Bibliography

[83] S. ATTAL, K. BURDZY, M. EMERY, Y. Hu: Sur quelques filtrations et

transformations browniennes. sem. Proba. XXIX, Lect. Notes in Maths. 1613, Springer (1995),56-69.

[84] J. AZEMA: Sur les fermes aleatoires, Sem. Proba. XIX, Lect. Notes in Maths. 1123, Springer (1985), 397-495.

[85] J. AZEMA, C. RAINER: Sur l'equation de structure d[X, X]t = dt - XtAXt, Sem. Proba. XXVIII, Lect. Notes in Maths. 1583, Springer (1994), 236-255.

[86] J. AZEMA, M. YOR: En guise d'introduction, Asterisque, 52-53, Temps locaux (1978),3-16.

[87] J. AZEMA, M. YOR: Etude d'une martingale remarquable. Sem. Proba. XXIII, Lect. Notes in Maths. 1372, Springer (1989), p.88-130

[88] J. AZEMA, M. YOR: Sur les zeros des martingales continues, Sem. Proba. XXVI, Lect. Notes in Maths. 1526, Springer (1992), 248-306.

[89] J. AZEMA, TH. JEULIN, F. KNIGHT, M. YOR: Le tMoreme d'arret en une fin d'ensemble previsible Sem. Proba. XXVII, Lect. Notes in Maths. 1557, Springer (1993),133-158.

[90] J. AZEMA, TH. JEULIN, F. KNIGHT, G. MOKOBODZKI, M. YOR: Sur les pro­

cessus croissants de type injectif, Sem. Proba. XXX, Lect. Notes in Maths. 1626, Springer (1996), 312-343.

[91] M.T. BARLOW: Study of a filtration expanded to include an honest time, Zeit. fur Wahr. 44 (1978), 307-323.

[92] M.T. BARLOW, S.D. JACKA, M. YOR: Inequalities for a pair of processes stopped at a random time, Proc. London Math. Soc. 52 (1986), 142-172.

[93] M.T. BARLOW, J.W. PITMAN, M. YOR: Une extension multidimensionnelle de la

loi de l'arc sinus, Sem. Proba. XXIII, Lect. Notes in Maths. 1372, Springer (1989), 294-314.

135

136 Bibliography

[94] M.T. BARLOW, J.W. PITMAN, M. YOR: On Walsh's Brownian motions, Sem. Proba. XXIII, Lect. Notes in Maths. 1372, Springer (1989), 275-293.

[95] M.A. BERGER, V.J. MIZEL: An extension of the stochastic integral, Ann.

Prob. 10, 2 (1982), 435-450.

[96] J. BERTOIN: Complements on the Hilbert transform and the fractional derivative

of Brownian local times, J. Math. Kyoto Univ., 30-4 (1990), 651-670.

[97] J. BERTOIN: Excursions of a BESo(d) and its drift term (0 < d < 1), Prob. Th. and ReI. Fields 84 (1990), 231-250.

[98] J. BERTOIN: Sur La decomposition de La trajectoire d 'un processus de Levy spectrale­ment positif en son minimum. Ann. Inst. H. Poincare, vol. 27, 4, (1991) 537-547.

[99] J. BERTOIN: How does a reflected one-dimensional diffusion bounce back?, Forum Math, 4, 6 (1992),549-565.

[100] J. BERTOIN: An extension of Pitman's theorem for spectrally positive Levy pro­

cesses, Ann. Proba , 20, 3, (1992), 1464-1483.

[101] J. BERTOIN: On the Hilbert transform of the local times of a Levy process, Bull.

Sci. Maths 119 (1995), 147-156.

[102] J. BERTOIN: Levy Processes, Cambridge Univ. Press (1996).

[103] PH. BlANE: Relations entre pont brownien et excursion normalisee du mouvement

brownien, Ann. 1. H.P,22, 1, (1986) 1-7.

[104] PH. BlANE: Mouvement brownien dans un cone et theoreme de Pitman. Stoch. Processes and their applications 53, (1994), 233-240.

[105] PH. BlANE, M. YOR: Valeurs principales associees aux temps locaux Browniens,

Bull. Sci. Maths., 2e serie, 111, 23-101, 1987.

[106] J.Y. CALAIS, M. GENIN: Sur les martingales locales continues indexees par ]0,00[. Sem. Proba. XVII, Lect. Notes in Maths. 986, Springer (1983), 162-178.

