C such that for the generalized solution u of (4.1)–(4.2) we

have

E|u(t) − u(s)|21 ≤ C|t− s| for all s, t ∈ [0, T ].

Proof. By the theorem on Ito’s formula (see Refs. [1] or [11]) from almost surely

u(t) = u0 +

∫ t

0

[Lu(s) + ψ(s)

]ds+

d1∑

k=1

∫ t

0

[Mku(s) + gk(s)

]dW k(s)

holds, as an equality in L2(Rd), for all t ∈ [0, T ], where

ψ(s, ·) := F (s, ·,∇u(s, ·), u(s, ·)).Due to (ii) from Theorem 4.3

E∣∣∣∫ t

s

[L(r)u(r) + ψ(r)

]dr∣∣∣2

1≤ E

(∫ t

s

|Lu(r) + ψ(r)|1 dr)2

≤ |t− s|E∫ t

s

|Lu(r) + ψ(r)|21 dr

≤ C |t− s|(E

∫ T

0

|u(t)|23 dt+E

∫ T

0

|ψ(t)|21 dt)

≤ C|t− s|



for all s, t ∈ [0, T ], where C is a constant. Furthermore, by Doob’s inequality

E

∣∣∣∣∫ t

s

[Mku(r) + gk(r)

]dW k(r)

∣∣∣∣2

1

≤ 4

∫ t

s

E|Mku(r) + gk(r)|21 dr

≤ C1|t− s|[1 +E

∫ T

0

|u(t)|22 dt]≤ C2|t− s|

for all s, t ∈ [0, T ], where C1 and C2 are constants. Hence

E|u(t) − u(s)|21 ≤2E∣∣∣∫ t

s

[Lu(r) + ψ(r)

)dr∣∣∣2

1

+ 2E∣∣∣d1∑

k=1

∫ t

s

[Mku(r) + gk(r)

]dW k(r)

∣∣∣2

1≤ C|t− s|,

and the proof of the corollary is complete.

The implicit scheme (2.14) applied to problem (4.1)-(4.2) reads as follows.

uτ (t0) = u0 ,

uτ (ti+1) = uτ (ti) +(Lτtiu

τ (ti+1) + F τti(uτ (ti+1)

)τ

+

d1∑

k=1

(Mτk,tiu

τ (ti) +Gτk,ti(uτ (ti))

)(W k(ti+1) −W k(ti)) , (4.15)

for 0 ≤ i < m , where

Lτtiv : =∑

|α|≤1,|β|≤1

Dα(aαβti (x)Dβv), Mτk,ti :=

∑

|α|≤1

bαk,tiDαv,

aαβti (x) : =1

τ

∫ ti+1

ti

aαβ(s, x) ds, (4.16)

bαk,0(x) = 0, bαk,ti+1(x) =

1

τ

∫ ti+1

ti

bk(s, x) ds, (4.17)

F τti(x, p, r) : =1

τ

∫ ti+1

ti

F (s, x, p, r) ds,

Gτk,0(x, r) : = 0, Gτk,ti+1(x, r) :=

∫ ti+1

ti

Gk(s, x, r) ds.

Definition 4.5. A random vector uτ (ti) : i = 0, 1, 2, ...,m is a called a gen-

eralized solution of the scheme (4.15) if uτ (t0) = u0, uτ (ti) is a W 12 (Rd)-valued

Fti-measurable random variable such that

E|uτ (ti)|21 <∞



and almost surely

(uτ (ti), ϕ) =∑

|α|≤1,|β|≤1

(−1)|α|(aαβti Dβuτ (ti), D

αϕ)τ + (F τti−1(∇uτti−1

, uτti−1), ϕ)τ

+∑

k

[ ∑

|α|≤1

(bαti−1

Dαuτti−1+Gk,ti−1(uτti−1

), ϕ)(W k(ti) −W k(ti−1)

]

for i = 1, 2, ...,m and all ϕ ∈ C∞0 (Rd), where (·, ·) is the inner product in L2(Rd).

From this definition it is clear that, using the operators A, Bk defined by (4.9),

we can cast the scheme (4.15) into the abstract scheme (2.14). Thus by applying

Theorem 2.6 we get the following theorem.

Theorem 4.6. Let (A1)-(A4) hold with l = 0. Then there exists an integer m0

such that (4.15) has a unique generalized solution uτ(ti) : i = 0, 1, ...,m for every

m ≥ m0. Moreover, there exists a constant C such that

E max0≤i≤m

|uτ (ti)|20 +E

m∑

i=1

|uτ (ti)|21 ≤ C

for all integers m ≥ m0.

To ensure condition (T1) to hold we impose the following assumption.

Assumption (H) There exists a constant C and a random variable ξ with finite

first moment such that for k = 1, 2, ..., d1

|Dγ(bαk (t, x) − bαk (s, x))| ≤ C|t− s|1/2 for all ω ∈ Ω, x ∈ Rd and |γ| ≤ l,

|gk(s) − gk(s)|2l ≤ ξ|t− s|

for all s, t ∈ [0, T ].

Now applying Theorem 3.4 we obtain the following result.

Theorem 4.7. Let (A1)-(A4) and (A6)-(A7) hold with l = 2. Assume (H) with

l = 0. Then (4.1)–(4.2) and (4.15) have a unique generalized solution u and uτ =

uτ (ti) : i = 0, 1, 2, ...,m, respectively, for all integers m larger than some integer

m0. Moreover, for all integers m > m0

E max0≤i≤m

|u(iτ) − uτ (iτ)|20 + E

m∑

i=1

|u(iτ) − uτ (iτ)|21τ ≤ Cτ, (4.18)

where C is a constant, independent of τ .

Proof. By Theorems 4.2 and 4.6 the problems (4.1)–(4.2) and (4.15) have a

unique solution u and uτ , respectively. It is an easy exercise to verify that As-

sumption (H) ensures that condition (T1) holds. By virtue of Corollary 4.4 condi-

tion (T2) is valid with ν = 1/2. Condition (T3) clearly holds by statement (i) of

Theorem 4.3. Now we can apply Theorem 3.4, which gives (4.18).



4.2. Linear stochastic PDEs

Let Assumptions (A1)–(A3) and (A7) hold and impose also the following condition

on F .

Assumption (A8) There exist a P ⊗ B(R)-measurable function f such that

F (t, x, p, r) = f(t, x), for all t, x, p, r,

and the derivatives in x of f up to order l are P ⊗B(R)-measurable functions such

that

|f(t)|2l ≤ ξ,

for all (ω, t), where ξ is a random variable with finite first moment.

Now equation (4.13) has become the linear stochastic PDE

du(t, x) =[Lu(t, x) + f(t, x)

]dt+

d1∑

k=1

[Mku(t, x) + gk(t, x)

]dW k(t), (4.19)

and by Theorem 3.4 we have the following result.

Theorem 4.8. Let r ≥ 0 be an integer. Let Assumptions (A1)–(A3) and (A7)–

(A8) hold with l := r + 2, and let Assumption (H) hold with l = r. Then there is

an integer m0 such that (4.19)–(4.2) and (4.15) have a unique generalized solution

u and uτ = uτ (ti) : i = 0, 1, 2, ...,m, respectively, for all integers m > m0.

Moreover,

E max0≤i≤m

|u(iτ) − uτ (iτ)|2r +E

m∑

i=1

|u(iτ) − uτ (iτ)|2r+1τ ≤ Cτ

holds for all m > m0, where C is a constant independent of τ .

Proof. For r = 0 the statement of this theorem follows immediately from Theo-

rem 4.7. For r > 0 set H = W r2 (Rd) and V = W r+1

2 (Rd) and consider the normal

triplet V → H ≡ H∗ → V ∗ based on the inner product (· , ·) := (· , ·)r in W r2 (Rd),

which determines the duality 〈· , ·〉 between V and V ∗. Using Assumptions (A3),

(A7) and (A8) with l = r, one can easily show that there exist a constant C and a

random variable ξ such that Eξ2 <∞ and∣∣∣

∑

|α|≤1,|β|≤1


)r

∣∣∣ ≤ C|v|r+1|ϕ|r+1,

d1∑

k=1

|(bαk (t)Dαv , ϕ)r|2 ≤ C|v|2r+1|ϕ|2r,

|(f(t) , ϕ

)r|2 ≤ ξ|ϕ|2r,

d1∑

k=1

|(gk(t) , ϕ)r|2 ≤ ξ|ϕ|2r



for all ω, t ∈ [0, T ] and v, ϕ ∈ W r+12 (Rd). Therefore the operators A(t, ·), Bk(t, ·)

defined for v, ϕ ∈ V by

〈A(t, v), ϕ〉 =∑

|α|≤1,|β|≤1


)r

+(f(t) , ϕ

)r,

(Bk(t, v) , ϕ) =(bαk (t)Dαv , ϕ

)r

+(gk(t) , ϕ

)r, (4.20)

are mappings from V into V ∗ and H , respectively, for each k and ω, t, such that

the growth conditions (2.6) and (2.7) hold. Thus we can cast the Cauchy problem

(4.19)– (4.2) into the evolution equation (1.1), and it is an easy to verify that

conditions (C1)–(C4) hold. Thus this evolution equation admits a unique solution

u, which clearly a generalized solution to (4.19)– (4.2). Due to assumptions (A1)–

(A3) and (A7)–(A8) by Theorem 1.1 in Ref. [10] u is a W r+2(Rd)-valued stochastic

process such that

E supt≤T

|u(t)|2r+2 +E

∫ T

0

|u(t)|2r+3 dt <∞.

Hence it is obvious that (T3) holds, and it is easy to verify (T2) with ν = 12

like it is done in the proof of Corollary 4.4. Finally, it is an easy exercise to

show that (T1) holds. Now we can finish the proof of the theorem by applying

Theorem 3.4.

From the previous theorem we obtain the following corollary by Sobolev’s em-

bedding from W r2 to Cq.

Corollary 4.9. Let q be any non-negative number and assume that the assumptions

of Theorem 4.8 hold with r > q + d2 . Then there exist modifications u and uτ of

u and uτ , respectively, such that the derivatives Dγu and Dγuτ in x up to order q

are functions continuous in x. Moreover, there exists a constant C independent of

τ such that

E max0≤i≤m

supx∈Rd

∑

|γ|≤q|Dγ

(u(iτ, x) − uτ (iτ, x)

)|2

+Em∑

i=1

supx∈Rd

∑

|γ|≤q+1

|Dγ(u(iτ, x) − uτ (iτ, x)

∣∣2τ ≤ Cτ. (4.21)

Acknowledgments

This paper was written while I. Gyongy was visiting the University of Paris 1. His

research is partially supported by EU Network HARP. The research of A. Millet is

partially supported by the research project BMF2003-01345 The authors wish to

thank the referee for helpful comments.



References

[1] N. Krylov and B. Rosovskii, Stochastic evolution equations, J.Soviet Mathematics.16, 1233–1277, (1981).

[2] E. Pardoux. Equations aux derivees partielles stochastiques nonlineares monotones.Etude de solutions fortes de type Ito, (1975). These Doct. Sci. Math. Univ. Paris Sud.

[3] B. L. Rozovskiı, Stochastic evolution systems. vol. 35, Mathematics and its Applica-tions (Soviet Series), (Kluwer Academic Publishers Group, Dordrecht, 1990). ISBN0-7923-0037-8. Linear theory and applications to nonlinear filtering, Translated fromthe Russian by A. Yarkho.

[4] I. Gyongy and A. Millet, On discretization schemes for stochastic evolution equations,Potential Anal. 23(2), 99–134, (2005). ISSN 0926-2601.

[5] N. V. Krylov, On Lp-theory of stochastic partial differential equations in the wholespace, SIAM J. Math. Anal. 27(2), 313–340, (1996). ISSN 0036-1410.

[6] C. Cardon-Weber, Cahn-Hilliard stochastic equation: existence of the solution andof its density, Bernoulli. 7(5), 777–816, (2001). ISSN 1350-7265.

[7] N. V. Krylov and B. L. Rozovskiı, Conditional distributions of diffusion processes,Izv. Akad. Nauk SSSR Ser. Mat. 42(2), 356–378, 470, (1978). ISSN 0373-2436.

[8] E. Pardoux. Filtrage non lineaire et equations aux derivees partielles stochastiquesassociees. In Ecole d’Ete de Probabilites de Saint-Flour XIX—1989, vol. 1464, LectureNotes in Math., pp. 67–163. Springer, Berlin, (1991).

[9] E. Pardoux, Stochastic partial differential equations and filtering of diffusion pro-cesses, Stochastics. 3(2), 127–167, (1979). ISSN 0090-9491.

[10] N. V. Krylov and B. L. Rozovskiı, The Cauchy problem for linear stochastic par-tial differential equations, Izv. Akad. Nauk SSSR Ser. Mat. 41(6), 1329–1347, 1448,(1977). ISSN 0373-2436.

[11] I. Gyongy and N. V. Krylov, On stochastics equations with respect to semimartin-gales. II. Ito formula in Banach spaces, Stochastics. 6(3-4), 153–173, (1981/82). ISSN0090-9491.


Chapter 12

Maximum Principle for SPDEs and Its Applications

Nicolai V. Krylov

School of Mathematics, University of Minnesota,

127 Vincent Hall, Minneapolis, MN, 55455, [email protected].

The maximum principle for SPDEs is established in multidimensional C1 do-mains. An application is given to proving the Holder continuity up to the bound-ary of solutions of one-dimensional SPDEs.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112 The maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

3 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

4 Proof of Theorems 2.5 and 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

5 Auxiliary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3256 Continuity of solutions of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

1. Introduction

The maximum principle is one of the most powerful tools in the theory of second-

order elliptic and parabolic partial differential equations. However, until now it did

not play any significant role in the theory of SPDEs. In this paper we show how

to apply it to one-dimensional SPDEs on the half line R+ = (0,∞) and prove the

Holder continuity of solutions on [0,∞). This result was previously known when

the coefficients of the first order derivatives of solution appearing in the stochastic

term in the equation obeys a quite unpleasant condition. On the other hand, if they

just vanish, then the Holder continuity was well known before (see, for instance,

Ref. [1] and the references therein).

To the best of our knowledge the maximum principle was first proved in Ref. [2]

(see also Ref. [3] for the case of random coefficients) for SPDEs in the whole space

by the method of random characteristics introduced there and also in Ref. [4]. Later

the method of random characteristics was used in many papers for various purposes,

for instance, to prove smoothness of solutions (see, for instance, Refs. [5–8] and the

references therein). It was very tempting to try to use this method for proving

the maximum principle for SPDEs in domains. However, the implementation of

311


312 N. V. Krylov

the method turns out to become extremely cumbersome and inconvenient if the

coefficients of the equation are random processes. Also, it requires more regularity

of solutions than actually needed. Here in Section 2 we state the maximum principle

in domains under minimal assumptions. We prove it in Section 4 by using methods

taken from PDEs after we prepare some auxiliary results in Section 3.

Section 6 contains an application of the maximum principle to investigating the

Holder continuity up to the boundary of solutions of one-dimensional SPDEs. Note

that, for instance, in Refs. [5, 6] and in many other papers that can be found from

our list of references the regularity properties are proved only inside domains. Quite

sharp regularity for solutions of SPDEs in multidimensional domains is established

in Ref. [9], it is stated in terms of appropriate weighted Sobolev spaces and, unfor-

tunately, do not imply even the pointwise continuity up to the boundary. It is worth

saying that we only deal with one-dimensional case and coefficients independent of

the space variable. In a subsequent paper we intend to treat the general case. In

Section 5 we introduce some auxiliary functions used in Section 6.

We denote by Rd the Euclidean space of points x = (x1, ..., xd),

Di =∂

∂xi.

For a domain D ⊂ Rd and we set W 12 (D) to be the closure of the set of infinitely

differentiable functions ϕ having finite norm

‖ϕ‖2W 1

2 (D) = ‖ϕ‖2L2(D) + ‖ϕx‖2

L2(D)

with respect to this norm. Here ϕx is the gradient of ϕ. By0

W 12(D) we denote the

closure of C∞0 (D) with respect to the norm ‖ · ‖W 1

2 (D). Our way to say that u ≤ v

on ∂D is that (u− v)+ ∈0

W 12(D). As usual, the summation convention is enforced

and writing N(....) is to say that the constant N depends and depends only on the

contents of the parentheses. Such constants may change from line to line.

2. The maximum principle

Let D be a domain in Rd of class C1loc and let (Ω,F , P ) be a complete probability

space with a given filtration (Ft, t ≥ 0) of σ-fields Ft ⊂ F complete with respect to

F , P .

We are investigating some properties of a function ut(x) = ut(ω, x) satisfying

(ϕ, ut) = (ϕ, u0) +

∫ t

0

(ϕ, σiks Dius + νks us + gks ) dmks

+

∫ t

0

(ϕ,Di(aijs Djus) + bisDius +Di(a

isus) − csus + fs +Dif

is) dVs. (2.1)

for all t ∈ [0,∞) and any ϕ ∈ C∞0 (D). Here mk

t , k = 1, 2, ..., are one-dimensional

continuous local Ft-martingales, starting at zero, Vt is a nondecreasing continuous


Maximum Principle for SPDEs 313

Ft-adapted process starting at zero, (ϕ, ·) is the pairing between a generalized func-

tion on D and a test function ϕ, the summation convention over repeated indices is

enforced, and the meaning of the remaining objects and further assumptions are de-

scribed below. We need some real-valued functions ξit(x), K1(t) > 0, and K2(t) ≥ 0

defined for i = 1, ..., d, t ∈ [0,∞), x ∈ Rd and also depending on ω.

We assume that aijt (x), bit(x), ait(x), ct(x), σikt (x), νkt (x), and gks are real-valued

functions defined for i, j = 1, ..., d, k = 1, 2, ..., t ∈ [0,∞), x ∈ Rd and also depending

on ω ∈ Ω.

Assumption 2.1. We suppose that, for any ω, 〈mi,mj〉t = 0 if i 6= j, and for any

k we have d〈mk〉t ≤ dVt.

Assumption 2.2. For all values of the arguments

(i) σi := (σi1, σi2, ...), ν := (ν1, ν2, ...), g := (g1, g2, ...) ∈ `2;

(ii) for all λ ∈ Rd

|∑

i

λiξi|2 ≤ K1(2aij − αij)λiλj ,

where αij = (σi, σj)`2 .

The case ξ ≡ 0 is not excluded and in this case Assumption 2.2 (ii) is just the

usual parabolicity assumption.

Assumption 2.3.

(i) The functions aijt (x), bit(x), ait(x), ct(x), σikt (x), νkt (x), ξit(x), K1(t), and

K2(t) are measurable with respect to (ω, t, x) and Ft-adapted for each x;

(ii) the functions aijt (x), bit(x), ait(x), ct(x), σikt (x), νkt (x), and ξit(x) are bounded;

(iii) for each ω, t the functions

ηit := ait − bit − (σit, νt)`2 − ξit

are once continuously differentiable on D, have bounded derivatives, and satisfy

Diηi − 2c+ |ν|2`2 ≤ K2 (2.2)

for all values of arguments;

(iv) for each ϕ ∈ C∞0 (D) the processes ϕft, ϕf

1t ,..., ϕfdt are L2(D)-valued and

ϕgt is an L2(D, `2)-valued Ft-adapted and jointly measurable; for all t ∈ [0,∞) and

ω ∈ Ω∫ t

0

(‖ϕfs‖2

L2(D) +∑

i

‖ϕf is‖2L2(D) + ‖ϕgs‖2

L2(D,`2)+K1(s) +K2(s)

)dVs <∞.

Assumption 2.4. For each ϕ ∈ C∞0 (D)

(i) the process ϕut = ϕut(ω) is L2(D)-valued, Ft-adapted, and jointly measur-

able;

(ii) for any ω

ϕut ∈W 12 (D) (dVt-a.e.);


314 N. V. Krylov

(iii) for each t ∈ [0,∞) and ω

∫ t

0

‖ϕus‖2W 1

2 (D) dVs <∞.

The above assumptions are supposed to hold throughout this section. Here is

the maximum principle saying, in particular, that if gk = f i = 0, f ≤ 0 and u ≤ 0

on the parabolic boundary of [0, T ] × D, then u ≤ 0 in [0, T ]. By the way, our

solutions are L2,loc(D)-valued functions of ω and t, so that for each ω and t an

equivalence class is specified. Naturally, if we write ut(ω) ≤ 0, or ut ≤ 0 we mean

that in the corresponding class there is a nonpositive function.

Theorem 2.5.

Let τ2 ≥ τ1 be stopping times, τ1 < ∞ for any ω. Suppose that, for any ω,

i = 1, ..., d, k = 1, 2, ...,

Iut>0gkt = Iut>0f

it = 0, u+

t ∈0

W 12(D), Iut>0ft ≤ 0

dVt-almost everywhere on (τ1, τ2) and suppose that uτ1 ≤ 0 for any ω. Then almost

surely ut ≤ 0 for all t ∈ [τ1, τ2] ∩ [τ1,∞).

The following comparison principle is a generalization of Theorem 2.5.

Theorem 2.6.

Let τ2 ≥ τ1 be stopping times, τ1 < ∞ for any ω. Let ρt ≥ 0, t ∈ [0,∞), be

a nondecreasing continuous Ft-adapted process and let ft, f1t ,..., f

dt , and gt satisfy

Assumption 2.3 (iv). Let ut be a process satisfying Assumption 2.4 and such that

equation (2.1) holds for all t ∈ [0,∞) and any ϕ ∈ C∞0 (D) with ft, f

1t ,..., f

dt , and

gt in place of ft, f1t ,..., f

dt , and gt, respectively.

Assume that, for any ω, (dVt-a.e.) on [τ1, τ2] we have

Iut>ρut (gt − ρtgt) = Iut>ρtut(fit − ρtf

it ) = 0, i = 1, ..., d,

Iut>ρtut(ft − ρtft) ≤ 0, Iut>ρtut ut ≥ 0, (ut − ρtut)+ ∈

0

W 12(D).

Finally, assume that uτ1 ≤ ρτ1 uτ1 for any ω.

Then almost surely ut ≤ ρtut for all t ∈ [τ1, τ2] ∩ [τ1,∞).

Corollary 2.7. Assume that, for any ω, (dVt-a.e.) on (τ1, τ2) ×D we have

Iut>1(νkt + gkt ) = Iut>1(f it + ait) = 0, i = 1, ..., d, k = 1, 2, ...

Iut>1f ≤ Iut>1c, (ut − 1)+ ∈0

W 12(D).

Also assume that uτ1 ≤ 1 for any ω. Then almost surely ut ≤ 1 for all t ∈ [τ1, τ2]∩[τ1,∞).



Indeed, it suffices to take ut ≡ 1, ρt ≡ 1 and observe that ut satisfies (2.1) with

f it = −ait, ft = ct, and gt = −νt in place of f it , ft, and gt, respectively.

This corollary generalizes the corresponding results of Refs. [2] and [3], where

νk = gk = f i = ai = 0.

Remark 2.8. Our equation has a special structure, which may look quite restric-

tive. In particular, we assume that the martingales mkt are mutually orthogonal.

The general case, actually, reduces to this particular one after using the fact that

one can always orthogonalize the martingales by using, for instance, the Gramm-

Schmidt procedure. This, of course, would change σ, ν, and g, and writing the

corresponding general conditions would only obscure the matter. Then passing

from mkt to (no summation in k)

∫ t

0

ρks dmks , ρks =

( dt

dt+ d〈mk〉t)1/2

allows one to have d〈mk〉t ≤ dt and adding after that t to Vt allows one to have

d〈mk〉t ≤ dVt. Again we should modify our coefficients but we will see in the proof

of Theorem 2.6 that this modification does not affect Assumption 2.2, which is an

assumption about parabolicity of our equation and not strict nondegeneracy.

3. Auxiliary results

In this section the notation ut is sometimes used for different objects than in Section

2.