[107] K.L. CHUNG: Excursions in Brownian motion, Ark Math. 14, 1976, 155-177.

[108] C. DELLACHERIE: Capacites et processus stochastiques, Springer (1972).

[109] C. DELLACHERIE, B. MAISONNEUVE, P.A. MEYER: Probabilites et potentiel, Chapitres XVII-XXIV: Processus de Markov (fin), Complements de Caleul stochas­

tique, Hermann (1992).

[110] C. DELLACHERIE, P.A. MEYER, M. YOR: Sur certaines proprietes des espaces Hi

et BMO, Sem. Proba. XII, Lect. Notes in Maths 649, 98-113, Springer (1978).

[111] V. DE LA PENA, N. EISENBAUM: Exponential Burkholder-Gundy inequalities,

Preprint (1996).

Bibliography 137

[112] P. DIACONIS, D. FREEDMAN: A dozen of de Finetti results in search of a theory,

Ann. LH.P., 23, (1987), Numero special Paul Levy, 397-423.

[113] C. DONATI-MARTIN, M. YOR: Some Brownian functionals and their laws. To appear in Ann. Prob. (1997).

[114] L. DUBINS, J. FELDMAN, M. SMORODINSKY, B. TSIREL'SON: Decreasing se­

quences of a-fields and a measure change for Brownian motion, Annals of Proba

24 (1996), 882-904.

[115] M. EMERY: On the Azema martingales, Sem. Proba. XXIII, Lect. Notes in

Maths. 1372, Springer (1989), 66-87.

[116] M. EMERY: Sur les martingales d'Azema (Suite), Sem. Proba. XXIV, Lect. Notes in Maths. 1426, Springer (1990), 442-447.

[117] M. EMERY: Quelques cas de representation chaotique, Sem. Proba. XXV, Lect. Notes in Maths. 1485, Springer (1991), 10-23.

[118] M. EMERY: On the chaotic representation property for martingales. To appear in the Proceedings of the Euler Institute (St. Petersburg). (1993).

[119] M. EMERY: Chaotic vs. predictable representation property. 1993 Summer Research

Institute / Stochastic analysis. Preprint (1993).

[120] P. FITZSIMMONS, R.K. GETOOR: On the distribution of the Hilbert transform of

the local time of a symmetric Levy process, Ann. Prob. 20, 3 (1992), 1484-1497.

[121] P. FITZSIMMONS, R.K. GETOOR: Limit theorems and variation properties for

fractional derivatives of the local time of a stable process, Ann. Inst.H.Poincare,

28,2, (1992),311-333.

[122] P. FITZSIMMONS, J.W. PITMAN, M. YOR: Markovian Bridges: Construction, Palm interpretation, and Splicing, in: Seminar on Stochastic Processes (eds. R. Bass, K. Burdzy), Birkhiiuser (1993).

[123] H. FOLLMER, P. IMKELLER: Anticipation cancelled by a Girsanov transformation:

a paradox on Wiener space, Ann. Inst. H. Poincare, Vol 29, 4, (1993), 569-586.

[124] M. FUKUSHIMA: A decomposition of additive functionals of finite energy, Nagoya Math. J. 74 (1979), 137-168.

[125] A. GOSWAMI, B.V. RAO: Conditional expectation of odd chaos given even,

Stochastics and Stochastic Reports 35, (1991),213-214.

[126] M. HITSUDA: Formula for Brownian partial derivatives, Second Japan-USSR Symp., Lect. Notes in Maths., Springer (1975),111-114.

[127] Y. Hu: Sur la representation des (Ft- = a{B;,s::::: t})-martingales, Sem. Prob. XXIX, Lect. Notes in Maths. 1613, Springer (1995), 290-296.

138 Bibliography

[128] Y. Hu: Sur le mouvement brownien: calculs de lois, etudes asymptotiques, filtra­tions, relations avec certaines equations paraboliques, These de l'Universite Paris

VI, January 1996.

[129] J.P. IMHOF: Density factorization for Brownian motion and the three-dimensional Bessel processes and applications, J. App. Proba. 21 (1984), 500-510.

[130] K. ITO: Extension of stochastic integrals, Proc. of Intern. Symp. SDE.

Kyoto (1976), 95-109.

[131] K. ITO, H.P. Mc KEAN: Diffusion processes and their sample paths, Sprin­

ger (1965).

[132] J. JACOD, PH. PROTTER: Time reversal of Levy processes, Ann. Prob., 16, 2,

(1988), 620-641.