Denote by R the set of real-valued functions convex r(x) on R such that

(i) r is continuously differentiable, r(0) = r′(0) = 0,

(ii) r′ is absolutely continuous, its derivative r′′ is bounded and left continuous, that

is usual r′′ which exists almost everywhere is bounded and there is a left-continuous

function with which r′′ coincides almost everywhere.

For r ∈ R by r′′ we will always mean the left-continuous modification of the

usual second-order derivative of r.

Remark 3.1. For each r ∈ R there exists a sequence rn ∈ R of infinitely differ-

entiable functions such that |rn(x)| ≤ N |x|2, |r′n(x)| ≤ N |x|, and |r′′n| ≤ N with

N <∞ independent of x and n, rn, r′n, r

′′n→r, r′r′′ on R.

Indeed, let ζ ∈ C∞0 (R) be a nonnegative function with support in (0, 1) and unit

integral. For ε > 0 define ζε(x) = ε−1ζ(x/ε) and rε(x) = r ∗ ζε(x)− r ∗ ζε(0)−xr′ ∗ζε(0). Then rε is infinitely differentiable, rε(0) = r′ε(0) = 0,

|r′′ε | = |r′′ ∗ ζε| ≤ sup |r′′| <∞.

In particular,

|r′ε(x)| = |∫ x

0

r′′ε (y) dy| ≤ N |x|, |rε(x)| = |∫ x

0

r′ε(y) dy| ≤ N |x|2.


316 N. V. Krylov

Finally, the convergences rε→r and r′ε→r′ follow by the continuity of r and r′ and

the convergence r′′ε→r′′ follows from the dominated convergence theorem, the left

continuity of r′′ and the formula

r′′ε (x) =

∫ 1

0

r′′(x − εy)ζ(y) dy.

In the following lemma the assumption that D is a locally smooth domain is not

used.

Lemma 3.2.

Let ut = ut(ω) be an L2(D)-valued process such that u0 is F0-measurable. Let

ft and gt = (g1t , g

2t , ...) be Ft-adapted and jointly measurable processes with values

L2(D) and L2(D, `2), respectively. Assume that for each t ∈ [0,∞) we have∫ t

0

(‖fs‖2L2(D) + ‖gs‖2

L2(D,`2)) dVs <∞ (3.1)

and for any ϕ ∈ C∞0 (D)

(ϕ, ut)L2(D) = (ϕ, u0)L2(D) +

∫ t

0

(ϕ, fs)L2(D) dVs +

∫ t

0

(ϕ, gks )L2(D) dmks . (3.2)

Then (i) ut is a continuous L2(D)-valued function (a.s.); (ii) for any r ∈ R(a.s.) for all t ∈ [0,∞)

‖r1/2(ut)‖2L2(D) = ‖r1/2(u0)‖2

L2(D) +

∫ t

0

hs dVs +mt, (3.3)

where

hs := (r′(us), fs)L2(D) + (1/2)‖(r′′)1/2(us)gs‖2L2(D,`2)

, (3.4)

gks :=

(d〈mk〉sdVs

)1/2

gks , mt :=

∫ t

0

(r′(us), gks )L2(D) dm

ks

and mt is a local martingale;

(iii) (a.s.) for t ∈ [0,∞)

‖u+t ‖2

L2(D) = ‖u+0 ‖2

L2(D) +

∫ t

0

hs dVs +mt, (3.5)

where

hs := 2(u+s , fs)L2(D) + ‖gsIus>0‖2

L2(D,`2),

mt := 2

∫ t

0

(u+s , g

ks )L2(D) dm

ks

and mt is a local martingale.



Proof. (i) Recall that the operation of stochastic integration of Hilbert space

valued processes is well defined. Therefore, the process

ut = u0 +

∫ t

0

fs dVs +

∫ t

0

gks dmks

is well defined as a continuous L2(D)-valued process. We also recall how the scalar

product interacts with integrals. Then it is seen that for any t and ϕ ∈ C∞0 (D) we

have (ϕ, ut) = (ϕ, ut) (a.s.). Since both parts are continuous in t, the equality holds

for all t at once (a.s.), and since C∞0 (D) is dense in L2(D), we have that ut = ut

for all t (a.s.). This proves (i). As a corollary we obtain that

supt≤T

‖ut‖L2(D) <∞, ∀T <∞ (a.s.). (3.6)

(ii) It suffices to prove (3.3) for infinitely differentiable r ∈ R. Indeed, for rnfrom Remark 3.1, passing to the limit in all term in (3.3) apart from mt presents

no problem at all in light of (3.6) and the dominated convergence theorem. Also

mt(n) :=

∫ t

0

(r′n(us), gks )L2(D) dm

ks→mt

uniformly in t on finite intervals in probability because

〈m(n) −m〉t =

∫ t

0

∑

k

((r′n − r′)(us), g

ks )2L2(D) d〈mk〉s

≤∫ t

0

∑

k

((r′n − r′)(us), g

ks )2L2(D) dVs

≤∫ t

0

‖(r′n − r′)(us)‖2L2(D)‖gs‖2

L2(D,`2)dVs→0

again owing to (3.6) and the dominated convergence theorem.

Thus, we may concentrate on the case that r is infinitely differentiable. Take

a symmetric ζ ∈ C∞0 (Rd) with support in the unit ball centered at the origin and

unit integral. For ε > 0 set ζε(x) = ε−dζ(x/ε) and for functions v = v(x) define

v(ε) = v ∗ ζε. Also set

Dε = x ∈ D : dist (x, ∂D) < ε.

According to (3.2) for any x ∈ Dε and t ≥ 0 we have

u(ε)t (x) = u

(ε)0 (x) +

∫ t

0

f (ε)s (x) dVs +

∫ t

0

gk(ε)s (x) dmks .

By Ito’s formula we have that on Dε

r(u(ε)t ) = r(u

(ε)0 ) +

∫ t

0

[r′(u(ε)s )f (ε)

s + (1/2)r′′(u(ε)s )|g(ε)

s |2`2 ] dVs


318 N. V. Krylov

+

∫ t

0

r′(u(ε)s )gk(ε)s dmk

s . (3.7)

Here, for each ε > 0, the integrands are smooth functions of x and their magnitudes

along with the magnitudes of each of their derivatives in Dε are majorized by a

constant (possibly depending on ε) times

‖fs‖L2(D) sups≤t

‖us‖L2(D) or ‖gs‖2L2(D,`2)

or ‖gks‖L2(D) sups≤t

‖us‖L2(D).

This and (3.1) and (3.6) allow us to use Fubini’s theorem while integrating through

(3.7) and conclude

‖r1/2(u(ε)t )‖2

L2(Dε) = ‖r1/2(u(ε)0 )‖2

L2(Dε) +

∫ t

0

[(r′(u(ε)s ), f (ε)

s )L2(Dε)

+(1/2)‖(r′′)1/2(u(ε)s )g(ε)

s ‖2L2(Dε,`2)

] dVs

+

∫ t

0

(r′(u(ε)s ), gk(ε)s )L2(Dε) dm

ks . (3.8)

Now we let ε ↓ 0. We use that for any function v ∈ L2(D)

‖v(ε)‖L2(Dε) ≤ ‖(vID)(ε)‖L2(Rd) ≤ ‖vID‖L2(Rd) = ‖v‖L2(D)

and v(ε)IDε→v in L2(D). In particular, gk(ε)s IDε→gks and u

(ε)s IDε→us implying

that r′(u(ε)s )IDε→r′(us) in L2(D) and

(r′(u(ε)s ), gk(ε)s )L2(Dε)→(r′(us), g

ks )L2(D)

for each k and dP × dVs-almost all (ω, s).

We also use (3.1) and (3.6) to assert that

∞∑

k=1

∫ t

0

supε∈(0,1)

|(r′(u(ε)s ), gk(ε)s )L2(Dε)|2 dVs

≤ N sups≤t

‖us‖2L2(D)

∫ t

0

‖gs‖2L2(D,`2)

dVs <∞.

As is easy to see this implies that the local martingale part in (3.8) converges to mt

as ε ↓ 0 in probability locally uniformly with respect to t.

Similar manipulations with other terms in (3.8) allow us to get (3.3). Since (3.5)

is just a particular case of (3.3), the lemma is proved.

Remark 3.3. Lemma 3.2 remains true if in the definition of R instead of requiring

r′′ to have a left-continuous modification we required it to have a right-continuous

one, and of course, in (3.4) used this right-continuous modification. This is seen

after replacing u with −u.



In case r(x) = (x+)2 the function r′′ has both right- and left-continuous mod-

ifications, so that in the definition of mt one can use 2Ius>0 or 2Ius≥0. It follows

that (a.s.) for any t

∫ t

0

‖gsIus=0‖2L2(D,`2)

dVs = 0.

Furthermore, since, for any v ∈ L2(D), ut + v has the same form as ut,

∫ t

0

‖gsIus=v‖2L2(D,`2)

dVs = 0.

Lemma 3.4. Let D be an arbitrary domain. Let ut be an L2(D)-valued F0-

measurable process such that for any ω

ut ∈ W 12 (D)

(dVt-a.e.) and for each T ∈ [0,∞) and ω

∫ T

0

‖ut‖2W 1

2 (D) dVt <∞. (3.9)

Let ft, f1t , ..., f

dt , and gt = (g1

t , g2t , ...) be Ft-adapted and jointly measurable processes

with values in L2(D) and L2(D, `2), respectively. Assume that for each t ∈ [0,∞)

we have∫ t

0

(‖fs‖2L2(D) +

∑

i

‖f is‖2L2(D) + ‖gs‖2

L2(D,`2)) dVs <∞, (3.10)

and for each t ∈ [0,∞), ϕ ∈ C∞0 (D), and ω

(ϕ, ut) = (ϕ, u0) +

∫ t

0

(ϕ, fs +Difis) dVs +

∫ t

0

(ϕ, gks ) dmks . (3.11)

Finally, assume that there is a compact set G ⊂ D such that

ut(x) = ft(x) = f it (x) = gkt (x) = 0

outside G. Then

(a) ut is a continuous L2(D)-valued function (a.s.); (b) (a.s.) for all t ∈ [0,∞)

‖u+t ‖2

L2(D) = ‖u+0 ‖2

L2(D) +

∫ t

0

hs dVs +mt, (3.12)

where

hs := 2(u+s , fs)L2(D) − 2(Ius>0Dius, f

is)L2(D) + ‖gsIϕus>0‖2

L2(D,`2),

mt := 2

∫ t

0

(u+s , g

ks )L2(D) dm

ks .


320 N. V. Krylov

Proof. Observe that (3.11) holds for all infinitely differentiable functions ϕ. Fur-

thermore, since ut ∈0

W 12(D) (dVt-a.e.) assertion (a) is well known (see, for instance,

Ref. [10], the references therein, and Remark 2.8).

To prove (b), take ε smaller than the distance between G and ∂D. Notice that,

owing to the symmetry of ζ, for ϕ ∈ C∞0 (D)

(ϕ(ε), ut) = (ϕ(ε), ut)L2(Rd) = (ϕ, u(ε)t )L2(Rd) = (ϕ, u

(ε)t )L2(D).

Therefore, it follows from (3.11) that for any ϕ ∈ C∞0 (D)

(ϕ, u(ε)t ) = (ϕ, u

(ε)0 ) +

∫ t

0

(ϕ, fεs ) ds+

∫ t

0

(ϕ, gk(ε)s ) dmks ,

where

fεs := f (ε)s +Dif

i(ε)s

is an L2(D)-valued function with norm that is locally square integrable against dVs.

By Lemma 3.2 for any r ∈ R

‖r1/2(u(ε)t )‖2

L2(D) = ‖r1/2(u(ε)0 )‖2

L2(D) +

∫ t

0

hεs dVs +mεt , (3.13)

where

mεt :=

∫ t

0

(r′(u(ε)s ), gk(ε)s )L2(D) dm

ks ,

hεs := (r′(u(ε)s ), fεs )L2(D) + (1/2)‖(r′′)1/2(u(ε)

s )g(ε)s ‖2

L2(D,`2)

= (r′(u(ε)s ), f (ε)

s )L2(D)−(r′′(u(ε)s )Diu

(ε)s , f i(ε)s )L2(D)+(1/2)‖(r′′)1/2(u(ε)

s )g(ε)s ‖2

L2(D,`2).

If r is infinitely differentiable, then by using (3.9) and (3.10) one easily passes

to the limit in (3.13) as ε→0. The argument is quite similar to the corresponding

argument in the proof of Lemma 3.2 and, for smooth r ∈ R, yields

‖r1/2(ut)‖2L2(D) = ‖r1/2(u0)‖2

L2(D) +

∫ t

0

hs dVs +mt,

where

mt =

∫ t

0

(r′(us), gks )L2(D) dm

ks ,

hs = (r′(us), fs)L2(D) − (r′′(us)Dius, fis)L2(D) + (1/2)‖(r′′)1/2(us)gs‖2

L2(D,`2).

Finally, as in the proof of Lemma 3.2 one easily passes from smooth r ∈ R to

arbitrary ones and gets (3.12) by taking r(x) = (x+)2. The lemma is proved.

Lemma 3.4 serves as an auxiliary tool to prove a deeper result.

Lemma 3.5. Let D be an arbitrary domain. Assume that for each ϕ ∈ C∞0 (D)

(i) ϕut is an L2(D)-valued process such that ϕu0 is F0-measurable;



(ii) for any ω

ϕut ∈ W 12 (D)

(dVt-a.e.) and for each T ∈ [0,∞) and ω∫ T

0

‖ϕut‖2W 1

2 (D) dVt <∞.

(iii) Let ft, f1t , ..., f

dt , and gt = (g1

t , g2t , ...) be Ft-adapted and jointly measurable

processes with values in L2(D) and L2(D, `2), respectively. Assume that for each

t ∈ [0,∞) and ϕ ∈ C∞0 (D) we have

∫ t

0

(‖ϕfs‖2L2(D) +

∑

i

‖ϕf is‖2L2(D) + ‖ϕgs‖2

L2(D,`2)) dVs <∞,

(ϕ, ut) = (ϕ, u0) +

∫ t

0

(ϕ, fs +Difis) dVs +

∫ t

0

(ϕ, gks ) dmks .

Then, for any ϕ ∈ C∞0 (D),

(a) ϕut is a continuous L2(D)-valued function (a.s.); (b) (a.s.) for all t ∈ [0,∞)

‖(ϕut)+‖2

L2(D) = ‖(ϕu0)+‖2L2(D) +

∫ t

0

hs dVs +mt,

where

hs := 2((ϕus)+, ϕfs − f isDiϕ)L2(D) − 2(Iϕus>0Di(ϕus), ϕf

is)L2(D)

+‖ϕgsIϕus>0‖2L2(D,`2)

, mt := 2

∫ t

0

(ϕu+s , ϕg

ks )L2(D) dm

ks .

Proof. Clearly, for any ϕ, η ∈ C∞0 (D) we have

(ϕ, ηut) = (ϕ, ηu0) +

∫ t

0

(ϕ, ηfs − f isDiη +Di(ηfis)) dVs +

∫ t

0

(ϕ, ηgks ) dmks . (3.14)

Therefore, ηut satisfies the assumptions of Lemma 3.4 with ηfs − f isDiη, ηf is,

and ηgks in place of fs, Difis, and gks , respectively.

By applying Lemma 3.4 to ηut in place of ut we get the result with η in place

of ϕ. This certainly proves the lemma.

4. Proof of Theorems 2.5 and 2.6

In this section the assumptions stated in Section 2 are supposed to be satisfied. We

use the fact that due to our hypothesis that D ∈ C1, there exist sequences ζn and

ζn of nonnegative C∞0 (D)-functions such that 0 ≤ ζn, ζn ≤ 1, ζn, ζn→1 in D as

n→∞ and for any v ∈0

W 12(D), i = 1, ..., d,

‖vDiζn‖L2(D) ≤ N(‖(1 − ζn)v‖L2(D) + ‖(1 − ζn)Dv‖L2(D)), (4.1)


322 N. V. Krylov

where N is independent of n and v (see, for instance, the proof of Theorem 5.5.2

in Ref. [11]). We also know (see, for instance, the proof of Lemma 2.3.2 in Ref. [12]

or Problem 17, Chapter 5 in Ref. [11]) that if v ∈W 12 (D), then v+ ∈W 1

2 (D) and

Div+ = Iv>0Div.

Proof of Theorem 2.5. Set

K = K1 +K2, ϕt =

∫ t

0

K(s) ds.

Take the sequences of nonnegative ζn, ζn ∈ C∞0 (D) from above. By Ito’s formula

and Lemma 3.5

‖ζnu+t ‖2

L2(D)e−ϕt = ‖ζnu+

0 ‖2L2(D) +

∫ t

0

hns dVs +mt(n),

where

eϕshns = I1s + I2s + I3s −K(s)‖ζnu+s ‖2

L2(D),

I1s = 2(ζnu

+s , ζn[fs + bisDius − csus] − [usa

is + aijs Djus + f is]Diζn

)L2(D)

,

I2s = −2(Iζnus>0Di(ζnus), ζn[usa

is + aijs Djus + f is]

)L2(D)

,

I3s = ‖ζnIζnus>0[σisDius + νsus + gs]‖2L2(D,`2)

,

mt(n) = 2

∫ t

0

e−ϕs(ζnu

+s , ζn[σiks Dius + νks us + gks ]

)L2(D)

dmks .

Since u+τ1 = 0 we have

e−ϕτ2∧t∨τ1 ‖ζnu+τ2∧t∨τ1‖2

L2(D) =

∫ t

0

Iτ2>s>τ1hns dVs + mt(n), (4.2)

where

mt(n) := mτ2∧t∨τ1(n) −mτ1(n)

is a local martingale.

Next we use the assumptions of the theorem and see that for dVs-almost all

s ∈ (τ1, τ2) we have

I1s ≤ 2(ζnu+s , ζn[bisDius − csus] − [usa

is + aijs Djus]Diζn)L2(D)

= 2(ζ2nu

+s , b

isDiu

+s − csu

+s )L2(D) + I4s

with

I4s = −2(ζnu+s Diζn, u

+s a

is + aijs Dju

+s )L2(D).



At this moment we recall (4.1) and observe that

(ζn|Diζn|, (u+s )2)L2(D) ≤ N‖ζnu+

s ‖L2(D)‖u+s Diζn‖L2(D).

Then we see that

I4s ≤ N(‖(1 − ζn)u+

s ‖L2(D) + ‖(1 − ζn)Du+s ‖L2(D)

)‖u+

s ‖W 12 (D),

where and below by N we denote various finite constants.

In I2s

Iζnus>0Di(ζnus) = Di(ζnu+s ) = u+

s Diζn + ζnDiu+s ,

so that

I2s = −2(ζ2nDiu

+s , u

+s a

is + aijs Dju

+s )L2(D) + I4s.

Next,

ζnIζnus>0 = ζnIus>0,

I3s ≤ ‖ζnIus>0[σisDius + νsus]‖2L2(D,`2)

= (ζ2nDiu

+s , α

ijs Dju

+s )L2(D)

+2(ζ2nDiu

+s , u

+s (σis, νs)`2)L2(D) + ‖ζn|νs|`2u+

s ‖2L2(D).

Also observe that certain parts of I2s and I3s can be combined if we use that

−2(ζ2nDiu

+s , a

ijs Dju

+s )L2(D) + (ζ2

nDiu+s , α

ijs Dju

+s )L2(D)

≤ −K−11 (s)‖ζnξisDiu

+s ‖2

L2(D).

It follows that for dVs-almost all s ∈ (τ1, τ2)

eϕshns ≤∫

D

[ζ2n(bis − ais + (σis, νs)`2)2u+

s Diu+s + ζ2

n(u+s )2(|νs|2`2 − 2cs)] dx

+N(‖(1 − ζn)u+

s ‖L2(D) + ‖(1 − ζn)Du+s ‖L2(D)

)‖u+

s ‖W 12 (D)

−K−11 (s)‖ζnξisDiu

+s ‖2

L2(D) −K(s)‖ζnu+s ‖2

L2(D).

Here

bis − ais + (σis, νs)`2 = −ξis − ηis

and we transform the integral of

ζ2n(−ηis)2u+

s Diu+s = −ηisζ2

nDi(u+s )2

by integrating by parts. Then we get that for dVs-almost all s ∈ (τ1, τ2)

eϕshns ≤∫

D

[−ζ2nξis2u

+s Diu

+s + ζ2

n(u+s )2(|νs|2`2 − 2cs +Diη

is)] dx


324 N. V. Krylov

+2

∫

D

(u+s )2ζnη

isDiζn dx+N

(‖(1− ζn)u+

s ‖L2(D) + ‖(1− ζn)Du+s ‖L2(D)

)‖u+

s ‖W 12 (D)

−K−11 (s)‖ζnξisDiu

+s ‖2

L2(D) −K(s)‖ζnu+s ‖2

L2(D).

We also use the fact that

| − ζ2nξis2u

+s Diu

+s | ≤ K−1

1 (s)|ζnξisDiu+s |2 +K1(s)ζ2

n(u+s )2,

|νs|2`2 − 2cs +Diηis ≤ K2(s).

Then we easily see that for dVs-almost all s ∈ (τ1, τ2)

eϕshns ≤ N(‖(1 − ζn)u+

s ‖L2(D) + ‖(1 − ζn)Du+s ‖L2(D)

)‖u+

s ‖W 12 (D)

+2

∫

D

(u+s )2ζnη

isDiζn dx ≤ N

(‖(1− ζn)u+

s ‖L2(D) +‖(1− ζn)Du+s ‖L2(D)

)‖u+

s ‖W 12 (D).

Now (4.2) yields

e−ϕτ2∧t∨τ1 ‖u+τ2∧t∨τ1‖2

L2(D) ≤ mt(n)

+N

∫ t

0

(‖(1 − ζn)u+s ‖L2(D) + ‖(1 − ζn)Du+

s ‖L2(D))‖u+s ‖W 1

2 (D) dVs. (4.3)

The integrals against dVs in (4.3) tend to zero as n→∞ by the dominated con-

vergence theorem. Since the sum of them with continuous local martingales is

nonnegative, the local martingales and the right-hand side of (4.3) tend to zero

uniformly on finite time intervals in probability (see, for instance, Ref. [13]). So

does the left-hand side and the theorem is proved.

Proof of Theorem 2.6. Obviously, ut = ρtut satisfies

(ϕ, ut) = (ϕ, u0) +

∫ t

0

(ϕ, σiks Dius + νks us + ρsgks ) dmk

s +

∫ t

0

(ϕ, us) dρs

+

∫ t

0


isus) − csus + ρsfs + ρsDif

is) dVs. (4.4)

We rewrite this equation introducing

Vt = Vt + ρt, pt =dρt

dVt, qt =

dVt

dVt,

(aijt , ait, b

it, ct) = qt(a

ijt , a

it, b

it, ct), (σikt , ν

kt ) = q

1/2t (σikt , ν

kt ),

ft = qtρtft + ptut, f it = qtρtfit , gkt = q

1/2t ρtg

kt .

We also set

mkt =

∫ t

0

q−1/2s dmk

s (0−1/2 := 0).



Notice that since d〈mk〉t ≤ dVt = qtdVt the last integral makes sense.

In this notation (2.1) and (4.4) are rewritten as

(ϕ, ut) = (ϕ, u0) +

∫ t

0

(ϕ, σiks Dius + νks us + q1/2s gks ) dmks

+

∫ t

0


isus) − csus + qsfs + qsDif

is) dVs,

(ϕ, ut) = (ϕ, u0) +

∫ t

0

(ϕ, σiks Dius + νks us + gks ) dmks

+

∫ t

0

(ϕ,Di(aijs Dj us) + bisDius +Di(a

isus) − csus + fs +Dif

is) dVs,

respectively. We subtract these equations, denote vt = ut − ut, and observe that

for any ω we have dVs-almost everywhere on (τ1, τ2) that

Ivs>0(q1/2s gks − gks ) = Ivs>0(qsfis − f is) = 0,

Ivs>0(qsfs − fs) = qsIvs>0(fs − ρsfs) − psusIvs>0 ≤ 0.

We also use the fact that the above versions of equations (2.1) and (4.4) satisfy the

same Assumptions 2.1, 2.2, 2.3, and 2.4 with qsξis and qsKi(s) in place of ξis and

Ki(s), respectively. Then we the desired result directly from Theorem 2.5. The

theorem is proved.

5. Auxiliary functions

Let C[0,∞) be the set of real-valued continuous functions on [0,∞). For x· ∈C[0,∞) set xs = x0 for s ≤ 0 and for n = 0, 1, 2, ... and t ≥ 0 introduce

∆−n (x·, t) = 2n/2 osc

[t−2−n,t]x· .

If c ∈ (0,∞), then define

M−n (x·, c, t) = #k = 0, ..., n : ∆−

k (x·, t) ≤ c.

For n negative we set M−n (x·, c, t) := 0. For c ≥ 0, d > 0, δ > 0 introduce

γ(c, d, δ) = 1 − P ( mint≤δ/2

wt ≤ −c− d/√

2, maxt≤δ/2

wt ≤ d− d/√

2).

As is easy to see

γ(c, d, δ) ≥ P (wt reaches d− d/√

2 before reaching − c− d/√

2)

=c+ d/

√2

c+ d> 1/

√2,


326 N. V. Krylov

so that 2 log2 γ(c, d, δ) > −1.

Set, for m = 0, 1, 2, ...,

Qm := Qm(x·) := (s, y) : s ≥ 0, xs < y < xs + 2−m/2.

Lemma 5.1. For m = 0, 1, 2, ..., t ≥ 0 and x ∈ (0, 2−m/2) introduce

rm(t, x) = rm(x·, t, x) = P (xt + x+ wτ√δ = xt−τ + 2−m/2),

where τ = infs > 0 : (t− s, xt + x+ ws√δ) 6∈ Qm. Then

rm(t, x) ≤ [γ(c, d, δ)]M−m+n(x·,c,t)−M−

m−1(x·,c,t)−k, (5.1)

where n = n(2m/2x/d), k = k(c+ d), and

n(y) = [(−2 log2 y)+], k(d) = 2 + [(2 log2 d)+].

Proof. Define

t = 2mt, x = 2m/2x, ws = 2m/2ws2−m , xs = 2m/2xs2−m .

Then as is easy to see rm(t, x) is rewritten as

P (xt + x+ wτ√δ = xt−τ + 1), (5.2)

where

τ = infs > 0 : (t− s, xt + x+ ws√δ) 6∈ Q0(x·) = 2mτ.

Since w· is a Wiener process, by Corollary 3.4 in Ref. [14] expression (5.2) is less

than

[γ(c, d, δ)]M−n (x·,c,t)−k,

where n = n(x/d). Here

M−n (x·, c, t) = #j = 0, ..., n : 2j/2 osc

[ t−2−j ,t ]x· ≤ c

= #j = 0, ..., n : 2(j+m)/2 osc[t−2−j−m,t ]

x· ≤ c

= #j = m, ...,m+ n : ∆−j (x·) ≤ c = M−

m+n(x·, c, t) −M−m−1(x·, c, t)

and the result follows. The lemma is proved.

Lemma 5.2. Let T ∈ (0,∞). Assume that

limm→∞

1

m+ 1inf

t∈[0,T ]M−m(x·, c, t) > α > 0.

Take constants p > 0 and ν so that

1 < νp < pχ+ 1 < 0, (5.3)

where χ = −2α log2 γ(c, d, δ). Then, for rm from Lemma 5.1 it holds that

supm≥0

supt∈[0,T ]

2−m(νp−1)/(2α)

∫ 2−m/2

0

1

xνprpm(t, x) dx <∞. (5.4)



Proof. By Lemma 5.1 for a constant N and γ = γ(c, d, δ)

rm(t, x) ≤ NγM−m+n−m,

where x ≤ 2−m/2, n = n(2m/2x/d), M−m+n = M−

m+n(x·, c, t). Furthermore,

n(2m/2x/d) ≥ (−2 log2(2m/2x) + 2 log2 d)+ − 1

≤ (−2 log2(2m/2x))+ −N = −m− 2 log2 x−N,

where N is a constant. Hence, m+ n ≥ −2 log2 x−N . Since obviously rm ≤ 1 we

have that

rm(t, x) ≤ 1 ∧ (Nγ−m+M−

−2 log2 x−N ).

By the assumption if x is small enough

M−−2 log2 x−N > α(−2 log2 x).

Therefore, for x ∈ (0, 2−m/2]

rm(t, x) ≤ 1 ∧ (Nγ−m−2α log2 x) = 1 ∧ (Nγ−mxχ). (5.5)

Next,

∫ 2−m/2

0

1

xνprpm(t, x) dx ≤

∫ ∞

0

1

xνp(1 ∧ (Nγ−mxχ))p dx

= γm(1−νp)/χ∫ ∞

0

1

xνp(1 ∧ (Nxχ))p dx,

where the last integral is finite owing to (5.3). This proves the lemma.

Let wt be a Wiener process with respect to a filtration Ft, t ≥ 0 of complete σ-

fields and let at and σt be bounded real-valued processes predictable with respect to

Ft, t ≥ 0 and such that at−σ2t ≥ δσ2

t , where δ ∈ (0,∞) is a constant, at−σ2t > 0

for all (ω, t) and for all ω∫ ∞

0

[at − σ2t ] dt = ∞.

Set Dx = ∂/∂x. For m = 0, 1, 2, ... we will be dealing with the SPDE

dv(t, x) = (1/2)atD2xv(t, x) dt+ σtDxv(t, x) dwt

in Bm = (0,∞) × (0, 2−m/2) with boundary conditions

v(t, 0) = 0, v(t, 2−m/2) = 1, t > 0, (5.6)

v(0, x) = 0, 0 < x < 1. (5.7)

Recall that by Theorem 2.1 in Ref. [14] there is a deterministic function α0(c),

c > 0, such that α0(c)→1 as c→∞ and with probability one for any T ∈ (0,∞)

limn→∞

inft∈[0,T ]

1

n+ 1M−n (w·, c, t) = α0(c).


328 N. V. Krylov

Theorem 5.3. For each m = 0, 1, 2, ... there is a function vm(t, x) = vm(ω, t, x)

defined on Ω × Bm such that

(i) vm(t, x) is Ft-measurable for each (t, x) ∈ Bm,

(ii) vm(t, x) is bounded and continuous in Bm \ (0, 2−m/2) for each ω,

(iii) derivatives of vm(t, x) of any order with respect to x are continuous in

Bm ∪ (0 × (0, 2−m/2)) for each ω,

(iv) equations (5.6) and (5.7) hold for each ω,

(v) almost surely, for any (t, x) ∈ Bm

vm(t, x) =

∫ t

0

(1/2)asD2xvm(s, x) ds+

∫ t

0

σsDxvm(s, x) dws,

(vi) for any T ∈ (0,∞), c, d > 0, p > 0, α > 0 such that α0(c√δ) > α, and ν

satisfying

1 < νp < χp+ 1, (5.8)

where χ = −2α log2 γ(c, d, 1), we have that with probability one

πT := supm≥0

supt∈[0,T ]

2−m(νp−1)/(2α)

∫ 2−m/2

0

1

xνpvpm(t, x) dx <∞. (5.9)

Proof. In Lemma 5.1 take δ = 1 and set vm(x·, xt + x, t) = rm(x·, t, x), where

rm is introduced in that lemma. Set

ψt =

∫ t

0

(as − σ2s ) ds, ξt =

∫ ϕt

0

σs dws, Ft = Fϕt ,

vm(t, x) = vm(ω, t, x) = vm(ξ·, t, x), vm(t, x) = vm(ω, t, x) = vm(ψt, x+ ξψt),

where ϕt = infs ≥ 0 : ψs ≥ t is the inverse function to ψt.

It is proved in Theorem 4.1 of Ref. [14] that v0 possesses properties (i)-(v). The

proof that this is also true for any m is no different.

Furthermore, it is well known that

√δ

∫ t

0

σs dws = wψ(t),

where wt is a Wiener process and

ψt = δ

∫ t

0

σ2s ds.

Hence ξt = δ−1/2wψ(ϕt)with

(ψ(ϕt))′ = δσ2

s/(as − σ2s )|s=ϕt ≤ 1.

It follows that for n = 0, 1, 2, ... we have

M−n (ξ·, c, t) ≥M−

n (w·, c√δ, ψ(ϕt)),



inft≤T

M−n (ξ·, c, t) ≥ inf

t≤TM−n (w·, c

√δ, t),

and with probability one

limn→∞

inft≤T

1

n+ 1M−n (ξ·, c, t) ≥ α0(c

√δ) > α.

Finally, for M = supω,t(at − σ2t ) we have

supt≤T

∫ 2−m/2

0

1

xνpvpm(t, x) dx ≤ sup

t≤MT

∫ 2−m/2

0

1

xνpvpm(t, x+ ξt) dx

= supt≤MT

∫ 2−m/2

0

1

xνpvpm(ξ·, t, x+ ξt) dx = sup

t≤MT

∫ 2−m/2

0

1

xνprpm(ξ·, t, x) dx.

After this it only remains to use Lemma 5.2. The theorem is proved.

Remark 5.4. Obviously, for any ε ∈ (0, 2−(m+2)/2) we have

vm(t, ·) ∈W 12 (ε, 2−m/2 − ε)

for any t ∈ [0,∞) and for any T ∈ (0,∞) we have∫ T

0

‖vm(t, ·)‖W 12 (ε,2−m/2−ε) dt <∞.

Furthermore, by using the deterministic and stochastic versions of Fubini’s theorem

one easily proves that for any ϕ ∈ C∞0 (0, 2−m/2) with probability one for all t ∈

[0,∞)

(ϕ, vm(t, ·) = (1/2)

∫ t

0

(ϕ, asD2xvm(s, ·)) ds+

∫ t

0

(ϕ, σsDxvm(s, ·)) dws.

6. Continuity of solutions of SPDEs

We take the processes at, σt as before Theorem 5.3 but impose stronger assumptions

on them.

Assume that there exist constants δ0, δ1 ∈ (0, 1] such that, for every (ω, t)

δ0 ≤ δ1at ≤ at − σ2t ≤ δ−1

0 ,

We will be dealing with solutions ut(x) of

dut = ((1/2)atD2xut + ft) dt+ (σtDxut + gt) dwt (6.1)

on R+ with zero initial condition. To specify the assumptions on f, g and the class

of solutions we borrow the Banach spaces Hγp,θ(τ) and Lp,θ(τ) from [15]. We also

denote by M the operator of multiplying by x. Recall that, for p ≥ 2, 0 < θ < p,

the norms in Hγp,θ(τ), γ = 1, 2, and Lp,θ(τ) are given by

‖v‖pLp,θ(τ) = E

∫ τ

0

∫ ∞

0

xθ−1|v(t, x)|p dxdt,


330 N. V. Krylov

‖v‖H1p,θ(τ) = ‖v‖Lp,θ(τ) + ‖MDxv‖Lp,θ(τ),

‖v‖H2p,θ(τ) = ‖v‖H1

p,θ(τ) + ‖M2D2xv‖Lp,θ(τ).

Given p ≥ 2, θ ∈ [p−1, p), any stopping time τ , f ∈M−1Lp,θ(τ), and g ∈ H1p,θ(τ)

by Theorem 3.2 in Ref. [15] equation (6.1) with zero initial condition has a unique

solution u ∈MH2p,θ(τ) and

‖M−1u‖H2p,θ(τ) ≤ N(‖Mf‖Lp,θ(τ) + ‖g‖H1

p,θ(τ)),

where N = N(p, θ, δ0, δ1).

We will also use Theorem 4.7 in Ref. [16], which implies that if u is a solution

of (6.1) of class MH2p,θ(τ) with zero initial condition and f ∈ M−1Lp,θ(τ), and

g ∈ H1p,θ(τ) and if there are numbers T ∈ (0,∞) and β such that

2/p < β ≤ 1, τ ≤ T,

then for almost any ω the function ut(x) is continuous in (t, x) (that is, has a

continuous modification) and

E supt≤τ

supx>0

|xβ−1+θ/put(x)|p ≤ NT βp/2(‖M−1u‖H2p,θ(τ) + ‖Mf‖Lp,θ(τ) + ‖g‖H1

p,θ(τ)),

where N = N(p, θ, β, δ0).

Everywhere below we take

p > 2.

Theorem 6.1. Let T ∈ (0,∞), c > 0, α ∈ (0, 1), θ > 0, µ be some constants such

that α0(c√δ1) > α,

θ0 < θ < p, µ < p(1 + 2 log2 γ(c)) − 2 = θ0 − 2 + 2p(1 − α) log2 γ(c),

where

γ(c) = γ(c, 1, 1), θ0 = p(1 + 2α log2 γ(c)) (> 0).

Let f ∈ M−1Lp,p−1(T ), g ∈ H1p,p−1(T ), and let u ∈ MH2

p,p−1(T ) be a solution of

(6.1) with zero initial condition.

Finally, assume that ft(x) = gt(x) = 0 for x ≥ 1 and f ∈ M−1Lp,µ(T ), g ∈H1p,µ(T ). Then there exist stopping times τn ↑ T , defined independently of f and g

such that, for each n, u ∈MH2p,θ(τn) and

‖M−1u‖pH2

p,θ(τn)≤ n(‖Mf‖p

Lp,µ(τn) + ‖g‖pH1

p,µ(τn)) (6.2)

Here is the result about the continuity of ut(x) we were talking about in the

introduction.

Remark 6.2. By Theorem 4.7 in Ref. [16] and Theorem 6.1 if we have a number

β ∈ (2/p, 1], then there exists a sequence of stopping times τn ↑ T such that

E supt≤τn

supx>0

|x−εut(x)|p <∞,



where ε = 1 − β − θ/p. Due to the freedom of choosing α, β, and θ, the number ε

can be made as close from the right as we wish to

1 − limα→α0(c

√δ1)

(2 + θ0)/p = −2/p− 2α0(c√δ1) log2 γ(c).

If we allow arbitrary p, then the rate of convergence of ut(x) to zero as x ↓ 0 is

almost

xε0 , ε0 = −2α0(c√δ1) log2 γ(c) > 0,

which is the same as we obtained for vm(t, x) (see (5.5)). Hence, the presence of f

and g does not spoil the situation too much.

It is also worth noting that f and g still may blow up near zero even if p is large.

When p is large we can take (µ − 1)/p as close to 1 + 2 log2 γ(c) as we wish and

then the integral∫ 1

0

xµ−1|xft(x)|p dx

converges if |ft(x)| blows up near x = 0 slightly slower than x−2(1+log2 γ(c)). Here

log2 γ(c)→0 as c→∞ and one can allow |ft(x)| to blow up almost as x−2.

However, when f and g become more irregular near 0, the rate with which the

solution goes to zero at 0 deteriorates. In connection with this it is interesting to

investigate what happens with ε0 as δ1 ↓ 0. Take an m so large that α0(m) > 1/2

and set c = mδ−1/21 − 1/2. Then for δ1 small we have α0(c

√δ1) > 1/2 and

ε0 ≥ − log2[1 − P ( mins≤1/2

ws ≤ −c− 1/√

2, maxs≤1/2

ws ≤ 1 − 1/√

2)),

ε0 ln 2 ≥ − ln[1 − P ( mins≤1/2

ws ≤ −c− 1/√

2, maxs≤1/2

ws ≤ 1 − 1/√

2))

∼ P ( mins≤1/2

wt ≤ −c− 1/√

2, maxs≤1/2

ws ≤ 1 − 1/√

2)

= P ( mins≤1/2

wt ≤ −c− 1/√

2) − P ( mins≤1/2

wt ≤ −c− 1/√

2, maxs≤1/2

ws ≥ 1 − 1/√

2)

and

P ( mins≤1/2

wt ≤ −c− 1/√

2, maxs≤1/2

ws ≥ 1 − 1/√

2) ≤ 2P ( mins≤1/2

wt ≤ −c− 1),

so that

P ( mins≤1/2

wt ≤ −c− 1/√

2, maxs≤1/2

ws ≤ 1 − 1/√

2)

≥ P ( mins≤1/2

wt ≤ −c− 1/√

2) − 2P ( mins≤1/2

wt ≤ −c− 1).


332 N. V. Krylov

Next, as a→∞

P ( mins≤1/2

ws ≤ −a) = P (|w1/2| ≥ a) =2√π

∫ ∞

a

e−x2

dx ∼ 1√πa−1e−a

2

and

limδ1↓0

[P ( mins≤1/2

wt ≤ −c− 1/√

2)

−2P ( mins≤1/2

wt ≤ −c− 1)](c+ 1/√

2)e(c+1/√

2)2 =1√π.

Hence

limδ1↓0

[mδ−1/21 em

2/δ1ε0] ≥ 1√π ln 2

.

This result may seem unsatisfactory since the guaranteed value of ε0 is extremely

small when δ1 is small. However, recall that by Remark 4.2 in Ref. [14] the best

possible rate with which the solutions go to zero for small δ1 is less than

(1 + κ)(2πδ1)−1/2e−1/(2δ1),

where κ > 0 is any number.

To prove Theorem 6.1, first we prove the following.

Lemma 6.3. Assume that, for an m = 0, 1, 2, ... we have ft(x) = gt(x) = 0 if

x ≤ 2−m/2. Then almost surely for all t ≤ T and x ∈ (0, 2−m/2)

|ut(x)| ≤ vm(t, x) sups≤t

|us(2−m/2)|. (6.3)

Proof. By Theorem 4.7 in Ref. [16] the function ut(x) is continuous in [0, T ] ×(0, 2−m/2) (a.s.) and therefore to prove (6.3) it suffices to prove that for each

ε ∈ (0, 2−(m+2)/2) almost surely for all t ≤ T and x ∈ D := (ε, 2−m/2 − ε)

|uεt (x)| ≤ vm(t, x) sups≤t

|uεs(2−m/2 − ε)| =: vm(t, x)ρεt , (6.4)

where uεt (x) = ut(x− ε). The function uεt satisfies (6.1) with f = g = 0 in (0, T ) ×(ε, 2−m/2 + ε) and in (0, T ) ×D. Furthermore, (a.s.) for almost any t ∈ (0, T ) we

have Dxut ∈ Lp(D) implying that the limit of uεt (x) as x ↓ ε exists. Since (a.s.) for

almost all t ∈ (0, T ) also (x − ε)−1uεt ∈ Lp(D), the limit is zero. As x ↑ 2−m/2 − ε

the situation is simpler and we see that (a.s.) for almost all t ∈ (0, T ) we have

limD3x→∂D

(uεt (x) − vm(t, x)ρεt )+ = 0.

Furthermore, (a.s.) for almost all t ∈ (0, T ) it holds that uεt ∈W 12 (D) and

∫ T

0

‖uεt‖2W 1

2 (D) dt <∞.



Combining this with Remark 5.4 we see that (a.s.) for almost all t ∈ (0, T ) we have

(uεt − vm(t, ·)ρεt )+ ∈W 12 (D), (uεt − vm(t, ·)ρεt )+ ∈

0

W 12(D) and

∫ T

0

‖(uεt − vm(t, ·)ρεt )+‖2W 1

2 (D) dt <∞.

By Theorem 2.6 we conclude that almost surely for all t ≤ T and x ∈ D

uεt (x) ≤ vm(t, x)ρεt .

By combining this with similar inequality for −uεt we obtain (6.4). The lemma is

proved.

Proof of Theorem 6.1. Clearly, we only need prove Theorem 6.1 for f and g

such that ft(x) = gt(x) = 0 for all ω, t if x is small. Then

f ∈M−1Lp,ϑ(T ), g ∈ H1p,ϑ(T ) (6.5)

for any ϑ.

According to Lemma 3.6 in Ref. [15], for each stopping time τn ≤ T , we have

u ∈MH2p,θ(τn) if u ∈MLp,θ(τn) and under this condition the left-hand side of (6.2)

is dominated by a constant N = N(θ, p, δ0, δ1) times

‖M−1u‖pLp,θ(τn) +E

∫ τn

0

∫ ∞

0

xθ−1|Ft(x)|p dxdt, (6.6)

where

Ft(x) := |xft(x)| + |gt(x)| + |xDxgt(x)|.Observe that obviously (α, γ(c) ≤ 1)

θ > θ0 > µ (6.7)

and since ft(x) = gt(x) = 0 for x ≥ 1, the integral involving Ft will increase if we

replace θ with µ. It follows that to prove the theorem, it suffices to estimate only

the lowest norm of u, that is to prove the existence of τn ↑ T such that

‖M−1u‖pLp,θ(τn) ≤ n[‖Mf‖p

Lp,µ(τn) + ‖g‖pH1

p,µ(τn)]. (6.8)

Next, take a ϑ ∈ [p − 1, p) such that ϑ > θ. For any stopping time τ ≤ T , by

Lemma 4.3 in Ref. [15] we have u ∈ MH2p,ϑ(τ) and by Theorem 3.2 of [15]

E

∫ τ

0

∫ ∞

0

xϑ−1|ut(x)/x|p dxdt ≤ N [‖Mf‖pLp,ϑ(τ) + ‖g‖p

H1p,ϑ(τ)

], (6.9)

where N = N(p, ϑ, δ0, δ1). As before on the right we can replace ϑ with µ. On the

left one can replace ϑ with θ if one restricts the domain of integration with respect

to x to x ≥ 1. Therefore (6.8) will be proved if we prove the existence of appropriate

stopping times τn such that

E

∫ τn

0

∫ 1

0

xθ−1|ut(x)/x|p dxdt ≤ n[‖Mf‖pLp,µ(τn) + ‖g‖p

H1p,µ(τn)]. (6.10)


334 N. V. Krylov

Take a nonnegative η ∈ C∞0 (R+) with support in (1, 4) such that the (1/2)-

periodic function on R

∞∑

k=−∞η(2x+k/2)

is identically equal to one. Introduce,

ηm(x) = η(2m/2x), (fmt, gmt) = (ft, gt)ηm.

Also introduce umt as solutions of class MH2p,p−1(T ) of (6.1) with zero initial

condition and fmt and gmt in place of ft and gt, respectively. Since only finitely

many fmt and gmt are not zero, we have

ut(x) =

∞∑

m=1

umt(x) = I1(t, x) + I2(t, x),

where

I1(t, x) :=

∞∑

m=1

umt(x)Ix≤2−m/2 , I2(t, x) :=

∞∑

m=1

umt(x)Ix>2−m/2 .

Estimating I2. Take a ϑ as above, set ε = (ϑ−θ)/(2p) and use Holder’s inequality

to obtain

|I2(t, x)|p ≤∞∑

m=1

2εpmupmt(x)Jp/q(x),

where

J(x) :=

∞∑

m=1

2−εqmIx>2−m/2 ≤ Nx2εq , Jp/q(x) ≤ Nxϑ−θ.

Then use (6.9) again to get

E

∫ τ

0

∫ 1

0

xθ−1|I2(t, x)/x|p dxdt

≤ N

∞∑

m=1

E

∫ τ

0

∫ ∞

0

2m(ϑ−θ)/2xϑ−1|umt(x)/x|p dxdt

≤ N∞∑

m=1

E

∫ τ

0

∫ ∞

0

2m(ϑ−θ)/2xϑ−1|Fmt(x)|p dxdt,

where

Fmt(x) = |xfmt(x)| + |gmt(x)| + |xDxgmt(x)|.Here we notice few facts, which will be also used in the future, that on the supports

of fmt(x) and gmt(x) we have x ∼ 2−m/2, 2m(ϑ−θ)/2Fmt(x) ∼ xθ−ϑFmt(x) and

Fmt(x) ≤ Ft(x)ηm(x)



where ηm(x) = ηm(x) + x2m/2|η′(2m/2x)|. Notice that the (1/2)-periodic function

∞∑

m=−∞ηpm(2y)

is bounded on R. Then we see that

E

∫ τ

0

∫ 1

0


≤ NE

∫ τ

0

∫ ∞

0

xθ−1|Ft(x)|p∞∑

m=1

ηpm(x) dxdt

≤ N [‖Mf‖pLp,θ(τ) + ‖g‖p

H1p,θ(τ)

]

for any τ ≤ T with a constant N under control. As above we can reduce θ in the

last expression to µ.