[133] S.D. JACKA, M. YOR: Inequalities for non-moderate functions of a pair of stochas­tic processes, Proc. London Math. Soc. (3), 67, p. 649-672, (1993).

[134] TH. JEULIN: Un tMoreme de J. W. Pitman. Sem. Proba. XIII, Lect. Notes in

Mathematics 721, p. 521-532, Springer (1979).

[135] TH. JEULIN: Semi-martingales et grossissements d'une filtration, Lecture Notes in

Mathematics 833, Springer (1980).

[136] TH. JEULIN: Application de la tMorie du grossissement Ii l'etude des temps locaux

browniens, in: "Grossissement de filtrations: Exemples et applications", Lecture

Notes in Mathematics 1118, Springer (1985), 197-304.

[137] TH. JEULIN, M. YOR (EDS.): Grossissement de filtrations: exemples et applica­tions. Lecture Notes in Mathematics 1118, Springer (1985).

[138] TH. JEULIN, M. YOR: Inegalite de Hardy, semi-martingales et faux-amis, Scm.

Proba. XIII, Lecture Notes in Mathematics 721, Springer (1979), 332-359.

[139] I. KARATZAS, S. SHREVE: Brownian motion and stochastic calculus, Springer

(1988).

[140] J. KENT: Some probabilistic properties of Bessel functions, Ann. Prob. 6 (1978),

760-770.

[141] F.B. KNIGHT: Inverse local times, positive sojourns, and maxima for Brownian motion, Colloque Paul Levy, Asterisque 157-158 (1988),233-247.

[142] F.B. KNIGHT: A remark on Walsh's Brownian motions, Colloque en l'honneur

de J.P. Kahane, Orsay (June 1993). In: The Journal of Fourier Analysis and

Applications, Special Issue, 1995, 317-324.

[143] F.B. KNIGHT, B. MAISONNEUVE: A characterization of stopping times. Annals

of Probability, 22, (1994), 1600-1606.

Bibliography 139

[144] J. LAMPERTI: Semi-stable stochastic processes, Trans. Amer. Math. Soc., 104,

(1962), 62-78.

[145] J. LAMPERTI: Semi-stable Markov processes I, Z. fiir Wahr., 22, (1972),205-225.

[146] J.F. LE GALL: Mouvement Brownien, cones et processus stables, Prob. Th. ReI.

Fields 76 (1987), 587-627.

[147] S. MOLCHANOV, E. OSTROVSKI: Symmetric stable processes as traces of degener­

ate diffusion processes, Theo. of Prob. and its App., vol XIV, No 1 (1969), 128-13l.

[148] MONOGRAPH: Exponential functionals and principal values related to Brownian

motion. To appear in :Biblioteca de la Revista Matematica Ibero-Americana (1997).

[149] T. MORTIMER, D. WILLIAMS: Change of measure up to a random time: theory,

J. App. Prob. 28, (1991),914-918.

[150] D. NUALART: Anticipative stochastic calculus, Bull. Sci. Maths, vol. 117 (1993),

49-62.

[151] D. NUALART, E. PARDOUX: Stochastic differential equations with boundary con­ditions. In "Stochastic Analysis and Applications" , Proceedings of the 1989 Lisbon

Conference, eds: A.B. Cruzeiro, J.C. Zambrini. Birkhaiiser (1991), 155-175.

[152] I. PIKOVSKI, I. KARATZAS: Anticipative Portfolio optimization, Advances in Ap­

plied Prob. , To appear (1996).

[153] J.W. PITMAN: One-dimensional Brownian motion and the three-dimensional

Bessel process, Adv. App. Prob. 7, (1975),511-526.

[154] J.W. PITMAN, M.YOR: Bessel processes and infinitely divisible laws, in: "Stochas­

tic Integrals", cd. D. Williams, Lect. Notes in Maths. 851, Springer (1981).

[155] J.W. PITMAN, M.YOR: A decomposition of Bessel bridges, Zeitschrift fUr Wahr, 59 (1982), 425-457.

[156] J.W. PITMAN, M.YOR: Dilatations d'espace-temps, rearrangements des trajec­

toires browniennes, et quelques extensions d 'une identiU de Knight, Comptes Ren­dus Acad. Sci. Paris, t. 316, Serie 1, (1993), 723-726.