Estimating I1. Here we will see how τn appear and how we get a substantial drop

from θ to µ. We have seen above that the smaller µ is the weaker the statement

of the theorem becomes. Therefore, we may concentrate on µ so close to p(1 +

2 log2 γ(c)) − 2 from below that

2 < βp := p(1 + 2 log2 γ(c)) − µ ≤ p.

Then

2/p < β ≤ 1.

Observe that

|I1(t, x)|p ≤( ∞∑

m=1

m−q)p/q ∞∑

m=1

mp|umt(x)|pIx≤2−m/2

≤ N | log2 x|p∞∑

m=1

|umt(x)|pIx≤2−m/2 .

It follows that for any θ′ < θ

E

∫ τ

0

∫ 1

0


≤ N

∞∑

m=1

E

∫ τ

0

∫ 2−m/2

0

| log2 x|pxθ−1|umt(x)/x|p dxdt ≤ NJ(τ),

where

J(τ) :=

∞∑

m=1

E

∫ τ

0

∫ 2−m/2

0

xθ′−1|umt(x)/x|p dxdt.


336 N. V. Krylov

By Theorem 4.7 in Ref. [16] and Theorem 3.3 in Ref. [17], for any τ ≤ T

E supt≤τ

supx>0

|xεumt(x)|p ≤ NT βp/2E

∫ τ

0

∫ ∞

0

xθ1−1|Fmt(x)|p dxdt, (6.11)

where N = N(p, δ0, δ1, β) and

θ1 := p− 1, ε := β − 1 + θ1/p = β − 1/p > 0.

Therefore,

E supt≤τ

|umt(2−m/2)|p ≤ 2m(βp−p+θ1)/2E supt≤τ

supx>0

|xεumt(x)|p

≤ NE

∫ τ

0

∫ ∞

0

2m(βp−p+θ1)/2xθ1−1|Fmt(x)|p dxdt

≤ N2m(βp−p+1)/2E

∫ τ

0

∫ ∞

0

|Fmt(x)|p dxdt.

Next, observe that, by Lemma 6.3 for x ∈ [0, 2−m/2] and t ≤ T ,

|umt(x)| ≤ vm(t, x) sups≤t

|ums(2−m/2)|.

Hence

J(τ) ≤∞∑

m=1

E supt≤τ

|umt(2−m/2)|∫ τ

0

∫ 2−m/2

0

xθ′−1|vm(t, x)/x|p dxdt

≤∞∑

m=1

2m(νp−1)/(2α)E supt≤τ

|umt(2−m/2)|∫ τ

0

πt dt,

where ν is defined according to

νp = p− θ′ + 1

and πt is introduced in Theorem 5.3. So far θ′ was only restricted to θ′ < θ, so that

νp > 1. Due to the assumption that θ > θ0 one can satisfy θ0 < θ′ < θ in which

case (5.8) holds. Then in light of Theorem 5.3 one can find stopping times τn ↑ Tsuch that

∫ τn

0

πt dt ≤ n.

Then

J(τn) ≤ nN

∞∑

m=1

2m(νp−1)/(2α)2m(βp−p+1)/2E

∫ τn

0

∫ ∞

0

|Fmt(x)|p dxdt.

As is easy to see the inequalities θ′ > θ0 and

βp− p+ (νp− 1)/α < −µ



are equivalent. Hence,

E

∫ τn

0

∫ 1

0

xθ−1|I1(t, x)/x|p dxdt ≤ NJ(τn)

≤ nN

∞∑

m=1

2m(1−µ)/2E

∫ τn

0

∫ ∞

0

|Fmt(x)|p dxdt

≤ nN

∞∑

m=1

E

∫ τn

0

∫ ∞

0

xµ−1|Fmt(x)|p dxdt

≤ nNE

∫ τn

0

∫ ∞

0

xµ−1|Ft(x)|p dxdt.

By combining this estimate with the estimate of I2, noticing that the above

constants N are independent of f and g and, if necessary, renumbering the sequence

τn we come to (6.10). This proves the theorem.

Acknowledgements

Several typos in the original version of the article were kindly pointed out by

Kyeong-Hun Kim and Hongjie Dong. The author is sincerely grateful for that.

Author’s work was partially supported by NSF Grant DMS-0140405.

References

[1] K.-H. Kim, Lq(Lp) theory and Holder estimates for parabolic SPDEs, StochasticProcess. Appl. 114(2), 313–330, (2004). ISSN 0304-4149.

[2] N. V. Krylov and B. L. Rozovskiı. On the first integrals and Liouville equationsfor diffusion processes. In Stochastic differential systems (Visegrad, 1980), vol. 36,Lecture Notes in Control and Information Sci., pp. 117–125. Springer, Berlin, (1981).

[3] N. V. Krylov and B. L. Rozovsky, On the characteristics of degenerate second orderparabolic ito equations, J. Soviet Math. 32(4), 336–348, (1986).

[4] H. Kunita, On backward stochastic differential equations, Stochastics. 6(3-4), 293–313, (1981/82). ISSN 0090-9491.

[5] S. Bonaccorsi and G. Guatteri, Stochastic partial differential equations in boundeddomains with Dirichlet boundary conditions, Stoch. Stoch. Rep. 74(1-2), 349–370,(2002). ISSN 1045-1129.

[6] S. Bonaccorsi and G. Guatteri. Classical solutions for SPDEs with Dirichlet boundaryconditions. In Seminar on Stochastic Analysis, Random Fields and Applications, III(Ascona, 1999), vol. 52, Progr. Probab., pp. 33–44. Birkhauser, Basel, (2002).

[7] G. Da Prato and L. Tubaro, Fully nonlinear stochastic partial differential equations,SIAM J. Math. Anal. 27(1), 40–55, (1996). ISSN 0036-1410.

[8] L. Tubaro, Some results on stochastic partial differential equations by the stochasticcharacteristics method, Stochastic Anal. Appl. 6(2), 217–230, (1988). ISSN 0736-2994.

[9] K.-H. Kim, On stochastic partial differential equations with variable coefficients inC1 domains, Stochastic Process. Appl. 112(2), 261–283, (2004). ISSN 0304-4149.


338 N. V. Krylov

[10] N. Krylov and B. Rosovskii, Stochastic evolution equations, J.Soviet Mathematics.16, 1233–1277, (1981).

[11] L. C. Evans, Partial differential equations. vol. 19, Graduate Studies in Mathematics,(American Mathematical Society, Providence, RI, 1998). ISBN 0-8218-0772-2.

[12] O. A. Ladyzhenskaya and N. N. Ural′tseva, Linear and quasilinear elliptic equa-tions. Translated from the Russian by Scripta Technica, Inc. Translation editor: LeonEhrenpreis, (Academic Press, New York, 1968).

[13] N. V. Krylov, Introduction to the theory of diffusion processes. vol. 142, Transla-tions of Mathematical Monographs, (American Mathematical Society, Providence,RI, 1995). ISBN 0-8218-4600-0. Translated from the Russian manuscript by ValimKhidekel and Gennady Pasechnik.

[14] N. V. Krylov, One more square root law for Brownian motion and its application toSPDEs, Probab. Theory Related Fields. 127(4), 496–512, (2003). ISSN 0178-8051.

[15] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constantcoefficients on a half line, SIAM J. Math. Anal. 30(2), 298–325 (electronic), (1999).ISSN 0036-1410.

[16] N. V. Krylov, Some properties of traces for stochastic and deterministic parabolicweighted Sobolev spaces, J. Funct. Anal. 183(1), 1–41, (2001). ISSN 0022-1236.

[17] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constantcoefficients in a half space, SIAM J. Math. Anal. 31(1), 19–33 (electronic), (1999).ISSN 0036-1410.


Chapter 13

On Delay Estimation and Testing for Diffusion Type Processes

Yury A. Kutoyants

Laboratoire de Statistique et Processus, Facult des Sciences, Universit du MaineAvenue Olivier Messiaen 72085 Le Mans CEDEX 9

[email protected]

We present a review of results concerning one example of parameter estimationproblem, which is the same time regular and singular depending on the chosenasymptotics. The observed process satisfies the linear stochastic differential equa-tion with the delayed trend coefficient and we are interested in the estimation ofthis delay. The trend coefficient depends on the unknown parameter (delay) assmooth as the Wiener process depends on time, i.e., we have a continuous butnot differentiable w.r.t. parameter statistical model. We study the asymptoticbehavior of the estimators and tests in two asymptotics: the first one correspondsto the small noise perturbation, when the diffusion coefficient tends to zero andthe second is large samples limit, when the time of observation tends to infinity.We show that the first asymptotics corresponds well to regular statistical experi-ments situation and the second one is typical for non regular (discontinuous type)statistical models.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3392 Estimation (small noise asymptotics) . . . . . . . . . . . . . . . . . . . . . . . . . . 343

2.1 Asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3452.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

3 Hypotheses Testing (small noise asymptotics) . . . . . . . . . . . . . . . . . . . . . . 3484 Estimation (large samples asymptotics) . . . . . . . . . . . . . . . . . . . . . . . . . 349

5 Hypotheses Testing (large samples asymptotics) . . . . . . . . . . . . . . . . . . . . 3526 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

1. Introduction

We consider several statistical problems concerning a one dimensional parameter ϑ

(called delay parameter) of the stochastic differential equation by the continuous

time observations X = Xt, 0 ≤ t ≤ T of the solution of stochastic differential

equation

dXt = S(Xt−ϑ) dt+ σ(Xt) dWt, Xs, s ≤ 0, 0 ≤ t ≤ T.

339


340 Yu. A. Kutoyants

Here S (x) and σ (x)2

are trend and diffusion coefficients, Wt, 0 ≤ t ≤ T is stan-

dard Wiener process and Xs are independent of the Wiener process initial values.

We are interested in asymptotic estimation and testing problems in two different

asymptotics. The first one corresponds to the dynamical system with small noise

(σ → 0), also called perturbed dynamical system and the second asymptotics is

typical for classical statistics large samples asymptotics (T → ∞). The first type of

asymptotics we realize on the linear model

dXt = −γ Xt−ϑ dt+ σ dWt, Xs = xs,−β ≤ s ≤ 0, 0 ≤ t ≤ T, (1.1)

and the second asymptotics is studied for the linear ergodic model (T → ∞)

dXt = −γ Xt−ϑ dt+ σ dWt, Xs = xs,−β ≤ s ≤ 0, 0 ≤ t ≤ T.

Therefore we have exactly the same statistical model in both cases. The unknown

parameter (delay) ϑ ∈ (α, β) = Θ, where 0 < α < β, γ > 0, and xs,−β ≤ s ≤ 0 is a

deterministic function (initial values). Note that this equation has a unique strong

solution for all γ, ϑ and σ ( [1], Theorem 4.6).

We study the properties of the maximum likelihood ϑσ,T and Bayesian ϑσ,T(for quadratic loss function and positive continuous prior density p (·)) estimators

defined by the usual equations

supϑ∈Θ

L(ϑ,XT

)= L

(ϑσ,T , X

T)

and

ϑσ,T =

∫

Θ

ϑp (ϑ|X) dϑ =

∫Θϑp (ϑ)L

(ϑ,XT

)dϑ∫

Θp (ϑ)L (ϑ,XT ) dϑ

where the log-likelihood ratio is ( [1], Theorem 7.7)

lnL(ϑ,XT

)= − γ

σ2

∫ T

0

Xt−ϑ dXt −γ2

2σ2

∫ T

0

X2t−ϑ dt.

The properties of the estimators are studied as σ → 0 and T → ∞ separately.

We present here a review of some results concerning delay parameter estimation

obtained in the works [2, 3] and [4–9]. The further results on parameter estimation

for linear stochastic differential equations with delay (in large samples asymptotics)

can be found in the works by Gushchin and Kuchler [10, 11]. The nonparametric

estimation problems in the case of integral-type drift are treated by Reiss [12, 13].

What indeed is surprising with these problems is the possibility to have regular

and singular estimation and testing problems for the same model. The first limit

(σ → 0) provides us statistically regular estimation and testing problems and the

large samples asymptotics (T → ∞) will be similar to the problems of change-point

type.

Let us remind the difference between regular and singular statistical problems.

We recall the well-known facts from classical estimation theory concerning regularity

conditions and properties of estimators [6]. Let Xn = X1, . . . , Xn be i.i.d. r.v.’s


Delay Estimation for Diffusion Processes 341

and Xj has density function f (ϑ, x), where ϑ ∈ Θ = (α, β). We denote by P(n)ϑ the

distribution of Xn. If we have to estimate ϑ ∈ Θ by observations Xn and describe

the properties of estimators as n → ∞, then there are at least two situations:

regular and non regular.

In regular situation the function√f (ϑ, x) is supposed to be differentiable w.r.t.

ϑ in L2-sense and the Fisher information

I (ϑ) =

∫ ∞

−∞

(∂f (ϑ, x)

∂ϑ

)2

f (ϑ, x)−1

dx

is positive continuous function of ϑ. Then the corresponding family of measuresP

(n)ϑ , ϑ ∈ Θ

is locally asymptotically normal (LAN), i.e., the normalized likeli-

hood ratio admits the representation

Zn (u) =dP

(n)ϑ+ u√

n

dP(n)ϑ

(Xn) = exp

u∆n (ϑ) − u2

2I (ϑ) + rn

,

where ∆n (ϑ) ⇒ ∆ (ϑ) ∼ N (0, I (ϑ)) and rn → 0 (see Ref. [14], [15]). Hence

Zn (u) =⇒ Z (u) = exp

u∆ (ϑ) − u2

2I (ϑ)

.

The notion of LAN is equivalent to the notion of regular statistical experiment.

If the family of measures of any statistical model is LAN, then we have immediately

the minimax lower bound on the risk of all estimators (Hajek-Le Cam bound). For

the quadratic loss function it is an asymptotically correct version of Rao-Cramer

bound: for all estimators ϑn and ϑ0 ∈ Θ

limδ→0

limn→∞

sup|ϑ−ϑ0|<δ

n Eϑ

(ϑn − ϑ

)2 ≥ I (ϑ0)−1. (1.2)

In this regular case (under some additional conditions) the maximum likelihood es-

timator (MLE) ϑn and the wide class of Bayesian estimators (BE) ϑn are consistent,

asymptotically normal√n(ϑn − ϑ

)=⇒ u,

√n(ϑn − ϑ

)=⇒ u (1.3)

where

supuZ (u) = Z (u) , u =

∆ (ϑ)

I (ϑ)∼ N

(0, I (ϑ)

−1),

and

u =

∫uZ (u) du∫Z (u) du

=∆ (ϑ)

I (ϑ)∼ N

(0, I (ϑ)

−1).

Moreover both are asymptotically efficient in the sense of the bound (1.2) (see

Ref. [15]), i.e., for the MLE we have

limδ→0

limn→∞

sup|ϑ−ϑ0|<δ

n Eϑ

(ϑn − ϑ

)2

= I (ϑ0)−1



for all ϑ ∈ Θ.

In non regular cases the situations are different. Note that there are many types

of non regularity (see, for example, Ref. [15], Chapters 5 and 6). Here we remind

one case only. Suppose that we have the shift parameter f (ϑ, x) = f (x− ϑ), where

the function f (·) has a jump at point x∗, i.e., f+ = f (x∗+), f− = f (x∗−) and

f+ − f− 6= 0. Then the normalized likelihood ratio (for u ≥ 0)

Zn (u) =dP

(n)ϑ+ u

n

dP(n)ϑ

(Xn) =⇒ Z (u) = exp

lnf−f+

π+ (u) − u (f− − f+)

where π+ (u) , u ≥ 0 is Poisson process of intensity f+. For u ≤ 0 we have lnZ (u) =

ln f+f−

π− (−u) − u (f− − f+), where π− (u) , u ≥ 0 is Poisson process of intensity

f−. The MLE and a wide class of Bayesian estimators are consistent, their limit

distributions are different

n(ϑn − ϑ

)=⇒ u, n

(ϑn − ϑ

)=⇒ u (1.4)

where the random variables u and u are defined by the relations

Z (u) = supuZ (u) , u =

∫uZ (u) du∫Z (u) du

and we have the convergence of their moments. The minimax lower bound for all

estimators ϑn and ϑ0 ∈ Θ is

limδ→0

limn→∞

sup|ϑ−ϑ0|<δ

n2Eϑ

(ϑn − ϑ

)2 ≥ Eϑ0 u2. (1.5)

Moreover, asymptotically efficient are Bayesian estimators only because we have

the strict inequality Eu2 > Eu2 (see Ref. [15], Chapter 5 for details).

Having the problem of delay parameter estimation by the observations (1.1)

we can ask the first question: to what statistical situation corresponds it, regular

or some non regular? Surprising, the answer depends on the type of the limit we

have and is the following: if we study the properties of estimators as σ → 0, then

the problem of estimation is regular (like (1.2), (1.3)) and if we are interested by

large samples properties (T → ∞) then the problem of parameter estimation is non

regular in statistical sense (like (1.4),(1.5)).

The particularity of this problem can be seen from the integral representation

of the trend coefficient

Xt−ϑ = X0 − γ

∫ t−ϑ

0

Xs−ϑ ds+ σWt−ϑ.

Hence Xt−ϑ it is as smooth w.r.t. ϑ as Wiener process w.r.t. time, i.e., we have

trend coefficient continuous but not differentiable w.r.t. ϑ for all t ∈ (0, T ].



2. Estimation (small noise asymptotics)

Let us fix the time of observation T and consider the problem of delay estimation

by observations

dXt = −γ Xt−ϑ dt+ σ dWt, Xs = xs,−β ≤ s ≤ 0, 0 ≤ t ≤ T (2.1)

where the diffusion coefficient σ2 is supposed to be small, i.e. we study the properties

of estimators ϑσ,T = ϑσ and ϑσ,T = ϑσ as T is fixed and σ → 0. The model (2.1)

can be considered as a small perturbation of the deterministic linear system

dxtdt

= −γ xt−ϑ, xs,−β < s ≤ 0, 0 ≤ t ≤ T. (2.2)

by white Gaussian noise of intensity σ2. The behavior of such stochastic systems

(without delays) statistical problems for them are well studied (see, e.g., [16], [5]).

Remind that if the observed process is

dXt = S (ϑ,Xt) dt+ σ dWt, X0 = x0, 0 ≤ t ≤ T (2.3)

where S (ϑ, x) is a smooth function, then the corresponding family of measures is

LAN. The lower (Hajek-Le Cam’s) bound is

limδ→0

limσ→0

sup|ϑ−ϑ0|<δ

Eϑ

(ϑσ − ϑ

σ

)2

≥ I (ϑ0)−1, I (ϑ) =

∫ T

0

Sϑ (ϑ, xt)2

dt. (2.4)

The behavior of estimators is entirely similar to that of the regular case

(1.2),(1.3), i.e., the MLE ϑσ and Bayesian estimators ϑσ are consistent, asymp-

totically normal

ϑσ − ϑ

σ=⇒ N

(0, I (ϑ)

−1),

ϑσ − ϑ

σ=⇒ N

(0, I (ϑ)

−1)

(2.5)

and asymptotically efficient in the sense of the bound (2.4).

If the trend coefficient is a discontinuous function of ϑ, say,

dXt =[S1 (Xt) 1t<ϑ + S2 (Xt) 1t≥ϑ

]dt+ σdWt, X0 = x0, 0 ≤ t ≤ T

where S2 (xt) − S1 (xt) = Vt 6= 0 and S1 (x) , S2 (x) are smooth functions of x, then

the situation changes. The normalized likelihood ratio

Zσ (u) =L(ϑ+ uσ2

V 2ϑ, Xσ

)

L (ϑ,Xσ)=⇒ Z (u) = exp

W (u) − |u|

2

, (2.6)

where W (·) is two-sided Wiener process. Let us introduce two random variables u

and u by the equations

Z (u) = supu∈R

Z(u), u =

∫Ru Z(u) du∫

RZ(u) du

. (2.7)



The lower bound on the risk of all estimators is

limδ→0

limσ→0

sup|ϑ−ϑ0|<δ

σ−4 V 4ϑ0

E(ϑσ − ϑ

)2 ≥ Eu2. (2.8)

The MLE ϑσ and BE ϑσ are consistent, have the following limit distributions

V 2ϑ

σ2

(ϑσ − ϑ

)=⇒ u,

V 2ϑ

σ2

(ϑσ − ϑ

)=⇒ u, (2.9)

and BE are asymptotically efficient in the sense of this bound (see Ref. [5], Section

5.1).

Note, that the random process Z (u) appeared as well in the change point prob-

lem for the model of signal in white Gaussian noise

dXt = S (t− ϑ) dt+ σ dWt, X0 = 0, 0 ≤ t ≤ T

where S (t) has a jump S (t∗+) − S (t∗−) 6= 0. In the asymptoics of small noise

(σ → 0) it was shown that the bound like (2.8) is valid and the estimators have

behavior similar (2.9) (see Ref. [15], Section 7.2). The values Eu2 and Eu2 were

calculated by several authors. Terent’ev [17]showed that Eu2 = 26 (see also Ref.

[15], Section 7.3). Ibragimov and Khasminski [15] obtained the first approximation

of the value Eu2 by Monte-Carlo simulation. Then Golubev [18] found an integral

representation of Eu2, which provided by numerical calculation the value Eu2 with

higher precision. Finally, Rubin and Song [19] showed that

Eu2 = 16 ζ (3) ,

where ζ (s) =∑∞

n=1 n−s is Riemann’s zeta function. As 16 ζ (3) = 19, 276± 0, 006

we see that Eu2 > Eu2 and the MLE is not asymptotically efficient.

In our case of observations (2.1) the derivative w.r.t. ϑ of the trend coefficient

−γXt−ϑ does not exist. Nevertheless, the family of measures is LAN, i.e.,

Zσ (u) =L (ϑ+ uσ,Xσ)

L (ϑ,Xσ)= exp

u∆σ (ϑ) − u2

2I (ϑ) + rσ

,

where ∆σ (ϑ) ∼ N (0, I (ϑ)) and rσ → 0 and the following quantity plays the role of

Fisher information:

I (ϑ) = γ2

∫ T

0

x2t−2ϑ dt

where we suppose for simplicity that the initial values xs, s ≤ 0 are given for s ∈[−2β, 0].

As the family is LAN we have the following minimax lower bound on the mean

square error of all estimators ϑσ (like (1.2),(2.4)).

limδ→0

limσ→0

sup|ϑ−ϑ0|<δ

E

(ϑσ − ϑ

σ

)2

≥ I (ϑ0)−1. (2.10)



Comparison with (1.2), (2.6) shows that I (ϑ) here indeed plays the role of Fisher

information. Then we define the asymptotically efficient estimators as estimators

for which we have equality in this inequality for all ϑ0 ∈ Θ.

Theorem 2.1. The MLE ϑσ and BE ϑσ are consistent, asymptotically normal

ϑσ − ϑ

σ=⇒ N

(0, I (ϑ)

−1),

ϑσ − ϑ

σ=⇒ N

(0, I (ϑ)

−1), (2.11)

we have the convergence of all moments of these estimators and both estimators

are asymptotically efficient.

Proof can be found in Ref. [4], (see also Ref. [5], Section 2.4). The proofs of this

theorem and the Theorems 4 and 5 below consist in the verification of the conditions

of two general theorems by Ibragimov and Khasminski [15]. These conditions are

given in terms of the normalized likelihood ratios Zσ (·) and ZT (·) respectively.