[157] J.W. PITMAN, M.YOR: Decomposition at the maximum for excursions and bridges

of one-dimensional diffusions. In: Ito's stochastic calculus and Probability theory,

eds: N. Ikeda, S. Watanabe, M. Fukushima, H. Kunita, Springer (1996),293-310.

[158] D. REVUZ, M. YOR: Continuous martingales and Brownian motion, Springer.

Second edition (1994).

[159] B. ROYNETTE, P. VALLOIS: Instabilite de certaines equations differentielles stochastiques non lineaires, Journal Funct. Anal. 130,2, (1995),477-523.

140 Bibliography

[160J J. RUIZ DE CHAVEZ: Le the-oreme de Paul Levy pour des mesures signees, Sem Proba. XVIII, Lect. Notes in Maths 1059, Springer (1984), 245~255.

[161J Y. SAISHO, H. TANEMURA: Pitman type theorem for one-dimensional diffusion

processes. Tokyo J. Math., 13, no. 2, p. 429~440 (1990).

[162J A.V. SKOROKHOD: On a generalization of a stochastic integral, Theo. Prob. Appl.

20 (1975), 219~233.

[163J L. SMITH, P. DIACONIS: Honest Bernoulli excursions, J. Appl. Proba. 25 (1988),

464~477.

[164J F. SPITZER: Some theorems concerning 2-dimensional Brownian motion, Trans.

Amer. Math. Soc. 87 (1958), 187~197.

[165J D.W. STROOCK: Probability theory: an analytic view, Cambridge Univ. Press

(1993).

[166J D.W. STROOCK, M. YOR: On extremal solutions of martingale problems, Ann.

Scient. ENS, Serie 4, t. 13, (1980), 95~164.

[167J K. TAKAOKA: On the martingales obtained by an extension due to Saisho, Tane­

mura and Yor of Pitman's theorem, to appear in Sem. Proba. XXXI, Lect. Notes in

Maths., Springer (1997).

[168J H. TANAKA: Time reversal of random walks in one-dimension. Tokyo J. Math.,

12, p. 159~174 (1989).

[169J H. TANAKA: Time reversal of random walk in ]Rd. Tokyo J. Math., 13, no. 2, p. 375~389 (1990).

[170J S. WATANABE: Generalized arc sine laws for one-dimensional diffusion processes and random walks, In: Stochastic Analysis, Proc. Symp. Pure Math., 57, 1995, 157~172.

[171J B. TSIREL'SON: An example of a stochastic differential equation having no strong

solution, Theor. Prob. Appl. 20 (1975), 427~430.

[172J P. VALLOIS: Sur la loi conjointe du maximum et de l'inverse du temps local du mou­vement brownien: application a un the-oreme de Knight, Stochastics and Stochastic

reports 35, 175~186 (1991).

[173J P. VALLOIS: Decomposing the Brownian path via the range process, Stoch. Pro­

cesses and their App., 55 (1995), 211~226.

[174J W. VERVAAT: A relation between Brownian bridge and Brownian excursion, Ann.

Prob. 7(1) (1979), 141~149.

[175J A.Y. VERETENNIKOV: On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR Sb. 39 (1981), 387~403.

Bibliography 141

[176] G.N. WATSON: A treatise on the theory of Bessel functions, Cambridge, Cambridge

University Press, 1944.

[177] D. WILLIAMS: Path decomposition and continuity of local time for one dimensional

diffusions I, Proc. London Math. Soc (3), 28, (1974),738-768.

[178] D. WILLIAMS: Brownian motion and the Riemann zeta function, In: Disorder in Physical Systems, Festschrift for J. Hammersley, eds. J. Grimmett and D. Welsh, Oxford (1990),361-372.

[179] T. YAMADA: On the fractional derivative of Brownian local times, J. Math. Kyoto Univ 25-1 (1985),49-58.

[180] T. YAMADA: On some limit theorems for occupation times of one-dimensional

Brownian motion and its continuous additive functionals locally of zero energy, J. Math. Kyoto Univ 26-2 (1986) 309-322.

[181] T. YAMADA: Representations of continuous additive functionals of zero energy

via convolution type transforms of Brownian local times and the Radon transform, Stochastics and Stochastics Reports, 46, 1, 1994, 1-15.

[182] T. YAMADA: Principal values of Brownian local times and their related topics. In: Ito's stochastic calculus and Probability theory, Springer (1996), 413-422.