Comparison of (2.6),(2.7) with (2.4),(2.5) shows that we have the regular prob-

lem of parameter estimation.

2.1. Asymptotic expansion

The asymptotic normality (2.6) can be proved in probability too, i.e., the estimators

can be written as

ϑσ = ϑ+ξ1√I (ϑ)

σ + o (σ) , ϑσ = ϑ+ξ1√I (ϑ)

σ + o (σ) ,

where σ−1o (σ) → 0 in probability and the r.v.

ξ1 = − γ√I (ϑ)

∫ T

0

xt−2ϑ dWt ∼ N (0, 1)

is defined on the same probability space. The proof of these representations can be

obtained using the method developed in Ref. [5], Section 3.1.

It is interesting to see the next after Gaussian terms of these expansions by the

powers of σ. In regular problems of parameter estimation by observations of the

process (2.3) with sufficiently smooth w.r.t. ϑ and x trend coefficient S (ϑ, x) such

expansion for the MLE has the form


σ +(a1ξ

21 + a2ξ1ξ2 + a3ξ1ξ3 + a4ξ4

)σ2 + o

(σ2),

where ξ1, ξ2, ξ3 are Gaussian r.v.’s and ξ4 is an Ito integral of a Gaussian process.

We see that the next after Gaussian term have to be of order σ2 (see Ref. [5], Section

3.1).

The following theorem shows that this term in the case of the model (2.3) is of

the order σ3/2 only.



Theorem 2.2. The MLE ϑσ admits the representation


σ − γ√T

2 I (ϑ)34

ζσ sgn (ξ1)√|ξ1|

σ3/2 + o(σ3/2

), (2.12)

where ζσ ∼ N (0, 1) and is asymptotically independent of ξ1 ∼ N (0, 1).

Proof. The proof can be found in Ref. [6]. It is based on the expansion of the

likelihood ratio by the powers of σ

lnZσ (u) = lnL(ϑ+ uσ,XT

)

L (ϑ,XT )

= −γσ

∫ T

0

[Xt−ϑ−uσ −Xt−ϑ] dWt −γ2

2σ2

∫ T

0

[Xt−ϑ−uσ −Xt−ϑ]2

dt

= u√

I (ϑ) ξ1 −u2

2I (ϑ) − γ

√|u|T ζσ σ

1/2 +O (σ) ,

where

ζσ =

∫ τ

0

η (t, u, σ) dWt, η (t, u, σ) =Wt−ϑ−uσ −Wt−ϑ√

T |u|σand τ is a stopping time

τ = inf

t :

∫ t

0

η (s, u, σ)2

ds ≥ 1

.

For construction of the r.v. ζσ we define η (s, u, σ) = 1 for s ∈ [T, T + 1] and we

introduce an independent Wiener process Ws for s ∈ [T, T + 1] (see Ref. [6], proof

of the theorem 1.19). Note that in Ref. [7] we used in the expansion the random

variable

ζ =

∫ T

0

η (t, u, σ) dWt,

which is asymptotically normal, but using the convergence∫ T

0

η (t, u, σ)2

dt −→ 1

proved there it is easy to see that ζ − ζσ → 0 in probability.

The similar expansion can be found for Bayesian estimator as well, but the final

expression is more cumbersome.

2.2. Generalizations

There are several direct generalizations of Theorem 2.1. We can call the model (2.3)

as shift-delay and consider as well the problem of scale-delay estimation for the

model of observations

dXt = −γ Xϑt dt+ σ dWt, X0 = x0, 0 ≤ t ≤ T,



where ϑ ∈ (0, 1). The study is quite close to (2.3) and yields the same properties of

MLE and BE with the Fisher information

I (ϑ) = γ2

∫ T

0

t2 x2ϑ2t dt,

i.e., these estimators are consistent, asymptotically normal (2.7) and asymptotically

efficient. The proof can be found in Ref. [4], [5], Section 2.4. The asymptotic

expansion like (2.8) can be obtained as well.

Another generalization can be done for nonlinear system

dXt = S (Xt−ϑ, t) dt+ σ b (Xt, t) dWt, Xs = xs,−β < s ≤ 0, 0 ≤ t ≤ T

where S (·) is a smooth function, then we have for the same estimators the asymp-

totic normality (2.7) with the Fisher information

I (ϑ) =

∫ T

0

S (xt−2ϑ, t− ϑ)2S

′(xt−ϑ, t)

2

b (xt, t)2 dt.

Here xt is solution of the equation

dxtdt

= S (xt−ϑ) , xs,−2β < s ≤ 0, 0 ≤ t ≤ T

and S′(x, t) = ∂

∂xS (x, t). The proof can be found in Ref. [2] (see also Ref. [5]).

It is interesting to see what happens if we have multiple delays in the linear

system

dXt =

k∑

j=1

λj Xt−τj dt+ σ dWt, Xs = xs, −β < s ≤ 0, 0 ≤ t ≤ T,

where ϑϑϑ = (λ1, . . . , λk , τ1, . . . , τk) is unknown parameter.

It is shown that the MLE ϑϑϑT is consistent, asymptotically normal and asymp-

totically efficient [20].

Let us consider the linear stochastic differential equation with integral-type

delay

dXt =

(∫ δ

0

Xt−s µ (ds)

)dt+ σ dWt, 0 ≤ t ≤ T. (2.13)

We suppose that the deterministic initial values Xs = xs, −β < s ≤ 0 are given.

Here the measure µ (·) is unknown. Then we can study the nonparametric estima-

tion of the function

f (t) =

∫ δ

0

xt−s µ (ds) , 0 ≤ t ≤ T



where xt = xt (µ) is solution of the equation (2.13) with σ = 0. We show that the

kernel-type estimator

fσ (t) =1

ψσ

∫ T

0

K

(τ − t

ψσ

)dXt

with ψσ = σ2/3 is uniformly consistent

limσ→0

supµ∈Θδ(L)

supa≤t≤b

Eµ

(fσ (t) − f (t)

)2

= 0

and asymptotically normal (see Ref. [9] for details). Here 0 < a < b < T and

Θδ (L) = µ : supp (µ) ⊂ [0, δ] ‖µ‖ ≤ L .

Using this nonparametric estimator we can estimate the measure µ (·) in the case

(2.4), i.e., when µ =∑k

i=1 λiδτ−i. Note that the minimum distance estimator of

the parameter ϑϑϑ = (λ1, . . . , λk , τ1, . . . , τk) is consistent and asymptotically normal

(see Refs. [8, 9]).

3. Hypotheses Testing (small noise asymptotics)

The observed process is always

dXt = −γ Xt−ϑ dt+ σ dWt, Xs = xs, s ≤ 0 0 ≤ t ≤ T

and we have to test the following two hypotheses

H0 : ϑ = 0, (no delay)

H1 : ϑ > 0.

Therefor the observed process under hypothesis H0 is Ornstein-Uhlenbeck. The

contiguous alternatives correspond to ϑ = uσ and we can rewrite the hypotheses

testing problem as :

H0 : u = 0, (no delay)

H1 : u > 0,

Let us fix some ε ∈ (0, 1) and denote by Kε the class of tests of asymptotic level

1− ε. The direct calculation shows that the limit (σ → 0) of the power function of

Neyman-Pearson test ϕσ ∈ Kε is

βNP (u) = Pζ > zε − u I

1/20

,

where ζ ∼ N (0, 1), zε is 1−ε quantil of N (0, 1) and the Fisher information I0 = I (0)

is

I (0) =x2

0 γ3

2

(1 − e−2γT

).



We call a test ϕσ ∈ Kε locally asymptotically uniformly most powerful if its

power function βσ (u) = Euϕ

σ (Xσ) satisfies the relation: for any test ϕσ ∈ Kε and

any K > 0

limσ→0

inf0≤u≤K

[βσ (u) − βσ (u)

]≥ 0.

Here βσ (u) = Euϕσ (Xσ).

Let us introduce three tests. The first one is score-function test

ϕ∗σ (Xσ) = χ∆σ>zε

,

where

∆σ = − γ

σ√

I0

∫ T

0

xt [dXt + γXt dt] .

Note that under H0 the statistic ∆σ ∼ N (0, 1).

The second test is the likelihood ratio test

ϕσ (Xσ) = χδσ>bε, δσ (Xσ) = sup

ϑ>0L (ϑ,Xσ) ,

where bε = expz2ε/2I0

.

The third is Wald’s test based on the MLE ϑσ :

ϕσ (Xσ) = χδσ>zεI

−1/20

, δσ (Xσ) =ϑσσ.

Theorem 3.1. The tests ϕ∗σ , ϕσ, ϕσ belong to Kε and are locally asymptotically

uniformly most powerful.

Proof. The proof follows from the local asymptotical normality of the family

of measures at the point ϑ = 0 and the standard arguments applied to regular

statistical experiments (see, e.g. [21]). It consists in the verification that the power

functions of these tests converge uniformly on compacts [0,K] to the limit power

function βNP (u) of the Neyman-Pearson test.

4. Estimation (large samples asymptotics)

Let us consider the same linear stochastic differential equation

dXt = −γ Xt−ϑ dt+ σ dWt, Xs = xs, −β < s ≤ 0, 0 ≤ t ≤ T

but now we study the properties of MLE ϑσ,T = ϑT and BE ϑσ,T = ϑT of the

delay ϑ in the asymptotics of large samples, i.e., as T → ∞. Suppose that γ > 0,

ϑ ∈(

0, π2γ

)= Θ, (β = π/2γ), then the process Xt has ergodic properties [22].

Remind the properties of parameter estimators for ergodic diffusion processes

dXt = S (ϑ,Xt) dt+ dWt, X0, 0 ≤ t ≤ T. (4.1)



If the function S (ϑ, x) is smooth w.r.t. ϑ, then the family of measures is LAN and

we have the lower bound on the risks of all estimators :

limδ→0

limT→∞

sup|ϑ−ϑ0|<δ

T Eϑ

(ϑT − ϑ

)2 ≥ I (ϑ0)−1, I (ϑ) =

∫S (ϑ, x)

2f (ϑ, x) dx,

where f (ϑ, x) is the density of invariant law. The properties of estimators are typical

for regular experiments, i.e., the MLE ϑT and BE ϑT are consistent, asymptotically

normal√T(ϑT − ϑ

)=⇒ N

(0, I (ϑ)

−1),

√T(ϑT − ϑ

)=⇒ N

(0, I (ϑ)

−1),

and asymptotically efficient (see Ref. [6]).

The properties of estimators change if the trend coefficient is not a smooth

function. For example, if S (ϑ, x) = S(x− ϑ), where S (x) has a finite jump at

some point x∗, i.e.; S (x∗+) − S (x∗ − +) 6= 0, then the rate of convergence of

estimators is T and not√T . To illustrate this let us consider a simple switching

ergodic diffusion process

dXt = −sgn (Xt − ϑ) dt+ dWt, X0, 0 ≤ t ≤ T. (4.2)

To describe the properties of estimators we reintroduce the random process (2.6)

Z(u) = exp

W (u) − 1

2|u|, u ∈ R,

where W (·) is two-sided Wiener process and corresponding two random variables

u and u defined by the equations (2.7).

Note that the normalized likelihood ratio

ZT (u) =L(ϑ+ u

4T , XT)

L (ϑ,XT )=⇒ Z (u) (4.3)

and this (together with some other estimates on the likelihood ratio) provides us

the lower bound like (2.8)

limδ→0

limT→∞

sup|ϑ−ϑ0|<δ

16T 2 Eϑ

(ϑT − ϑ

)2 ≥ Eu2. (4.4)

Moreover, the MLE and BE are consistent, have the following limit distributions

4T(ϑT − ϑ

)=⇒ u, 4T

(ϑT − ϑ

)=⇒ u, (4.5)

and the BE are asymptotically efficient in the sense of this bound (see [6], Section

3.4).

Let us return to delay estimation problem. First we verify that the normalized

likelihood ratio

ZT (u) =L(ϑ+ u

γ2T , XT)

L (ϑ,XT )=⇒ Z (u) = exp

W (u) − |u|

2

. (4.6)



Then we obtain the lower bound: for all estimators ϑT we have

limδ→0

limT→∞

sup|ϑ−ϑ0|<δ

γ4 T 2 Eϑ

(ϑT − ϑ

)2 ≥ Eu2. (4.7)

Therefore we can define the asymptotically efficient estimators as estimators for

which we have equality in this inequality for all ϑ0 ∈ Θ.

Theorem 4.1. The MLE ϑT and BE ϑT are consistent, have the following limits

in distribution

γ2 T(ϑT − ϑ

)=⇒ u, γ2 T

(ϑT − ϑ

)=⇒ u (4.8)

and BE are asymptotically efficient.

For the proof see Kuchler and Kutoyants [3] and Kutoyants [6], Section 3.3. Remind

how the weak convergence of the random function ZT (·) to Z (·) yields the weak

convergence (4.8) of the MLE [15]. Let us denote ϑ0 the true value of the parameter.

Then we can write

P(T )ϑ0

γ2 T

(ϑT − ϑ0

)< z

= P(T )ϑ0

ϑT < ϑ0 +

z

γ2 T

= P(T )ϑ0

sup

ϑ<ϑ0+z

γ2T

L(ϑ,XT

)> sup

ϑ≥ϑ0+z

γ2T

L(ϑ,XT

)

= P(T )ϑ0

sup

ϑ<ϑ0+z

γ2T

L(ϑ,XT

)

L (ϑ0, XT )> supϑ≥ϑ0+ z

γ2T

L(ϑ,XT

)

L (ϑ0, XT )

= P(T )ϑ0

supu<z

ZT (u) > supu≥z

ZT (u)

= P

(T )ϑ0

supu<z

ZT (u) − supu≥z

ZT (u) > 0

−→ P

supu<z

Z (u) − supu≥z

Z (u) > 0

= P u < z ,

where we changed the variable ϑ = ϑ0 + uγ2T and used the convergence (4.6).

Comparison of (4.7), (4.8) with (4.4), (4.5) shows that the delay estimation

problem is singular and similar to space change point estimation problem.

Let us suppose that the observed process

dXt = −γ Xt−τ dt+ σ dWt, Xs = xs, −2π

β< s ≤ 0, 0 ≤ t ≤ T

has two unknown parameters γ ∈ (α, β) , α > 0 and τ ∈(

0, 2πβ

)and we have to esti-

mate the both parameters simultaneously, i.e. we have two-dimensional parameter

ϑϑϑ = (γ, τ). We need the further notation.

• normalizing matrix

ϕϕϕT =

(T−1/2, 0

0 , T−1

).



• Fisher information of the component γ

I (ϑϑϑ) =

∫ ∞

0

x2t dt

where xt is fundamental solution of the deterministic equation

dxtdt

= −γ xt−τ , xs = 0, −2π

β< s < 0, x0 = 1,

• the random vectors www =(v, uγ2

)and www =

(v, uγ2

)where the random variables

u and u are the same as before and v is independent of u and u Gaussian r.v.

N(

0, I (ϑϑϑ)−1)

The lower bound is given by the following inequality: for any ϑ0 ∈ Θ = (α, β)×(0, 2π

β

)and any estimator ϑ

T

limδ→0

limT→∞

sup|ϑ−ϑ0|<δ

Eϑ

∣∣∣ϕ−1T

(ϑT− ϑ

)∣∣∣2

≥ I (ϑϑϑ)−1

+Eu2

γ4.

We call an estimator ϑ∗T

asymptotically efficient if for all ϑ0 ∈ Θ

limδ→0

limT→∞

sup|ϑ−ϑ0|<δ

Eϑ

∣∣∣ϕ−1T

(ϑ∗T− ϑ

)∣∣∣2

= I (ϑϑϑ)−1

+Eu2

γ4.

The asymptotic properties of the MLE ϑT

=(γT, τT

)and BE ϑ

T= (γT , τT )

are described in the following theorems.

Theorem 4.2. The MLE ϑT

and BE ϑT

of the parameter ϑ are consistent, have

different limit distributions

ϕ−1

T

(ϑT− ϑ

)=⇒ w, ϕ−1

T

(ϑT− ϑ

)=⇒ w,

for any p > 0 the moments Eϑ

∣∣ϕ−1

T

(ϑT− ϑ

) ∣∣p and Eϑ

∣∣ϕ−1

T

(ϑT− ϑ

) ∣∣p converge.

Moreover, the BE are asymptotically efficient.

For the proof see Ref. [6], Section 3.3.

From this theorem it follows that the parameters γ and τ are estimated with

the different rates, say,

√T (γT − γ) =⇒ v, T (τT − τ) =⇒ u

γ2.

5. Hypotheses Testing (large samples asymptotics)

The observed process is

dXt = −γ Xt−ϑ dt+ σ dWt, Xs = xs, s ≤ 0 0 ≤ t ≤ T



and we have to test the following two hypotheses

H0 : ϑ = 0, (no delay)

H1 : ϑ > 0,

The contiguous alternatives correspond to ϑ = u/γT :

H0 : u = 0, (no delay)

H1 : u > 0,

We have no asymptotically uniformly most powerful test and describe the like-

lihood ratio test only. This test is defined as follows

ϕT(XT)

= χδT>1ε, δT

(XT)

= supϑ>0

L(ϑ,XT

)

L (0, XT ).

Its properties are given in the following theorem.

Theorem 5.1. The likelihood ratio test ϕT belongs to Kε and its power function

βT (u, ϕT ) = P

ζ + max [η, ξ] > ln

1

ε− u

2

+ o (1) , (5.1)

where the random variables ζ, η and ξ are independent and

ζ ∼ N (0, u) , Fξ (x) = 1 − e−x, x ≥ 0 (5.2)

Fη (x) = Φ

(x√u

+

√u

2

)+ e−x

[1 − Φ

(x√u−

√u

2

)], (5.3)

where Φ (x) is Gaussian N (0, 1) distribution function.

Proof. Remind that the normalized likelihood ratio ZT (u) converges to the process

Z (u) (see (4.6)), hence

P0

δT(XT)>

1

ε

= P0

supu≥0

ZT (u) >1

ε

−→ P

supu≥0

Z (u) >1

ε

= P

supu≥0

[W (u) − u

2

]> ln

(1

ε

)= ε

because the random variable ξ = supu≥0

[W (u) − u

2

]has exponential distribution.

Therefore ϕT ∈ Kε. The proof of (5.1)-(5.3) can be found in Ref. [6], Section 5.2.

6. Discussion

The difference of the properties of estimators and tests in these two different limits

can be explain as follows. The properties of estimators depend strongly of the local



structure of the likelihood ratio. Let us write it as

Zσ,T (u) =L (ϑ+ ϕσ,Tu,X)

L (ϑ,X)= exp

−γ∫ T

0

[Xt−ϑ−ϕσ,Tu −Xt−ϑ

]

σdWt

−γ2

2

∫ T

0

[Xt−ϑ−ϕσ,Tu −Xt−ϑ

σ

]2dt

,

where ϕσ,T → 0.

Using equation (1.1) we obtain the representation

Xt−ϑ−ϕσ,Tu −Xt−ϑσ

= −γσ

∫ t−ϑ−ϕσ,Tu

t−ϑXs−ϑ ds+Wt−ϑ−ϕσ,Tu −Wt−ϑ.

If we put ϕσ,T = σ → 0, then

Xt−ϑ−ϕσ,Tu −Xt−ϑσ

−→ γ u xt−2ϑ

because the stochastic process Xt converges uniformly in t ∈ [0, T ] to the determin-

istic function xt = xt (ϑ), which is solution of the differential equation

dxtdt

= −γ xt−ϑ, xs, s ≤ 0.

Therefore the normalized likelihood ratio converges

Zσ (u) =⇒ Z (u) = exp

uγ

∫ T

0

xt−2ϑ dWt −u2γ2

2

∫ T

0

x2t−2ϑ dt

and we have regular estimation and testing problems. The contribution of the term

Wt−ϑ−ϕσ,Tu −Wt−ϑ in this case is asymptotically negligible.

If we fix σ and consider T → ∞ with ϕσ,T = T−1, then

Xt−ϑ− uT−Xt−ϑ

σ= −γ

σ

∫ t−ϑ− uT

t−ϑXs−ϑ ds+Wt−ϑ− u

T−Wt−ϑ

∼ −γuT

Xt−2ϑ +

√u

Tη (t, ϑ, u) ,

where Eϑ η (t, ϑ, u)2

= 1. We see that the main part of the contribution is due to

the Wiener process and as the Wiener process is not differentiable w.r.t. time, the

statistical problems became non regular.

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[16] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems. vol.260, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences], (Springer-Verlag, New York, 1984). ISBN 0-387-90858-7.Translated from the Russian by Joseph Szucs.

[17] A. S. Terent’yev, Probability distribution of a time location of an absolute maximumat the output of a sinchronized filter, Radioengineering and Electronics. 13(4), 652–657, (1968).

[18] G. K. Golubev, Computation of efficiency of maximum-likelihood estimate when ob-serving a discontinuous signal in white noise, Problems Inform. Transmission. 15(3),61–69, (1979).

[19] H. Rubin and K. S. Song, Exact computation of the asymptotic efficiency of maximumlikelihood estimators of a discontinuous signal in a Gaussian white noise, Ann. Statist.23(3), 732–739, (1995). ISSN 0090-5364.

[20] Y. A. Kutoyants, T. Mourid, and D. Bosq, Estimation parametrique d’un processus dediffusion avec retards, Ann. Inst. H. Poincare Probab. Statist. 28(1), 95–106, (1992).ISSN 0246-0203.

[21] G. G. Roussas, Contiguity of probability measures: some applications in statistics.



(Cambridge University Press, London, 1972). Cambridge Tracts in Mathematics andMathematical Physics, No. 63.

[22] U. Kuchler and B. Mensch, Langevin’s stochastic differential equation extended by atime-delayed term, Stochastics Stochastics Rep. 40(1-2), 23–42, (1992). ISSN 1045-1129.


Chapter 14

On Cauchy-Dirichlet Problem for Linear Integro-Differential

Equation in Weighted Sobolev Spaces

Remigijus Mikulevicius and Henrikas Pragarauskas∗

Department of Mathematics, University of Southern CaliforniaLos Angeles, CA 90089-2532

[email protected]

We study the Cauchy-Dirichlet problem in a smooth bounded domain for linearparabolic integro-differential equations. Sufficient conditions are derived underwhich the problem has a unique solution in weighted Sobolev classes. The resultcan be used in the regularity analysis of certain functionals arising in the theoryof Markov processes.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

2 Notation and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

3 Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

3.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

3.2 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

1. Introduction

The paper is devoted to the Cauchy-Dirichlet problem for integro-differential equa-

tions associated with d-dimensional Markov process Xs,xt , t ≥ s, defined by Ito

stochastic differential equation

dXt = σ(t,Xt) dWt + b(t,Xt) dt+

∫

|y|≤1

y q(dt, dy) +

∫

|y|>1

y p(dt, dy), (1.1)

Xs = x,

where Wt is a standard d-dimensional Wiener process, p(dt, dy) is the jump measure

of Xt with the compensator π(t,Xt, dy)dt, and q(dt, dy) = p(dt, dy)−π(t,Xt, dy)dt

is the corresponding martingale measure. A very simple example is the process Xt

∗Institute of Mathematics and Informatics, Akademijos 4, Vilnius, Lithuania, [email protected]

357


358 R. Mikulevicius and H. Pragarauskas

satisfying the equation

dXt = σ(t,Xt) dWt + b(t,Xt) dt+ dZt, (1.2)

Xs = x,

where Zt is a d-dimensional α-stable process, α ∈ (0, 2). In this case π(t, x, dy) =

dy/|y|d+α.In many problems arising in the theory of Markov processes it is important to

study the smoothness properties of the functionals

v(s, x) = E

∫ τs,x∧T

s

f(t,Xs,xt ) dt,

where τs,x is the first exit time of the process Xs,xt from a domain G ⊆ Rd. If v is

a sufficiently smooth function, it is a solution to Cauchy-Dirichlet problem

∂tu+Au+ f = 0, in (0, T ) ×G,

u(T, x) = 0, x ∈ Rd, (1.3)

u(t, x) = 0, t ∈ [0, T ], x /∈ G,

where

Au(t, x) = aij(t, x)∂2iju(t, x) + bi(t, x)∂iu(t, x)

+

∫[u(t, x+ y) − u(t, x) − ∂iu(t, x)yi1|y|≤1] π(t, x, dy),

aij(t, x) = (1/2)σ(t, x)σ∗(t, x), 1|y|≤1 is the indicator function of y ∈ Rd : |y| ≤1, and the implicit summation convention over repeated indices is assumed.

The problem of such type (including the case of nonlinear equations) was consid-

ered by a number of authors (see e.g. Refs. [1–4], and references therein) in Sobolev

and Holder spaces under certain restrictive assumptions on π(t, x, dy). In [3], only

finite number of jumps can occur outside of D in finite time, and in Ref. [2] very

restrictive assumptions on the order of jumps are imposed. In Ref. [5], general re-

sults were obtained in weighted Sobolev classes for a differential operator L. We

extend some results in Ref. [5] for the integro-differential operator Lu and show for

a bounded domain G ⊆ Rd that the assumptions on π(t, x, dy) can be considerably

relaxed. The results in Refs. [3] and [2] do not apply for the equation (1.2) which

is obviously covered by our Theorem 2.1.

The results obtained can be also used in the analysis of non-linear equations

arising in the optimal control theory of Markov processes.

The main result of the paper is presented in Section 2 and proved in Section 3.


Cauchy-Dirichlet Problem for an Integro-Differential Equation 359

2. Notation and main result

We will consider a bounded domain G in Rd of class C1u. In other words (see

Ref. [5]), there are two numbers r0,K0 ∈ (0,∞) and an increasing function ω0(ε)

on [0,∞) such that ω0(ε) ↓ 0 as ε ↓ 0 so that for each x0 ∈ ∂G, there exists a one-

to-one continuously differentiable mapping Ψ of Br0(x0) = y ∈ Rd : |y−x0| < r0onto a domain D ⊆ Rd having the following properties:

(i) D+ = Ψ (Br0(x0) ∩G) ⊆ Rd+ = x ∈ Rd : x1 > 0 and Ψ(x0) = 0;

(ii) Ψ (Br0(x0) ∩ ∂G) = D ∩ x ∈ Rd : x1 = 0;

(iii) |Ψ(x)| + |∂Ψ(x)| ≤ K0 on Br0(x0) and |Ψ−1(y1) − Ψ−1(y2)| ≤ K0|y1 − y2|for all y1, y2 ∈ D;

(iv) for x1, x2 ∈ Br0(x0), we have |∂Ψ(x1) − ∂Ψ(x2)| ≤ ω0(|x1 − x2|).Denote ρ(x) =dist(x, ∂G), x ∈ G; ρ(x) = 0, if x /∈ G. For p > 1, θ ∈ R, define

Lp,θ(G) = Lp(G; ρθ−d(x) dx) the space of all measurable functions u(x) on G such

that∫

G

|u(x)|pρθ−d(x) dx <∞.

Let Hγp,θ(G) be the space of all measurable function u(x) on G such that

u, ρ∂u, . . . ργ∂γu ∈ Lp,θ(G), where γ = 0, 1, 2.

Denote δ = θ−dp , |u|pp,G =

∫G|u(x)|p dx. We mostly are interested in the spaces

H0p,θ+p(G), H1

p,θ(G), H2p,θ−p(G) with the norms

|u|H0p,θ+p(G) = |uρ1+δ|p,G,

|u|H1p,θ(G) = |uρδ|p,G + |∂uρ1+δ|p,G,

|u|H2p,θ−p(G) = |uρ−1+δ|p,G + |∂uρδ|p,G + |∂2uρ1+δ|p,G.

Let T > 0. We introduce the following spaces Hγp,θ(T,G), γ = 0, 1, 2, of functions

u = u(t, x) with the norm

|u|Hγp,θ(T,G) =

∫ T

0

|u(t, ·)|pHγ

p,θ(G)dt

1/p

.

Let H1p,θ(T,G) be the space of all functions u ∈ H2

p,θ−p(T,G) such that

u(t, x) =

∫ t

0

f(s, x) ds, 0 ≤ t ≤ T,

in a generalized sense, where f ∈ H0p,θ+p(T,G), which means that for all ϕ ∈ C∞

0 (G),

∫u(t, x)ϕ(x) dx =

∫ t

0

∫f(s, x)ϕ(x) dxds, t ≤ T.



For u ∈ H1p,θ(T,G), we define

|u|H1p,θ(T,G) = |u|H2

p,θ−p(T,G) + |f |H0p,θ+p(T,G).

Let us introduce the operators

Lu(t, x) = aij(t, x)∂2iju(t, x) + bi(t, x)∂iu(t, x) − r(t, x)u(t, x), (t, x) ∈ [0, T ] ×G,

where a = (aij)1≤i,j≤d is symmetric and nonnegative definite, and

Iu(t, x) =

∫∇2yu(t, x)π(t, x, dy), (t, x) ∈ [0, T ] ×G,

where

u(t, x) =

u(t, x), (t, x) ∈ [0, T ] ×G,

0, 0 ≤ t ≤ T, x /∈ G,

and

∇2yu(t, x) = u(t, x+ y) − u(t, x) − ∂iu(t, x)yi1|y|≤1.

The summation convention that repeated indices indicate summation from 1 to

d is followed here as it will throughout.

We will need the following assumptions.

A1. The domain G is bounded of class C1u (see Ref. [5])

A2. There exist two positive numbers 0 < c < C such that

c|ξ|2 ≤ aijξiξj ≤ C|ξ|2

for all ξ ∈ Rd, and

lim|x−y|→0

sup0≤t≤T

|aij(t, x) − aij(t, y)| = 0.

A3. The functions bi(t, x), r(t, x) are measurable and bounded;

A4. Let p > 1, d− 1 < θ < d− 1 + p.

The following two types of assumptions will be used for the measurable family

of non-negative measures π(t, x, dy) on Rd\0.B. (The case of a dominating measure). There are measures π(t, dy), t ∈ [0, T ]

such that π(t, x, dy) ≤ π(t, dy), t ∈ [0, T ], x ∈ G, and

sup0≤t≤T

∫|y|2 ∧ 1π(t, dy) < ∞,

limε→0

sup0≤t≤T

∫

|y|≤ε|y|2π(t, dy) = 0.

C. (“general case”) There is a constant N such that∫

|y|2 ∧ 1 π(t, x, dy) ≤ N



and

limε→0

sup0≤t≤T,x∈G

∫

|y|≤ε|y|2π(t, x, dy) = 0.

We consider a linear equation

∂tu(t, x) = Lu(t, x) + Iu(t, x) + h(t, x), (t, x) ∈ [0, T ] ×G, (2.1)

u(0, x) = 0.

A function u ∈ H1p,θ(T,G) is called a solution of (2.1), if for every ϕ ∈ C∞

0 (G)

and all t ∈ [0, T ],

∫u(t, x)ϕ(x) dx =

∫ t

0

∫[Lu(s, x) + Iu(s, x) + h(s, x)]ϕ(x) dxds.

Our main results are the following two statements. The first one holds for the

case of a dominating measure.

Theorem 2.1. Assume the assumptions A1-A4 and B are satisfied. Then for each

h ∈ H0p,θ+p(T,G) there is a unique solution u ∈ H1

p,θ(T,G) of (2.1). Moreover,

there is a constant C independent of h such that

|u|H2p,θ−p(T,G) ≤ C|h|H0

p,θ+p(T,G).

The next statement holds under the assumption C for π(t, x, dy).

Theorem 2.2. Assume p > d, θ ≤ p and the assumptions A1-A4 and C hold.

Then for each h ∈ H0p,θ+p(T,G) there is a unique solution u ∈ H1

p,θ(T,G) of (2.1).

Moreover, there is a constant C independent of h such that


p,θ+p(T,G).

3. Proof of the main results

Consider a partial differential equation

∂tu(t, x) = Lu(t, x) + h(t, x), (t, x) ∈ [0, T ] ×G, (3.1)

u(0, x) = 0, x ∈ G.

According to Theorem 2.10 in Ref. [5], the following statement holds.

Theorem 3.1. Assume A1- A4 are satisfied. Then for each h ∈ H0p,p+θ(T,G) there

is a unique solution u ∈ H1p,θ(T,G) of (3.1). Moreover,

|u|H2p,θ−p(T,G) ≤ NeNT |h|H0

p,θ+p(T,G),

where N is independent of T and h.



Assume V t : H2p,θ−p(G) → H0

p,p+θ(G), t ≥ 0, is a family of linear operators such

that the following assumption holds.

L. (i) For any u ∈ H2p,θ−p(T,G), the function V u(t, x) = V tu(t, x) is a measur-

able function on GT = [0, T ] ×G

(ii) For each ε > 0 there is a constant µε such that

|V tu|H0p,θ+p(T,G) ≤ ε|u|H2

p,θ−p(G) + µε|u|H1p,θ(G),

for all u ∈ H2p,θ−p(G), t ≥ 0.

We will show that under the assumptions A1-A4 and L the equation

∂tu(t, x) = Lu(t, x) + V tu(t, x) + h(t, x), (t, x) ∈ [0, T ] ×G, (3.2)

u(0, x) = 0, x ∈ G.

has a unique solution u ∈ u ∈ H1p,θ(T,G).

According to Lemma 2.6 in Ref. [5], there is a function ψ ∈ C∞(G) such that

(i) ψ and ∂ψ are continuous on G, |∂ψ(x)| ≥ 1 on ∂G;

(ii) ψ(x) > 0 for any x ∈ G, and ψ = 0 on ∂G; for each ε > 0 the function ψ is

bounded away from zero on the set x ∈ G : ρ(x) ≥ ε;

(iii) for any integer k ≥ 0 we have

supGρk(x)|∂k+1ψ(x)| <∞. (3.3)

(iv) there are two constants 0 < N1 < N2 such that

N1ρ(x) ≤ ψ(x) ≤ N2ρ(x), x ∈ G. (3.4)

Remark 3.2. a) Since ρ and ψ are comparable ((3.4) holds), we obtain equivalent

norms of the spaces if the distance ρ is replaced by its regularized version ψ.

b) The inequality (3.4) implies that for each ε > 0 there is a constant Cε such

that

1 ≤ ε

ψ+ Cεψ. (3.5)

We will need the following estimate of H1p,θ(G) norm.

Lemma 3.3. For each ε > 0 there is a constant Cε such that for all v ∈ H2p,θ−p(G)

|v|H1p,θ(G) ≤ ε|v|H2

p,θ−p(G) + Cε|vρ1+δ|p,G.

Proof. Let v ∈ H2p,θ−p(G) and u = vψ1+δ . We have

∂iu = ∂ivψ1+δ + (1 + δ)vψδ∂iψ,

∂2iju = ∂2

ijvψ1+δ + (1 + δ)∂ivψ

δ∂jψ + (1 + δ)∂jvψδ∂iψ

+(1 + δ)δvψδ−1∂jψ∂iψ + (1 + δ)vψδ∂2ijψ.



For each ε > 0 there is a constant Cε such that |∂u|p,G ≤ ε|∂2u|p,G + Cε|u|p,G.

Therefore, using the properties of ψ, for each ε we can find a constant Cε > 0 such

that

|∂iu|p,G ≤ ε(|∂2vψ1+δ |p,G + |∂ivψδ |p,G + |vψδ−1|p,G

)

+Cε|vψ1+δ |p,G.

By (3.5), for each ε > 0 there is a constant Cε such that

|v|H1p,θ(G) ≤ C(|vψδ |p,G + |∂vψ1+δ |p,G) ≤ C(|vψδ |p,G + |∂u|p,G)

≤ ε|u|H2p,θ−p(G) + Cε|uψ1+δ |p,G,

and the statement follows.

Now we prove the existence and uniqueness of the solution to (3.2).

Proposition 3.4. Assume A1- A4 and L are satisfied. Then for each h ∈H0p,p+θ(T,G) there is a unique solution u ∈ H1

p,θ(T,G) of (3.2). Moreover,


p,θ+p(T,G),

where C is independent of h.

Proof. For τ ∈ [0.1] define the operators Aτu = Lu + τV u. Assume u ∈H1p,θ(T,G) solves the equation

∂tu(t, x) = Aτu(t, x) + h(t, x), (t, x) ∈ [0, T ] ×G, (3.6)

u(0, x) = 0, x ∈ G,

where h ∈ H0p,p+θ(T,G). We will prove that there is a constant independent of h, u

and τ such that


p,θ+p(T,G), (3.7)

Then by Theorem 3.1 , there is a constant C independent of τ, h and u such

that for all t ≤ T

|u|pH2

p,θ−p(t,G)≤ C(|h|p

H0p,θ+p(t,G)

+ |V u|pH0

p,θ+p(t,G)).

Therefore, by Lemma 3.3, there is a constant C such that for all t ≤ T,

|u|pH2

p,θ−p(t,G)≤ C(|h|p

H0p,θ+p(t,G)

+ |ρ1+δu|pp,Gt), (3.8)



where |f |pp,Gt=∫ t0

∫G |f(s, x)|pdxds. On the other hand, multiplying both sides of

the equation (3.6) by ρ1+δ we have, by Lemma 3.3, (3.8) and assumption L,

|ρ1+δu(t, ·)|pp,G

≤ C

∫ t

0

[2∑

k=0

|ρ1+δ∂ku(s, ·)|pp,G + |ρ1+δV su(s, ·)|pp,G + |ρ1+δh(s, ·)|pp,G] ds

≤ C

∫ t

0

[u(s, ·)|pH2

p,θ−p(G)+ |ρ1+δu(s, ·)|pp,G + |ρ1+δh(s, ·)|pp,G] ds

≤ C

∫ t

0

[|ρ1+δu(s, ·)|pp,G + |ρ1+δh(s, ·)|pp,G] ds,

t ≤ T. Therefore (3.7) follows by Gronwall’s lemma and (3.8).

Consider the mappings Tτ : H1p,θ(T,G) → H0

p,θ+p(T,G) defined by

u(t, x) =

∫ t

0

f(s, x) ds 7→ f −Aτu,

where f ∈ H0p,θ+p(T,G). Obviously, there is a constant independent of τ such that

|Tτu|H0p,θ+p(T,G) ≤ C|u|H1

p,θ(T,G).

On the other hand, there is a constant C independent of τ such that for all u ∈H1p,θ(T,G),

|u|H1p,θ(T,G) ≤ C|Tτu|H0

p,θ+p(T,G).

Indeed, for u(t, x) =∫ t0 f(s, x) ds (f ∈ H0

p,θ+p(T,G), u ∈ H1p,θ(T,G)) we have

∂tu = Aτu+ (f −Aτu) and by (3.7),

|u|H1p,θ(T,G) ≤ C(|f −Aτu|H0

p,θ+p(T,G) + |Aτu|H0p,θ+p(T,G))

≤ C(|f −Aτu|H0p,θ+p

(T,G) + |u|H2p,θ−p

(T,G))

≤ C|f −Aτu|H0p,θ+p(T,G) = C|Tτu|H0

p,θ+p(T,G).

Since by Theorem 3.1, T0 is an onto map, all Tτ , τ ∈ [0, 1] are onto maps as well by

Theorem 5.2 in Ref. [6].

We will show that under the assumption B or C,

Itu(x) = Iu(t, x) =

∫∇2yu(x)π(t, x, dy) ∈ H0

p,θ+p(G),

if u ∈ H2p,θ−p(G), and Itu, t ≥ 0, satisfies the assumption L.




The following lemma shows that under B, the assumptions L for holds for Iu.

Therefore Theorem 2.1 follows immediately by Proposition 3.4.

Lemma 3.5. Assume B holds and u ∈ H2p,θ−p(G). Then for each ε > 0 there is Cε

such that for all t ≥ 0

|ρ1+δItu|p,G ≤ ε|u|H2p,θ−p(G) + Cε|u|H1

p,θ(G).

Proof. Assume u ∈ H2p,θ−p(G). Then

Iu(x) =

∫

Γ1ε

...+

∫

Γ2ε

. . .+

∫

|y|>ε. . . = (I1 + I2 + I3)u(x),

where

Γ1ε = y ∈ Rd : x+ y ∈ G, |y| ≤ ε,

Γ2ε = y ∈ Rd : x+ y /∈ G, |y| ≤ ε.

10. We start with I1u and split it into two integrals

I1u(x) =

∫

Γ1ε

... =

∫

Γ11ε

...+

∫

Γ12ε

... = (I11 + I12)u(x),

where Γ11ε = Γ1

ε ∩ y : |y| ≤ ρ(x), Γ12ε = Γ1

ε ∩ y : |y| > ρ(x).

For any τ ∈ (0, 1), y ∈ Γ11ε we have ρ(x + τy) ≥ ρ(x) − τ |y| ≥ (1 − τ)ρ(x).

Therefore

ρ(x)

ρ(x+ τy)≤ (1 − τ)−1.

If ε is sufficiently small,

ρ1+δ(x)|∇2yu(t, x)| = ρ1+δ(x)|

∫ 1

0

(1 − τ)∂2iju(x+ τy)yiyj dτ |

≤ N |y|2∫ 1

0

(1 − τ)|(ρ1+δ∂2u)(x+ τy)|[

ρ(x)

ρ(x+ τy)

]1+δdτ.



By Holder inequality,

|ρ1+δI11u(x)|pp,G

≤ N

∫

G

∫

Γ11ε

∫ 1

0

(1 − τ)−δ |(ρ1+δ∂2u)(x+ τy)| dτ |y|2π(t, dy)

pdx

≤ N

∫

G

∫

Γ11ε

∫ 1

0

|(ρ1+δ∂2u)(x+ τy)|p dτ |y|2π(t, dy)

×∫

Γ11ε

∫ 1

0

(1 − τ)−δqdτ |y|2π(t, dy)

pq

dx

≤ N |ρ1+δ∂2u|pp,G

∫

|y|≤ε|y|2π(t, dy)

1+ pq ∫ 1

0

(1 − τ)−δqdτ

pq

,

where δq = θ−dp and p

p−1 = θ−dp−1 < 1, because θ < d− 1 + p. Thus

|ρ1+δI11u|p,G ≤ Nγ(ε)|ρ1+δ∂2u|p,G ≤ Nγ(ε)|u|H2p,θ−p(G),

where

γ(ε) = supt>0

∫


∫ 1

0

(1 − τ)−δqdτ

pq

→ 0,

as ε→ 0, by assumption B.

If y ∈ Γ12ε , we have ρ(x+ y) ≤ ρ(x) + |y| ≤ 2|y|, and the estimate is straightfor-

ward:

ρ1+δ(x)|u(x + y)| ≤ ρ1+δ(x)ρ1−δ(x+ y)|(ρδ−1u)(x+ y)|

≤ 2|y|2|(ρδ−1u)(x+ y)|,

ρ1+δ(x)|u(x)| ≤ ρ2(x)|(ρδ−1u)(x)| ≤ |y|2|(ρδ−1u)(x)|,

ρ1+δ(x)|∂u(x)||y| ≤ |y|2|(ρδ∂u)(x)|,

and

|ρ1+δI12u(x)|pp,G ≤ C|u|pH2

p,θ−p(G)

(∫


)p.

20. Consider now I2u. Obviously,

|I2u(x)| ≤ |u(x)|π(t,Γ2ε) + |∂u(x)|

∫

Γ2ε

|y| π(t, dy).



If y ∈ Γ2ε, then ε ≥ |y| > ρ(x) and

|(ρ1+δI2u

)(x)| ≤ C

(|(ρ−1+δu)(x)| + |(ρδ∂u)(x)|

)supt

∫

|y|≤ε|y|2π(t, dy).

Thus

|ρ1+δI2u|p,G ≤ Cκ(ε)|u|H2p,θ−p(G),

where κ(ε) = supt∫|y|≤ε |y|2π(t, dy).

30. Finally, we handle I3u. It follows that

|I3u(x)| ≤∫

|y|>ε|u(x+ y)|π(t, dy) + |u(x)|π(t, |y| > ε)

+|∂u(x)|∫

1≥|y|>ε|y|π(t, dy).

First we estimate

Ju(x) =

∫

|y|>ε|u(x+ y)|π(t, dy).

For ε0 > 0 and z ∈ G we have

|u(z)| = |u(z)ρ−1+δ(z)|ρ1−δ(z)1ρ(z)≤ε0 + |u(z)ρδ(z)|ρ−δ(z)1ρ(z)>ε0

≤ ε1−δ0 |u(z)ρ−1+δ(z)| + ε−δ0 |u(z)ρδ(z)|.Therefore,

Ju(x) ≤∫

|y|>ε|u(x+ y)|

(ε1−δ0 ρ−1+δ(x+ y) + ε−δ0 ρδ(x+ y)

)π(t, dy).

By Minkowski’s inequality,

|Ju|p,G ≤ supt>0

∫

|y|>επ(t, dy)

(ε1−δ0 |uρ−1+δ|p,G + ε−δ0 |uρδ|p;G

)

≤ Nε−2(ε1−δ0 |uρ−1+δ|p,G + ε−δ0 |uρδ|p;G

).

Since G is bounded,

|ρ1+δJu|p,G ≤ N |Ju|p,G.The remaining terms

J2u(x) = |u(x)|π(t, |y| > ε , J3u(x) = |∂u(x)|∫

1≥|y|>ε|y|π(t, dy)

are estimated in the following way:

ρ1+δ(x)J2u(x) ≤ Nρδ(x)|u(x)|π(t, |y| > ε) ≤ Nε−2|(ρδu)(x)|,

ρ1+δ(x)J3u(x) ≤ Nε−1|(ρ1+δ∂u)(x)|.



Therefore,

|ρ1+δJ2u|p,G ≤ Nε−2|u|H1p,θ

(G),

|ρ1+δJ3u|p;G ≤ Nε−1|u|H1p,θ(G).

By choosing in these estimates a small ε > 0 and then a small ε0 we obtain the

statement

So, the statement of Theorem 2.1 follows by Proposition 3.4.


We will use the maximal functions. So, the function u needs to be bounded. The

following embedding theorem holds in weighted Sobolev spaces.

Proposition 3.6. (see Proposition 2.2 in Ref. [7]) If γp > d, then

|ρ θpu|∞,G ≤ N |u|Hγ

p,θ(G).

In particular,

|ρ θp−1u|∞,G ≤ N |u|H2

p,θ−p(G),

and u is bounded on G, if p ≥ θ.

Denote

Tyu(x) =|∇2

yu(x)||y|2 , x ∈ G, y ∈ Rd,

and define the maximal function

Mf(x) = supR>0

R−d∫

|z|≤Rf(x+ z) dz.

Lemma 3.7. Let u ∈ C∞0 (G), p > d

2 ∨ 1. Then there is a constant N such that for

all x ∈ G

sup|y|≤ 1

4ρ(x)

|Tyu(x)|p ≤ Nρ−(1+δ)p(x)[M(|∂2uρ1+δ|p

)(x) + M

(|∂uρδ|p

)(x)

+M(|uρ−1+δ|p

)(x)].

Proof. According to [4] and [1], for all x, y ∈ Rd,

(Tyu(x))p ≤ NM

(|∂2u|p

)(x). (3.9)

Fix x ∈ G. Let ϕ ∈ C∞0 (Rd), ϕ(z) = 1, if |z| ≤ 1

2 , ϕ(z) = 0, if |z| > 1 and

ϕr(z) = ϕ(z − x

r).



Obviously, Tyu(x) = Ty (uϕr) (x), if |y| ≤ 12r. By (3.9), we have for |y| ≤ 1

2r ≤14ρ(x)

(Tyu(x))p

= (Ty(uϕr)(x))p ≤ NM

(|∂2(uϕr

)|p)(x)

≤ NM(|∂2uϕr|p + |∂u · ∂ϕr|p + |u∂2ϕr|p

)(x).