[183] CH. YOEURP: ThCoreme de Girsanov generalise, et grossissement d'une filtra­

tion., In: Grossissements de filtrations: exemples et applications (ref. [137], above) Springer (1985),172-196.

[184] M. YOR: Sur la transformee de Hilbert des temps locaux browniens et une extension

de laformule d'Ito, sem. Proba. XVI, Lecture Notes in Maths. 920, Springer (1982),

238-247.

[185] M. YOR: Inegalites de martingales continues arretees Ii un temps quelconque, I, II, In : Grossissements de filtrations: exemples et applications, (ref. [137]' above) Springer (1985),110-171.

[186] M. YOR: Une extension markovienne de l'algebre des lois beta-gamma, C.R.A.S. Paris, Serie I, 257-260, (1989).

[187] M. YOR: Tsirel'son's equation in discrete time, Prob. Th. and ReI. Fields 91 (1992)

135-152.

[188] A.K. ZVONKIN: A transformation of the phase space of a diffusion process that

removes the drift, Math. USSR Sb. 22 (1974), 129-149.

[189] B. TSIREL'SON: Walsh process filtration is not Brownian, Preprint Aug. 1996.

[190] A. BORODIN, P. SALMINEN: Handbook of Brownian motion: facts and formulae,

Birkhiiuser (1996).

Index

Bridge: Bessel - 1-36, II-15 Brownian - 1-2, II-18

pseudo- - 1-125

Brownian motion: perturbed - II-129 randomized - II-91 skew - 1-99 Walsh - II-I07

Chaos: Wiener - I-I Wiener - decomposition 1-27

Decomposition: canonical - II-34

Doob-Meyer - 11-65 non-canonical- 11-118 semimartingalc - 1-47

Decomposition of paths: Vervaat - II-17 Williams - II-33

Distribution: Arc sine - 1-99, 11-9 Beta - 1-100

Gamma - 1-100 Hartman-Watson - 1-61

Enlargements of filtrations: initial - II-33 progressive - II-41

Equation: conditional - II-73 Langevin's - 1-11

142

Skorokhod's reflection - 1-116

structure - II-102

Equivalence:

Levy's - 11-78

past and future - II-113

Excursion: Ito's - measure 1-30

Master formulae of - theory 1-30

- measures 11-15

normalized Brownian - 11-14

- theory 11-9

Filtration:

Brownian - II-117

Goswami-Rao - 11-115

Formula: agreement - II-14

balayage - II-61 Feynman-Kac - 1-86

integration by parts - I-53 Ito's - II-58

Levy's stochastic area - 1-16

Tanaka's - 1-107

Function:

confluent hypergeometric - 1-48 gamma - II-12

moderate - II-51

modified Bessel - 1-60

non-moderate II-54

Riemann zeta - 11-11 theta - 11-11

Young - II-54

Functional:

additive - 1-121,11-7

Brownian - 1-15

- equation 11-11

quadratic - 1-18

skew-multiplicative - 1-31

Identity:

Chung's - II-16

Ciesielski-Taylor - I-50

Jacobi's - 11-12

Knight's - 1-124, 11-19

Kolmogorov-Smirnov's - 11-16

Index:

- of a Bessel process II-26

- of a stable process 11-4

Inequality:

Burkholder-Gundy - II-51

Fefferman - II-57

Hardy's - 1-9

Information:

loss of - 11-114

Integral:

multiple Wiener - 1-15, II-81

stochastic - 11-32 stochastic - representation 11-104

Intertwining: 1-74,1-84, II-88

Lace: Brownian - 1-64

Lemma:

Jeulin's - 11-39

"Poincare's" - II-55

Levy:

- equivalence 1-102, 11-78

- exponent 1-71

- measure II-40

- process 11-35

INDEX

Local times: Brownian - 1-27

intersection - 1-95

Martingale: Azema's first - 11-80 Azema's second - 11-80 BMO- - II-58 Emery's - 11-87

parabolic - 11-88

spider - 11-109

Meander: Brownian - 1-41

generalized - 1-41, II-127

Norm: Luxemburg - II-54 Orlicz - II-54

Number: Gauss linking - 1-87

self-linking - 1-94

winding - 1-88

Occupation:

density of - formula 11-49 - measure 1-12 - times formula 1-27

Options: Asian - 1-68, 11-128

Polynomials: Hermite - II-82

Laguerre - 1-5

Principle: transfer - 1-27, 1-35

Process: Bessel - 1-28 Cauchy - 1-64, 11-39 Dirichlet - II-3 increasing - II-65

143

144

injective - II-72 Levy - 1-75

optional - II-62

Ornstein-Uhlenbeck - 1-16 predictable - II-62

progressively measurable - II-62

stable - II-40

Property:

regeneration - II-75

scaling - II-20, II-34

strong Markov - II-24

Quantiles: Brownian II-128

Relation: Imhof's - 1-42, II-44

Representation: Ito's martingale - II-62

Lamperti's - of a semi-stable Markov process II-93

Pitman's - of the BES(3) process 1-28, II-118

stochastic integral- II-83, II-104 Vervaat's - of the Brownian

excursion II-16

INDEX

Reversal: time - 1-28 Williams' time - II-36

Semimartingale: 1-3 - decomposition 1-47

Set: end of a predictable - II-108

saturated - II-107 Sheet:

Brownian - II-126

Snake: Brownian - II-75, II-128

Space-time: - harmonic function 1-9, II-121

Supermartingale: Azema - II-41

Supremum:

- of Brownian bridge II-16 - of Brownian excursion II-16 - of Brownian motion 1-102

Theorem: Knight's - 1-107

Time: random - II-41, II-52 stopping - II-41

PROBABILITY THEORY • STATISTICS

LM • Lectures in Mathematics - ETH Zürich

M. Vor, I)niversite Pierre et Marie Curie, Paris, France

Some Aspects of Brownian Motion Part I: Some Special Functionals

1992. 148 pages. Softcover ISBN 3-7643-2807-X

The present notes represent approximately the first half of the lectures given by the author in the Nachdiplomvorlesung at the ETH (winter term 1991-92). Each chapter in the book is devoted to a particular dass of Brownian functionals: Gaussian subspaces of the Gaussian space of Brownian motion • Brownian quadratic functionals • Brownian local times • Exponential functionals of Brownian motion with drift • Winding numbers of one or several points, or straight lines, or curves • Time spent by Brownian motions below a multiple of its one-sided supremum

Roughly, half of the text consists of new results ; hence these notes may be placed midway between an advanced crash course on Brownian motion, and a complement to existing texts, to which precise references are given throughout. This volume will be of interest to researchers either in probability theory or in more applied fields , such as polymer physics or mathematical finance.

" ... the aim of the book ... to calculate as explicitly as possible the laws of Brownian functionaLs, ... has been magnificently achieved ... . the work will be a highly valuable resource for afficianados of Brownian motion. "

f. Kn ight, Univ . of Urbana, Illinois SIAM Review

" ... This book is a masterful account of the computation of laws of several important fuctionals of Brownian motion. It is largely based on author's own contribution to this jield over the last decade ... . "

For orders originating trom an aver the world ex ce pt USA and Canada: Birkhäuser Verlag AG P.O Box 133 CH-4010 Basel/Switzerland Fax: +41/61/205 07 92 e·mail: farnik@birkhauser.ch

For orders originating in the USA and Canada: Birkhäuser 333 Meadowland Parkway USA-Secaurus, NJ 07094-2491 Fax: +1 201 348 4033 e~mail: orders@birkhauser.com

JOURN AL OF THE INSTITUTE OF SCIENCE NOV. -D EC. 1994

Birkhäuser Birkhäuser Verlag AG Basel· Boston · Berlin

VISIT DUR HDMEPAGE http://www.birkhauser.ch

PA • Probability and Its Applications

R. Aebi, University of Berne, Switzerland

Schrödinger Diffusion Processes 1996. 194 pages. Hardcover ISBN 3-7643-5386-4

In 1931 Erwin Schrödinger considered the following prob­lem: A huge e/oud of independent and identical partie/es with known dynamics is supposed to be observed at finite ini-tial and final times. What is the "most probable" state of the e/oud at intermedi­ate times?

The present book provides a general yet comprehensive discourse on Schrödinger's question. Key roles in this investigation are played by conditional diffusion pro­cesses, pairs of non-linear integral equations and interacting particles systems. The introductory first chapter gives some historical background, presents the main ideas in a rather simple discrete setting and reveals the meaning of inter­mediate prediction to quantum mechanics. In order to answer Schrödinger's ques­tion, the book takes three distinct approaches, dealt with in separate chapters: transformation by means of a multiplicative functional, projection by means of relative entropy, and variation of a functional associated to pairs of non-linear integral equations.