For r = 12ρG(x),

M(|∂2uϕr|p

)(x) = sup

R>0R−d

∫

|z|≤R|∂2u(x+ z)|pϕpr(x+ z) dz

≤ supR>0

R−d∫

|z|≤R∧r|(∂2uρ1+δ)(x+ z)|pρ−(1+δ)p(x+ z) dz

≤ Nρ−(1+δ)p(x)M(|∂2uρ1+δ|p

)(x),

because ρ(x+ z) ≥ 12ρ(x), |z| ≤ r = 1

2ρ(x).

Similarly we get

M (|∂u · ∂ϕr|p) (x) ≤ Nr−p supR>0

∫

|z|≤R∧r|(∂uρδ)(x+ z)|pρ−δp(x+ z) dz

≤ Nρ−(1+δ)p(x)M(|∂uρδ|p

)(x),

and

M(|u∂2ϕr|p

)(x) ≤ Nr−2p sup

R>0R−d

∫

|z|≤R∧r|(uρ−1+δ)(x+ z)|pρ(1−δ)p(x+ z) dz

≤ Nρ−(1+δ)p(x)M(|uρ−1+δ|p

)(x).

Using the properties of maximal functions (see Ref. [8]) we obtain the following

obvious statement.

Corollary 3.8. If u ∈ H2p,θ−p(G), p > d

2 ∨ 1, then

|ρ1+δTu|p,G ≤ N |u|H2p,θ−p(G),

where

Tu(x) = sup|y|≤ 1

4ρ(x)

Tyu(x).

Proof. For each p > d2 ∨ 1, there are p > d

2 ∨ 1 and ε > 0 such that p = p(1 + ε).

By Lemma 3.7,

ρ(1+δ)p(x) sup|y|≤ 1

4 ρ(x)

|Tyu(x)|p ≤ N [M(|∂2uρ1+δ|p

)(x) + M

(|∂uρδ|p

)(x)

+M(|uρ−1+δ|p

)(x)].



Therefore

∫

G

(ρ(1+δ)p(x) sup|y|≤ 1

4ρ(x)

|Tyu(x)|p)1+εdx

≤ C(

∫

G

|∂2uρ1+δ|p(1+ε)dx+

∫

G

|∂uρδ|p(1+e) dx +

∫

G

|uρ−1+δ|p(1+ε)dx).

In order to prove that the assumption L holds for I tu, the following estimate is

needed as well.

Lemma 3.9. Assume θ ≤ p, r > 0, u ∈ C∞0 (G). Then there is a constant N such

that for all x ∈ G

sup|y|>r,

ρ(x+y)>0

|u(x+ y)|p|y|2p ≤ Nr−(1+δ)p[M

(|∂2uρ1+δ|p

)(x)

+M(|∂uρδ|p

)(x) + M

(|uρ−1+δ|p

)(x)].

Proof. Fix x ∈ G and denote U(y) = u(x+ y). Then

sup|y|>r,

ρ(x+y)>0

|u(x+ y)||y|2 = sup

t>r

1

t2supy∈Γt

|y|=1

|U(ty)|, (3.10)

where

Γt = y : ρ(x+ ty) > 0.

For y ∈ Γt, let ρt(y) = dist(y, ∂Γt). Notice that

dist(y, ∂Γt) = infz,ρ(x+tz)=0

|y − z| =1

tinf

z,ρ(x+tz)=0|ty − tz|

=1

tinf

z,ρ(x+tz)=0|(x+ ty) − (x + tz)|

=1

tρ(x+ ty).



Let ϕ ∈ C∞0 (Rd), ϕ ≥ 0, ϕ(y) = 1, if |y| ≤ 1, and ϕ(y) = 0, if |y| > 2. By

Proposition 3.6 for U(ty)ϕ(y), y ∈ Γt, we have

|U(t·)|∞;B1∩Γt ≤ |U(t·)ϕ|∞,Γt ≤ N |U(t·)ϕ|H2p,θ−p(Γt)

≤ N [|∂2(U(t·)ϕ)ρ1+δt |p,Γt + |∂(U(t·)ϕ)ρδt |p,Γt

+|U(t·)ϕρ−1+δt |p,Γt ]

≤ N [|∂2U(t·)ρ1+δt |p,B2∩Γt + |∂U(t·)ρδt |p,B2∩Γt

+|U(t·)ρ−1+δt |p,B2∩Γt ],

where BR = y : |y| < R . Using the change of variable ty = z, we obtain

|∂2U(t·)ρ1+δt |pp,B2∩Γt

=

∫

B2∩Γt

t2p|∂2u(x+ ty)|pt−(1+δ)pρ(1+δ)p(x+ ty) dy

= t(1−δ)pt−d∫

B2t∩G|∂2u(x+ z)|pρ(1+δ)p(x+ z) dz

≤ Nt(1−δ)pM(|∂2uρ1+δ|p

)(x),

and

|∂U(t·)ρδt |pp,B2∩Γt=

∫

B2∩Γt

tp|∂u(x+ ty)|pt−δpρδp(x+ ty) dy

= t(1−δ)pt−d∫

B2t∩G|∂u(x+ z)|pρδp(x+ z) dz

≤ Nt(1−δ)pM(|∂uρδ|p

)(x).

Also,

|U(t·)ρ−1+δt |pp,B2∩Γt

=

∫

B2∩Γt

|u(x+ ty)|pt(1−δ)pρ(−1+δ)p(x + ty) dy

= t(1−δ)pt−d∫

B2t∩G|u(x+ z)|pρ(−1+δ)p(x+ z) dz

≤ Nt(1−δ)pM(|uρ−1+δ|p

)(x).



These estimates and (3.10) imply that

sup|y|>r,

ρ(x+y)>0

|u(x+ y)|p|y|2p = sup

t>r

1

t2p|U(t·)|p∞;S1∩Γt

≤ supt>r

1

t2pNt(1−δ)p(M...+ M...+ M...)

≤ Nr−(1+δ)p(M...+ M...+ M...)

and the statement follows.

The Lemma below completes the proof of Theorem 2.2. It shows that I tu

satisfies the assumption L.

Lemma 3.10. Assume u ∈ H2p,θ−p(G), p > d, θ ≤ p and the assumption C holds.

Then for each ε > 0 there is a constant Cε > 0 such that

|ρ1+δIu|p,G ≤ ε|u|H2p,θ−p(G) + Cε|u|H1

p,θ(G).

Proof. Fix ε > 0 and split the integral

Iu(x) =

∫

|y|≤ε∧ 14ρ(x)

∇2yu(x)π(t, x, dy) +

∫

ε≥|y|>ε∧ 14ρ(x)

...+

∫

|y|>ε...

= I1u(x) + I2u(x) + I3u(x).

10. Estimate of I1u. Obviously,

|I1u(x)| ≤ γ(ε) sup|y|≤ε∧ 1

4ρ(x)

|∇2yu(x)||y|2 ,

where γ(ε) = supt>0,x∈G∫|y|≤ε |y|2 π(t, x, dy). By Lemma 3.7,

|ρ1+δ(x)I1u(x)|p ≤ (Nγ(ε))p

[M(|∂2uρ1+δ|p

)(x) + M

(|∂uρδ|p

)(x)

+M(|uρ−1+δ|p

)(x)]

for p > d2 ∨ 1. Therefore using properties of the maximal functions (see Ref. [8] and

the proof of Corollary 3.8 above)

|ρ1+δI1u|p;G ≤ Nγ(ε)|u|H2p,θ−p(G).

20. Estimate of I2u. We have

|I2u(x)| ≤∫

Γε(x)

|u(x+ y)|π(t, x, dy)| + |u(x)|π(t, x,Γε(x))

+|∂u(x)|∫

Γε(x)

|y|π(t, x, dy)

= I21u(x) + I22u(x) + I23u(x),



where Γε(x) = y : ε ≥ |y| > 14ρ(x).

We have

I21u(x) ≤ supy∈Γε(x),ρ(x+y)>0

|u(x+ y)||y|2

∫

|y|≤ε|y|2π(t, x, dy)

≤ Nγ(ε) sup|y|> 1

4ρ(x),ρ(x+y)>0

|u(x+ y)||y|2 .

Therefore by Lemma 3.9,

|ρ1+δ(x)I21u(x)|p ≤ (Nγ(ε))p [M(|∂2uρ1+δ|p

)(x) + M

(|∂uρδ|p

)(x)

+M(|uρ−1+δ|p

)(x)],

and by properties of maximal functions (see Ref. [8] and the proof of Corollary 3.8

above))

|ρ1+δI21u|p;G ≤ Nγ(ε)|u|H2p,θ−p(G).

Obviously,

|I22u(x)| + |I23u(x)| ≤ Nγ(ε)(ρ−2(x)|u(x)| + ρ−1(x)|∂u(x)|

)

and therefore

|ρ1+δI22u|p,G + |ρ1+δI23u|p,G ≤ Nγ(ε)(|ρ−1+δu|p,G + |ρδ∂u|p,G

)

≤ Nγ(ε)|u|H2p,θ−p(G).

Thus

|ρ1+δI2u|p;G ≤ Nγ(ε)|u|H2o,θ−p(G).

30. Estimate of I3u. We have

|I3u(x)| ≤∫

|y|>ε|u(x+ y)|π(t, x, dy) + |u(x)|π(t, x, |y| > ε)

+|∂u(x)|∫

1≥|y|>ε|y|π(t, x, dy) (3.11)

≤ N(ε−2|u|∞;G + ε−1|∂u(x)|).For ε0 > 0 and z ∈ G,

|u(z)| = |u(z)ρθp−1(z)|ρ1− θ

p (z)1ρ(z)≤ε0

+|u(z)ρθp (z)|ρ− θ

p (z)1ρ(z)>ε0 (3.12)

≤ ε1− θ

p

0 |uρ θp−1|∞;G + ε

− θp

0 |uρ θp |∞;G.



It follows from (3.11) and (3.12) that

|ρ1+δ(x)I3u(x)| ≤ Nε−2ε1− θ

p

0 |uρ θp−1|∞;Gρ

1+δ(x)

+Nε−2ε− θ

p

0 |uρ θp |∞;Gρ

1+δ(x)

+Nε−1|∂u(x)|ρ1+δ(x).

By embedding theorem (we use p > d when γ = 1),

|ρ1+δI3u|p;G ≤ Nε−2ε1− θ

p

0 |u|H2p,θ−p(G) +Nε−2ε

− θp

0 |u|H1p,θ(G)

+Nε−1|u|H1p,θ(G).

Choosing a small ε > 0 and then a small ε0 we obtain our statement.

Acknowledgement

We are very grateful to our referee for valuable comments.

References

[1] J.-M. Bony, Probleme de Dirichlet et semi-groupe fortement fellerien associes a unoperateur integro-differentiel, C. R. Acad. Sci. Paris Ser. A-B. 265, A361–A364,(1967).

[2] F. Gimbert and P.-L. Lions, Existence and regularity results for solutions of second-order, elliptic integro-differential operators, Ricerche Mat. 33(2), 315–358, (1984).

[3] M. G. Garroni and J.-L. Menaldi, Green functions for second order parabolic integro-differential problems. vol. 275, Pitman Research Notes in Mathematics Series, (Long-man Scientific & Technical, Harlow, 1992). ISBN 0-582-02156-1.

[4] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Holder spaces, Liet. Mat. Rink. 32(2), 299–331,(1992). ISSN 0132-2818.

[5] K.-H. Kim and N. V. Krylov, On the Sobolev space theory of parabolic and ellipticequations in C1 domains, SIAM J. Math. Anal. 36(2), 618–642 (electronic), (2004).ISSN 0036-1410.

[6] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order.vol. 224, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences], (Springer-Verlag, Berlin, 1983), second edition. ISBN 3-540-13025-X.

[7] S. V. Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains,Stochastics Stochastics Rep. 68(1-2), 145–175, (1999). ISSN 1045-1129.

[8] E. M. Stein, Singular integrals and differentiability properties of functions. PrincetonMathematical Series, No. 30, (Princeton University Press, Princeton, N.J., 1970).


Chapter 15

Strict Solutions of Kolmogorov Equations in Hilbert Spaces and

Applications

Giuseppe Da Prato

Scuola Normale Superiore56126, Pisa, Italy

[email protected]

Consider a stochastic differential equation in a Hilbert space H, with the associ-ated differential operator K0 determined by the coefficients in the equation, andthe corresponding Markov semigroup Pt. The paper investigates the problem ofconstructing a core for the (weak) generator K of Pt in the space of real, uniformlycontinuous, and bounded functions on H, and studies the relation between K andK0. The paper proposes an approach based on the concept of a strict solutionof the corresponding Kolmogorov equation: a suitable assumption of existence ofstrict solutions imply good results on the core, and the assumption is verified inan SPDE example.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3752 Existence of cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3793 Invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

4.1 Estimates for Xx(t, x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3844.2 Estimates for Xxx(t, x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3864.3 Estimates of TR[Xx,x(t, x)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3874.4 Estimates of Ptϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

1. Introduction

We are concerned with a stochastic differential equation in a separable Hilbert space

H (norm | · |, inner product 〈·, ·〉) of the following form

dX(t) = (AX(t) + b(X(t))dt+ σ(X(t))dW (t), t ≥ 0,

X(0) = x ∈ H,

(1.1)

where A : D(A) ⊂ H→H is the infinitesimal generator of a strongly continuous

semigroup etA in H and the mappings b : D(b) ⊂ H→H and σ : H→L(H) are

(generally) nonlinear. Moreover, W (t) is a cylindrical Wiener process in H .

375


376 G. Da Prato

We shall assume that problem (1.1) has a unique mild solution X(t, x) (a con-

cept that has to be made precise in any specific situation) which is adapted and

continuous in mean square. We shall denote by Pt the transition semigroup

Ptϕ(x) = E[ϕ(X(t, x))], ϕ ∈ Cb(H), x ∈ H, (1.2)

where Cb(H) denotes the Banach space of all real uniformly continuous and bounded

functions in H endowed with the norm

‖ϕ‖0 = supx∈H

|ϕ(x)|, ϕ ∈ Cb(H).

The Kolmogorov equation corresponding to (1.1) looks like

ut(t, x) =1

2Tr [(σσ∗)(x)uxx(t, x)] + 〈Ax+ b(x), ux(t, x)〉,

for x ∈ D(A) ∩D(b),

u(0, x) = ϕ(x), x ∈ H.

(1.3)

As well known, a candidate for the solution of (1.3) is given by

u(t, x) = E[ϕ(X(t, x))], t ≥ 0, x ∈ H.

Let us give some notations. We shall denote by C1b (H) the space of all real functions

in H which are continuous together with their Frechet derivatives of the first order.

It is a Banach space with the norm ‖ϕ‖1 = ‖ϕ‖0 + ‖ϕx‖0. Spaces Ckb (H), k > 1,

are defined similarly. Moreover we denote by ek a complete orthonormal system

in H .

Definition 1.1. A strict solution of (1.3) is a function u : [0,∞)×H→R such that

(i) u is continuous in [0,∞) ×H and u(0, x) = ϕ(x), x ∈ H .

(ii) For all t ≥ 0, u(t, ·) ∈ C1b (H) and there exist the second derivatives of u,

ux,x(t, ·)(h, k) in all directions h, k ∈ H.

(iii) For all t > 0 the function

H→R, x 7→ Tr [(σσ∗)(x)uxx(t, x)] =

∞∑

k=1

uxx(t, x)(σ(x)ek , σ(x)ek)

belongs to Cb(H).

(iv) For all x ∈ H , u(·, x) ∈ C1((0,+∞)).

(v) For all (t, x) ∈ (0,+∞) ×D(A) ∩D(b), equation (1.3) is fulfilled.

We are also interested in the elliptic equation

λϕ− 1

2Tr [(σσ∗)(x)ϕxx(t, x)] − 〈Ax + b(x), ϕx(t, x)〉 = f(x) (1.4)

where λ > 0 and f ∈ Cb(H) are given. The following definition is the elliptic

counterpart of Definition 1.1.

Definition 1.2. A strict solution of (1.4) is a function ϕ : H→R such that


Strict Solutions of Kolmogorov Equations 377

(i) ϕ ∈ C1b (H) and there exist the second derivatives of ϕ, ϕx,x(x)(h, k) in all

directions h, k ∈ H.

(ii) For all x ∈ D(A) ∩D(b), equation (1.4) is fulfilled.

For this equation a candidate solution is given, as well known, by

ϕ(x) =

∫ +∞

0

e−λtPtf(x)dt, x ∈ H. (1.5)

Finally, we define the Kolmogorov operator K0 setting

K0ϕ(x) =1

2Tr [(σσ∗)(x)ϕxx(x)] + 〈Ax + b(x), ϕx(x)〉, ϕ ∈ D(K0),

D(K0) =ϕ ∈ C1

b (H) : conditions (i) of Definition 1.2 holds.

(1.6)

It is well known that the semigroup Pt is not strongly continuous in Cb(H) in

general (in fact in all interesting situations). However, we can define its infinitesimal

generator by proceeding as in Ref. [1]. For any λ > 0 and any f ∈ Cb(H) we set

Fλ(f)(x) =

∫ ∞

0

e−λtPtf(x)dt, x ∈ H. (1.7)

Then the following result holds.

Proposition 1.3. For any f ∈ Cb(H) and any λ > 0 we have Fλ(f) ∈ Cb(H) and

‖Fλ(f)‖0 ≤ 1

λ‖f‖0. (1.8)

Moreover there exists a unique closed operator K : D(K) ⊂ Cb(H)→Cb(H) such

that for any λ > 0 and any f ∈ Cb(H) we have Fλ(f) = (λ−K)−1f.

Proof. Let first take f in C1b (H). Then for all x, y ∈ H we have

|Fλ(f)(x) − Fλ(f)(y)| ≤∫ ∞

0

e−λt E(|f(X(t, x)) − f(X(t, y))|)dt

≤ ‖f‖1

∫ ∞

0

e−λt E|X(t, x) −X(t, y)|dt.

Since X(t, x) is mean square continuous, we see that Fλ(f) ∈ Cb(H). Moreover,

it is obvious that (1.8) holds. Since C1b (H) is dense in Cb(H) we can conclude, by

a straightforward argument, that Fλ(f) ∈ Cb(H) for all f ∈ Cb(H) and that (1.8)

holds.

Now, by a direct computation, we see that Fλ fulfills the resolvent identity

Fλ − Fµ = (µ− λ)FλFµ, λ, µ > 0.

Since for every f ∈ Cb(H)

limλ→∞

λFλ(f)(x) = limλ→∞

∫ +∞

0

e−τP τλf(x)dτ = f(x), x ∈ H,


378 G. Da Prato

F is one-to-one. So, by a classical result, see e.g. Ref. [2], there exists a unique closed

operator K : D(K) ⊂ Cb(H)→Cb(H) such that for any λ > 0 and any f ∈ Cb(H)

we have Fλ(f) = (λ −K)−1f.

The goal of this paper is to study strict solutions both of equations (1.3) and

(1.4) and to investigate the relationship between the “abstract” generator K and

the “concrete” differential operator K0 defined by (1.6).

We shall consider situations where K is an extension of K0 (see Remark 2.4

below) and K0 determines K, more precisely that D(K0) is a core for K (in a sense

to be made precise). This fact is important in order to prove several properties of

the operator K (starting from the analogous ones for the more accessible operator

K0), as we shall explain in section 3 below.

A natural idea to find strict solutions of equation (1.4) when f ∈ C3b (H) is to

check, by successive differentiations, that the candidate function ϕ given by (1.5)

fulfills (1.4). This method works easily provided b and σ are of class C3, since in

this case X(t, ·) is of class C2 for any t ≥ 0, see Ref. [3]. But in all interesting

applications b and σ are not even of class C1 so that, proving that u is a strict

solution, can be much more involved. In section 4 we shall present an application to

a very simple but non trivial stochastic partial differential equation. We plan to take

in consideration more interesting stochastic PDEs in successive papers. We quote

in this direction the paper [4] where this method was used for the 3D-stochastic

Navier-Stokes. However, in that case a much weaker notion (than Definition 1.1)

of strict solution was used.

We end this section by giving a useful characterization of D(K) similar to that

proved in Ref. [5].

Proposition 1.4. Let ϕ ∈ D(K). Then we have

limh→0+

1

h(Phϕ(x) − ϕ(x)) = Kϕ(x), for all x ∈ H (1.9)

and

suph∈(0,1]

∥∥∥∥1

h(Phϕ− ϕ)

∥∥∥∥0

< +∞. (1.10)

Conversely, if there exists ϕ, g ∈ Cb(H) such that

limh→0+

1

h(Phϕ(x) − ϕ(x)) = g(x), for all x ∈ H (1.11)

and (1.10) holds we have ϕ ∈ D(K) and Kϕ = g.



Proof. Assume that ϕ ∈ D(K). Fix λ > 0 and set f = λϕ−Kϕ. Then Fλ(f) = ϕ

and for any h > 0 and any x ∈ H we have

Phϕ(x) = PhFλ(f)(x) = eλh∫ +∞

h

e−λsPsf(x)ds.

It follows that

DhPhϕ(x)|h=0 = λ

∫ +∞

0

e−λsPsf(x)ds − f(x)

= λFλ(f)(x) − f(x) = KFλ(f)(x) = Kϕ(x),

and so (1.9) follows. Moreover, since

|Phϕ(x) − ϕ(x)| ≤ (eλh − 1)

∣∣∣∣∫ +∞

h

e−λsPsf(x)ds

∣∣∣∣

+

∣∣∣∣∣

∫ h

0

e−λsPsf(x)ds

∣∣∣∣∣ ≤ ‖f‖0

[eλh − 1

λe−λh +

1 − e−λh

λ

]≤ ch,

where c is a suitable positive constant, we see that (1.10) follows as well.

Assume now that there exists ϕ, g ∈ Cb(H) such that (1.11) is fulfilled. Since

clearly for any x ∈ H

d

dtPtϕ(x) = lim

h→0

1

h(Pt+hϕ(x) − Ptϕ(x)) = Ptg(x),

we have

Fλ(ϕ)(x) = − 1

λ

∫ t

0

Ptϕ(x)de−λt

=1

λϕ(x) +

1

λ

∫ t

0

e−λtPtg(x)dt

=1

λϕ(x) +

1

λFλ(g)(x).

Therefore

Fλ(ϕ)(x) =1

λϕ(x) +

1

λFλ(g)(x),

which implies ϕ ∈ D(K) and Kϕ = g.

2. Existence of cores

When the dimension of H is infinite it is well known that C2b (H) is not dense in

Cb(H), see Ref. [6]. For this reason we introduce a notion of convergence weaker


380 G. Da Prato

than that of Cb(H), see Ref. [5]. We say that a sequence ϕn ⊂ Cb(H) is π-

convergent to ϕ ∈ Cb(H) if limn→∞

ϕn(x) = ϕ(x) for all x ∈ H and there exists c > 0

such that |ϕn(x)| ≤ c for all n ∈ N and x ∈ H .

Proposition 2.1. Let fn ⊂ Cb(H) be π-convergent to a function f in Cb(H).

Then Fλ(fn) is π-convergent to f .

Proof. The conclusion follows from the mean square continuity of X(t, x) and the

dominated convergence theorem.

In the next definition we follow Ref. [5].

Definition 2.2.

(i) A subspace Z of Cb(H) is said to be π-dense in Cb(H) if for any ϕ ∈ Cb(H)

there exists a sequence ϕn ⊂ Z which is π–convergent to ϕ.

(ii) A subspace Y of D(K) is a π-core for K if for any ϕ ∈ D(K) there exists a

sequence ϕn ⊂ Y which is π–convergent to ϕ and such that the sequence

Kϕn is π–convergent to Kϕ.

It is easy to see that for any k ∈ N, Ckb (H) is π-dense in Cb(H), see e.g. Refs. [5]

and [3].