The book presumes a graduate level of knowledge in mathematics or physics and represen,ts a relevant and demanding application oftoday's advanced probability theory. For orders originating frorn all aver the world except USA and Canada: Birkhäuser Verlag AG P.O Box 133 CH-4010 BaseljSwitzerland Fax: +4 1/ 61/ 205 07 92 e-mail: farn ik@birkhauser.ch

For orders originating in the USA and Canada: Birkhäuser 333 Meadowland Parkway USA-Secaurus. NJ 07094-249 1 Fax: +1 201 3484033 e-mail: orders@birkhauser.com

VISIT DUR HDMEPAGE http://www.birkhauser.ch

Birkhäuser Birkhäuser Verlag AG Basel · Boston · Berlin

PA • Probability and Its Applications

B. Fristedt / L. Gray, University of Minnesota, MN, USA

A Modern Approach to Probability Theory 1996. 776 pages. Hardcover ISBN 3-7643-3807-5

Students and teachers of mathematics, theoretical statistics and eco­nomics will find in this textbook a comprehensive and modern approach to probability theory, providing them with the background and tech-niques necessary to go from the beginning graduate level to the point of specialization in research areas of current interest. It presupposes only a rigorous advanced calculus or undergraduate real analysis course, together with a small amount of elementary linear algebra. The authors introduce the basic objects of probability theory (random variables, distributions and distribution functions, expec­tations, independence) , at the same time developing concepts from measure theory as required. They

then proceed through the standard topics in the subject, including la ws of large numbers, charac­teristic functions, centrallimit theorems, conditioning, and random walks. The latter part of the book concerns stochastic processes in both discrete and continuous time, with individual chapters being devoted to martingales, renewal sequences, Markov processes, exchangeable sequences, stationary se­quences, point processes, Levy processes, interacting particle systems, and diffusions. The treatment of these topics is sufficiently advanced to bridge the gap between standard material and specialized research monographs.

The book contains numerous examples and over 1000 exercises, illustrating the richness and variety that exists in the subject, from sophisticated results in gambling theory to concrete calculations in­volving random sets.

In order to actively involve the student in the mathematical theory, a portion of the exercises re­quest proofs of some of the easier results. All of the problems are designed to help the student pro­ceed beyond mere rote learning of theorems and proofs to a deep intuitive feel for the far-reaching im­plications of the theory. Solutions are provided for approximately 25% of the exercises.

Far orders originating frorn alt over the world except USA and Canada: Birkhäuser Verlag AG P.G Box 133 CH-4010 BaseljSwitzerland Fax: +41/61/205 07 92 e-mail: farnik@birkhauser.ch

For orders originating in the USA and Canada: Birkhäuser 333 Meadowland Parkway USA·Secaurus, NJ 07094-2491 Fax: +1 201 348 4033 e-mail: orders@birkhauser.com

VISIT DUR HDMEPAGE http://www.birkhauser.ch

Birkhäuser Birkhäuser Verlag AG Basel ' Boston ' Berlin

PA • Probability and Its Applications

A. Borodin, St. Petersburg, Russia / P. Salminen, Abo Akademi University, Turku, Finland

Handbook of Brownian Motion Facts and Formulae 1996. 476 pages. Hardcover ISBN 3-7643-5463-1

The purpose of this book is to provide

an easy reference to a large number of

facts and formulae associated with

Brownian motion. The book consists

of two parts. The first part, dealing

with theory, is devoted mainly to pro­

perties of linear diffusions in general

and Brownian motion in particular.

Results are given mainly without pro­

ofs. The second part is a table of dis­

tributions of functionals of Brownian

motion and related processes. The

collection contains more than 1500

numbered formulae.

This book is of value as basic reference

material for researchers, graduate stu­

dents, and people doing applied work

in Brownian motion and diffusions. It

can also be used as a source of expli­

cit examples when teaching stocha­

stic processes.

For orders originating from alt aver the wortd except USA and Canada: Birkhäuser Verlag AG P.O Box 133 CH-4010 Basel/Switzerland Fax: +41/61/205 07 92 e-mail: farnik@birkhauser.ch

For orders originating in the USA and Canada: Birkhäuser 333 Meadowland Parkway USA-Secaurus, NJ 07094-2491 Fax: + 1 201 348 4033 e-mail: orders@birkhauser.com

VISIT DUR HDMEPAGE http://www.birkhauser.ch

Birkhäuser Birkhäuser Verlag AG Basel ' Boston' Ber lin