We notice that there are many π-cores for K. It is enough to choose a π-dense

linear subspace R(H) in Cb(H) and set

Γ :=⋃

λ>0

(λ−K)−1R(H). (2.1)

In fact Γ is π-dense in Cb(H) and consequently in D(K), endowed with the graph

norm, by Proposition 2.1.

Concerning existence of π-cores, we shall make the following basic assumption.

Hypothesis 2.3.

(i) K is an extension of K0.

(ii) There exists a linear subspace R(H) which is π-dense in Cb(H) and such that

for any f ∈ R(H), ϕ = Fλ(f) is a strict solution of (1.4).

Then we define Γ by (2.1).

Remark 2.4. Assumption 2.3-(i) holds provided the following Ito formula is fulfilled

for any ϕ ∈ C2b (H),

E[ϕ(X(t, x))] = ϕ(x) + E

∫ t

0

K0ϕ(X(s, x))ds, x ∈ H, t ≥ 0.

In fact, from Proposition 1.4 it follows that D(K0) ⊂ D(K) and

K0ϕ = Kϕ.



Let ϕ ∈ Γ; we do not know if ϕ2 ∈ Γ; however we can show that ϕ2 ∈ D(K0).

We have in fact

Lemma 2.5. Assume that Hypothesis 2.3 is fulfilled. Then for all ϕ ∈ Γ we have

ϕ2 ∈ D(K0) and

K0(ϕ2)(x) = 2ϕ(x) K0ϕ(x) + |σ∗(x)Dϕ(x)|2, x ∈ D(A) ∩D(b). (2.2)

Proof. Let ϕ ∈ Γ, λ > 0. Then f = λϕ−Kϕ belongs to R(H) by assumption. We

know by Hypothesis 2.3 that ϕ ∈ D(K0). Let now x ∈ D(A)∩D(b). Then we have

〈Ax,D(ϕ2(x))〉 = 2ϕ(x)〈Ax,Dϕ(x)〉,

〈b(x), D(ϕ2(x))〉 = 2ϕ(x)〈b(x), Dϕ(x)〉,

and

Tr [(σσ∗)(x)D2(ϕ2(x))] = 2ϕ Tr [(σσ∗)(x)D2(ϕ(x))] + 2|σ∗(x)Dϕ(x)|2 .

Consequently ϕ2 ∈ D(K0) and (2.2) follows from a straightforward computation.

3. Invariant measures

In this section we assume, besides Hypothesis 2.3 that there exists an invariant

measure ν for Pt, that is a probability measure in (H,B(H)) a such that∫

H

Ptfdν =

∫

H

fdν, f ∈ Cb(H). (3.1)

We shall prove the basic integration by parts formula (3.4) below. We recall that

this formula is the main tool in order to define the Sobolev space W 1,2(H, ν), for

studying regularity properties of the domain D(K) of K and also to prove the

Poincare and log-Sobolev inequalities. We will not deal with these problems in the

present paper; several results in this direction can be found in Refs. [7], [8] and

references therein.

Let λ > 0. Multiplying both sides of (3.1) by e−λt and integrating with respect

to t over (0,+∞), yields∫

H

R(λ,K)fdν =1

λ

∫

H

fdν. (3.2)

Let now ϕ ∈ D(K) and λ > 0. Set f = λϕ − Kϕ so that ϕ = R(λ,K)f. Then,

taking into account (3.2), it follows that∫

H

ϕdν =

∫

H

R(λ,K)fdν =1

λ

∫

H

fdν =1

λ

∫

H

(λϕ −Kϕ)dν,

aB(H) is the σ-algebra of all Borel subsets of H.


382 G. Da Prato

which implies

∫

H

Kϕdν = 0, for all ϕ ∈ D(K). (3.3)

It is well known that Pt can be uniquely extendible to a strongly continuous semi-

group of contractions in L2(H, ν) still denoted Pt; let K2 be its infinitesimal gener-

ator.

Remark 3.1. Assume that Hypothesis 2.3 is fulfilled. Then, by the dominated

convergence theorem and a standard monotone classes argument, it follows that

R(H) is dense in L2(H, ν). Consequently, Γ is a core for K2.

We now consider the mapping

γ : Γ ⊂ D(K2)→L2(H, ν), ϕ 7→ |σ∗Dϕ|2,

where D(K2) is endowed with the graph norm.

Proposition 3.2. The mapping γ is continuous and uniquely extendible to D(K2).

Moreover the following identity holds

∫

H

ϕ Kϕdν = −1

2

∫

H

|σ∗Dϕ|2dν, ϕ ∈ D(K2). (3.4)

Proof. Let first ϕ ∈ Γ. Since the measure ν is invariant for Pt we have by Lemmas

2.5 and identity (3.3)

∫

H

K0(ϕ2)dν = 0.

Then, integrating both sides of identity (2.2) yields

∫

H

ϕ K0ϕdν = −1

2

∫

H

|σ∗Dϕ|2dν, ϕ ∈ Γ. (3.5)

Now let ϕ ∈ D(K2). Since Γ is a core for K2, there exists a sequence ϕn ⊂ Γ

convergent to ϕ in L2(H, ν) and such that Kϕn is convergent to Kϕ in L2(H, ν).

Then by (3.5) we have

∫

H

|σ∗(x)D(ϕn − ϕm)|2dν = −2

∫

H

(ϕn − ϕm) K0(ϕn − ϕm)dν,

so the sequence σ∗Dϕn is Cauchy in L2(H, ν) and the conclusion follows.



4. Application

We consider here the following reaction-diffusion equation in the Hilbert space H =

L2(0, 2π).

dX(t, ξ) = [(D2ξX(t, ξ) −X(t, ξ)) + g(X(t, ξ))]dt+ dW (t, ξ),

t ≥ 0, ξ ∈ [0, 2π],

X(t, 0) = X(t, 2π), DξX(t, 0) = DξX(t, 2π), t ≥ 0,

X(0, ξ) = x(ξ), ξ ∈ [0, 2π],

(4.1)

where g ∈ C3b (R) and W is a cylindrical Wiener process in H which will be specified

below.

Let us write problem (4.1) in the abstract form (1.1). Define

Ax(ξ) = D2ξx(ξ) − x(ξ), ξ ∈ [0, 2π], x ∈ D(A)

D(A) = x ∈ H2(0, 2π) : x(0) = x(2π), Dξx(0) = Dξx(2π)

and

b(x)(ξ) = g(x(ξ)), x ∈ H, ξ ∈ [0, 2π].

Denote by (ek)k∈Z the complete orthonormal system of L2(0, 2π) b,

ek(ξ) =1√2π

eikξ , ξ ∈ [0, 2π], k ∈ Z

and define

W (t) =∑

k∈Z

βk(t)ek,

where βk(t)k∈Z is a family of standard Brownian motions mutually independent

in a filtered probability space (Ω,F , (Ft)t≥0,P). We have

Aek = −(1 + k2)ek, k ∈ Z

and

‖etA‖ ≤ e−t, t ≥ 0.

It is well known that problem (4.1) has a unique mild solution X(t, x), see e.g.

Ref. [9]. We recall that a mild solution of equation (4.1) is a stochastic processbHere we are dealing with the usual complexification of L2(0, 2π).


384 G. Da Prato

X ∈ CW ([0, T ], H) such that

X(t, x) = etAx+

∫ t

0

e(t−s)Ab(X(s, x))dW (s) +

∫ t

0

e(t−s)AdW (s), t ≥ 0, x ∈ H.

(4.2)

Here CW ([0, T ], H) is the space of all mean square continuous adapted stochastic

process X(·) defined in [0, T ] and taking values in H , endowed with the norm

‖X‖CW ([0,T ],H) =

(supt∈[0,T ]

E(|X(t)|2)

)1/2

.

Notice that, in spite of the fact that the function g is of class C3, the Nemitskii

operator b is not of class C1 (except when g is constant). However, b is differen-

tiable in all directions of L2(0, 2π) and it is twice differentiable in all directions of

L4(0, 2π). So, the Kolmogorov equation cannot be solved by the standard method

from Chapter 7 in Ref. [3].

We notice that the existence of a strict solutions of the Kolmogorov equation

corresponding to (4.1) have been studied (in a more general setting) in Ref. [10]

using the Bismut-Elworthy formula.

4.1. Estimates for Xx(t, x)

For any h ∈ H, t ≥ 0, x ∈ H , we denote by Xx(t, x)h : = ηh(t, x) the directional

derivative of Xx(t, x) in the direction of h. It is well known that ηh(t, x) does exist,

(see Ref. [11]) and that it is the mild solution of the problem

d

dtηh(t, x) = Aηh(t, x) + f ′(X(t, x))ηh(t, x), ηh(0, x) = h, (4.3)

that is

ηh(t, x) = etAh+

∫ t

0

e(t−s)Af ′(X(s, x))ηh(s, x)ds. (4.4)

Proposition 4.1. We have

|ηh(t, x)|2L2(0,2π) ≤ e−2(1−‖g‖1)t|h|2L2(0,2π), t ≥ 0, h ∈ H. (4.5)

Proof. In fact, multiplying scalarly both sides of (4.3) by ηh(t, x) and integrating

over [0, 2π] yields,

1

2

d

dt|ηh(t, x)|2L2(0,2π) = 〈Aηh(t, x), ηh(t, x)〉L2(0,2π)

+〈g′(X(t, x))ηh(t, x), ηh(t, x)〉L2(0,2π)

≤ (−1 + ‖g‖1)|ηh(t, x)|2L2(0,2π)

and so, the conclusion follows from a classical comparison result.



Assume that one has to estimate the series∞∑

k=1

|ηh(t, x)|2L2(0,2π).

To this purpose the estimate (4.5) will not be enough. For this reason we prove

another estimate.


|ηh(t, x)|2L2(0,2π) ≤ |etAh|L2(0,2π) + ‖g‖1

∫ t

0

e(‖g‖1−1)(t−s)|esAh|L2(0,2π)ds. (4.6)

Proof. By (4.4) we have

|ηh(t, x)|L2(0,2π) ≤ |etAh|L2(0,2π) +

∫ t

0

|e(t−s)Ag′(X(s, x))ηh(s, x)|L2(0,2π)ds

≤ |etAh|L2(0,2π) + ‖g‖1

∫ t

0

e−(t−s)|ηh(s, x)|L2(0,2π)ds.

Now, setting

ϕ(t) = et|ηh(t, x)|L2(0,2π),

we have

ϕ(t) ≤ et|etAh|L2(0,2π) + ‖g‖1

∫ t

0

ϕ(s)ds.

The conclusion follows from the Gronwall lemma.

We are going now to estimate, for further use, the L2 norm of [ηh(t, x)]2. Write

1

2

d

dt[ηh(t, x)]2 = Aηh(t, x) ηh(t, x) + g′(X(s, x))[ηh(s, x)]2.

Since

A([ηh(t, x)]2) = 2Aηh(t, x) ηh(t, x) + |Dηh(t, x)|2,

we deduce that

d

dt[ηh(t, x)]2 ≤ A([ηh(t, x)]2) − |Dηh(t, x)|2 + 2‖g‖1[ηh(t, x)]2

≤ A([ηh(t, x)]2) + 2‖g‖1[ηh(t, x)]2.

Therefore

[ηh(t, x)]2 ≤ etA(h2) + 2‖g‖1

∫ t

0

e(t−s)A[ηh(s, x)]2ds.

Arguing as in the proof of Proposition 4.2, we find the following result.


386 G. Da Prato


|[ηh(t, x)]2|L2(0,2π) ≤ |etA(h2)|L2(0,2π) +

∫ t

0

e(2‖g‖1−1)(t−s)|esA(h2)|L2(0,2π)ds. (4.7)

Using the well known ultracontractivity type estimate

|etAf |L2(0,2π) ≤ ct−1/4e−t|f |L1(0,2π), (4.8)

we find the result

Corollary 4.4. There exist c1 > 0, κ1 > 0 such that

|[ηh(t, x)]2|L2(0,2π) ≤ c1(1 + t−1/4)eκ1t |h|2L2(0,2π). (4.9)

Proof. We have in fact by (4.8)

|etA(h2)|L2(0,2π) ≤ ct−1/4e−t|h2|L1(0,2π)

= ct−1/4e−t|h|2L2(0,2π),

and the conclusion follows from (4.7).

4.2. Estimates for Xxx(t, x)

We set now

ζh(t, x) = 〈Xxx(t, x)h, h〉, t ≥ 0, x, h ∈ H.

ζh does exist and fulfills the equation (see e.g. Ref. [11])

d

dtζh(t, x) = Aζh(t, x) + g′(X(t, x))ζh(t, x) + g′′(X(t, x))[ηh(t, x)]2,

ζh(0, x) = 0.

(4.10)

Proposition 4.5. There exist c2 > 0 and κ2 > 0 such that

|ζh(t, x)|2L2(0,2π) ≤ c2eκ2t|h|4L2(0,2π). (4.11)

Proof. Multiplying both sides of (4.10) by ζh(t, x) and integrating with respect to

ξ over [0, 2π] yields

1

2

d

dt|ζh(t, x)|2L2(0,2π) ≤ ‖g‖1|ζh(t, x)|2L2(0,2π)

+‖g‖2

∫ 1

0

[ηh(t, x)]2|ζh(t, x)|dξ

≤ (‖g‖1| +1

2‖g‖2

2)|ζh(t, x)|2L2(0,2π)

+1

2|[ηh(t, x)]2|L2(0,2π).



It follows that

|ζh(t, x)|2L2(0,2π) ≤∫ t

0

e(2‖g‖1|+‖g‖22)(t−s)|[ηh(s, x)]2|2L2(0,2π)ds.

Now, using (4.9), we find

|ζh(t, x)|2L2(0,2π) ≤ c21

∫ t

0

e(2‖g‖1|+‖g‖22)(t−s)(1 + s−1/4)2e2κ1t |h|4L2(0,2π)ds

and the conclusion follows.

4.3. Estimates of TR[Xx,x(t, x)]

We want here to estimate the norm of the vector trace of Xx,x(t, x), namely

TR[Xx,x(t, x)] : =

∞∑

k=1

〈Xx,x(t, x)ek , ek〉 =

∞∑

k=1

ζek (t, x).

Let us set

T (t, x) = TR[Xx,x(t, x)], t ≥ 0, x ∈ H.

Setting in (4.10) h = ek and summimg up on k we see that T (t, x) fulfills the

equation

d

dtT (t, x) = AT (t, x) + g′(X(t, x))T (t, x) + g′′(X(t, x))Z(t, x),

T (0, x) = 0,

(4.12)

where

Z(t, x) =

∞∑

k=1

[ηek (t, x)]2.

Proposition 4.6. There exist c3 > 0 and κ3 > 0 such that

|TR (Xx,x(t, x))|2L2(0,2π) ≤ c3eκ3t. (4.13)

Proof. By (4.7) we have, taking into account that e2k =√

2π e2k,

|[ηek (t, x)]2|L2(0,2π) ≤ |etA(e2k)|L2(0,2π) +

∫ t

0

e(2‖g‖1−1)(t−s)|esA(e2k)|L2(0,2π)ds

≤ (2π)2e−4k2t + (2π)2∫ t

0

e(2‖g‖1−1)(t−s)e−4k2sds.

Summing up on k yields

|Z(t, x)|2 ≤ (2π)2∞∑

k=1

e−4k2t + (2π)2∫ t

0

e(2‖g‖1−1)(t−s)∞∑

k=1

e−4k2sds.


388 G. Da Prato

Since, for a suitable constant c > 0,∞∑

k=1

e−4k2t ≤ ct−1/2,

the conclusion follows.

4.4. Estimates of Ptϕ

In this subsection we show that Hypothesis 2.3 is fulfilled by choosing R(H) =

EA(H), where EA(H) is the linear span of all real and imaginary parts of the expo-

nential functions ϕh,

ϕh(x) := ei〈h,x〉, x ∈ H, h ∈ D(A).

It is easy to see that the space EA(H) is dense in L2(H, ν), see e.g. Ref. [8].

Let ϕ ∈ EA(H) and set u(t, x) = Ptϕ(x). Then for any h ∈ H we have

〈ux(t, x), h〉 = E[〈Dϕ(X(t, x)), ηh(t, x)〉

], (4.14)

〈uxx(t, x)h, h〉 = E[〈D2ϕ(X(t, x))ηh(t, x), ηh(t, x)〉

]

+E[〈Dϕ(X(t, x)), ζh(t, x)〉

].

(4.15)

By (4.5) it follows that

|〈ux(t, x), h〉| ≤ ‖ϕ‖1e−(1−‖g‖1)t|h|, (4.16)

for all h ∈ H . Moreover by (4.11) we deduce that

|〈uxx(t, x)h, h〉| ≤ c1/22 e

12 κ2t|h|2, (4.17)

for all h ∈ H . Now by (4.17) it follows easily that ϕ ∈ C1b (H) and

|ux(t, x)| ≤ ‖ϕ‖1e−(1−‖g‖1)t. (4.18)

It remains to estimate TR[uxx(t, x)]. We have

TR [uxx(t, x)] = Tr [E(X∗x(t, x)D2ϕ(X(t, x))X∗

x(t, x))]

+E [〈Dϕ(X(t, x)),TR[Xxx(t, x)〉] .(4.19)

It follows, by (4.5) and (4.13) that

TR [uxx(t, x)] = ‖Tr D2ϕ‖0 + ‖ϕ‖1c3eκ3t. (4.20)

Moreover, recalling that for all ϕ ∈ E(H) we have that Dϕ(x) ∈ D(A) for any

x ∈ H and that ADϕ ∈ Cb(H), we have

TA(u(t, x)) = 〈Ax, ux(t, x〉 = E[〈ADϕ(X(t, x)), ηh(t, x)〉

], x ∈ D(A).

So u(t, x) fulfills conditions (i)–(iii) of Definition 1.1. We can now prove

Proposition 4.7. u(t, x) is a strict solution of (1.3).



Proof. Let us introduce Galerkin approximations. For any n ∈ N we denote by Pnthe projector

Pnx =n∑

k=1

〈x, ek〉ek, x ∈ H

and set An = APn. Then we consider the approximating equation

dXn(t, x) = (PnAXn(t, x) + Png(PnX

n(t, x)))dt + PndW (t),

Xn(0, x) = Pnx,

(4.21)

which has a unique solution Xn(t, x). We define also

ηhn(t, x) = Xnx (t, x)h, ζhn(t, x) = Xn

xx(t, x)(h, h).

The following results are standard.

limn→∞

Xn(·, x) = X(·, x), in CW ([0, T ], H),

limn→∞

ηhn(t, x) = ηh(t, x), in CW ([0, T ], H),

and

limn→∞

ζhn(t, x) = ζn(t, x), in CW ([0, T ], H),

for any h ∈ H .

Moreover un(t, x) = E[ϕ(Xn(t, x))] is a strict solution of the approximating

Kolmogorov equation

unt (t, x) =1

2Tr [unxx(t, x)] + 〈APnx, uNx (t, x)〉 + 〈Png(Pnx), uNx (t, x)〉,

un(0, x) = ϕ(Pnx), x ∈ H.

(4.22)

Letting n→∞ we see that

limn→∞

un(t, x) = u(t, x), uniformly in t, x ∈ H,

limn→∞

〈PnAx, un(t, x)〉 = 〈x,Au(t, x)〉, uniformly in t, x ∈ H,

and

limn→∞

Tr [unxx(t, x)] = Tr uxx(t, x), uniformly in t, x ∈ H.

Thus

limn→∞

d

dtun(t, x) =

d

dtu(t, x) uniformly in t, x ∈ H

and (1.3) is fulfilled.


390 G. Da Prato

In an analogous way, using inequalities (4.16), (4.17), (4.20), we can prove the

result.

Proposition 4.8. Set

ϕ(x) =

∫ ∞

0

e−λtu(t, x)dt.

Then ϕ is a strict solution of (1.4).

References

[1] S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Fo-rum. 49(3), 349–367, (1994). ISSN 0037-1912.

[2] K. Yosida, Functional analysis. Die Grundlehren der Mathematischen Wis-senschaften, Band 123, (Academic Press Inc., New York, 1965).

[3] G. Da Prato and J. Zabczyk, Second order partial differential equations in Hilbertspaces. vol. 293, London Mathematical Society Lecture Note Series, (Cambridge Uni-versity Press, Cambridge, 2002). ISBN 0-521-77729-1.

[4] G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equa-tions, J. Math. Pures Appl. (9). 82(8), 877–947, (2003). ISSN 0021-7824.

[5] E. Priola, On a class of Markov type semigroups in spaces of uniformly continuousand bounded functions, Studia Math. 136(3), 271–295, (1999). ISSN 0039-3223.

[6] A. S. Nemirovski and S. M. Semenov, The polynomial approximation of functions inhilbert spaces, Mat. Sb. (N.S.). 92, (1973).

[7] G. Da Prato, A. Debussche, and B. Goldys, Some properties of invariant measuresof non symmetric dissipative stochastic systems, Probab. Theory Related Fields. 123(3), 355–380, (2002). ISSN 0178-8051.

[8] G. Da Prato, Kolmogorov equations for stochastic PDEs. Advanced Courses in Math-ematics. CRM Barcelona, (Birkhauser Verlag, Basel, 2004). ISBN 3-7643-7216-8.


[10] S. Cerrai. Classical solutions for Kolmogorov equations in Hilbert spaces. In Seminaron Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999), vol. 52,Progr. Probab., pp. 55–71. Birkhauser, Basel, (2002).

[11] S. Cerrai, Second order PDE’s in finite and infinite dimension. vol. 1762, LectureNotes in Mathematics, (Springer-Verlag, Berlin, 2001). ISBN 3-540-42136-X. A prob-abilistic approach.


Author Index

Blomker, D., 71

Borcea, L., 91

de Bouard, A., 113

Brzezniak, Z., 135

Cadenillas, A., 169

Chigansky, P., 197

Crisan, D., 221

Cvitanic, J., 169

Debbi, L., 135

Debussche, A., 113

Decreusefond, L., 249

Duan, J., 71

Flandoli, F., 263

Ghazali, S., 221

Gyongy, I., 281

Krylov, N. V., 1, 311

Kutoyants, Yu. A., 339

Liptser, R., 197

Mikulevicius, R., 357

Millet, A., 281

Nualart, D., 249

Papanicolaou, G., 91

Pragarauskas, H., 357

Da Prato, G., 375

Romito, M., 263

Rozovskii, B. L., 1

Tsogka, C., 91

Zapatero, F., 169

391




Subject Index

Bismut-Elworthy-Li formula, 271

Chapman-Kolmogorov equation, 267

coercivity, 282

coherent interferometry, 104

comparison principle for SPDEs, 314

cubature formula, 230

decoherence parameters, 104

effort, 172

exponential tightness, 200

filtering problem, 2, 239

fractional Brownian motion, 249

fractional Laplacian, 140

γ-radonifying operator, 157

generalized Stieltjes integral, 252

Gronwall’s lemma, 157

Hadamard’s formula, 35, 60

Hajek-Le Cam bound, 341

hemicontinuity, 282

homogeneous noise, 113

hyperviscous Burgers equation, 84

incomplete market, 182

Ito-Liouville equation, 97

Kirchhoff migration, 103

Kolmogorov equation, 5

LDP, 199

locally asymptotical normality, 341

Markov kernel, 267

irreducible, 268

stochastically continuous, 268

strong Feller, 268

martingale problem, 266

maximum principle for SPDEs, 314

mean energy, 72

measurability, 10

m-perfect family, 228

mild solution, 137

monotonicity, 30, 35, 282

Moore-Penrose pseudoinverse, 214

normal triple, 281

pointwise multiplier, 164

project, 172

Rao-Cramer bound, 341

rate function, 199

soliton, 114

stochastic parabolicity, 302

Stokes problem, 275

strict solution, 376

strong parabolicity, 59

Wald’s test, 349

Weyl derivative, 251

Wigner transform, 96

393

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Stochastic Differential - 213.230.96.51:8090