Viscosity Solutions of Nonlinear Degenerate Parabolic Equations and Several Applications

Yi Zhan

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

@Copyright by Yi Zhan 1999

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For the Cauchy problem of a class of fully nonlinear degenerate parabolic equations.

this paper studies the existence,uniqueness and regularity of viscosity solutions: these

rcsul ts apply t O Hamilton-Jacobi-Beliman (HJB for short) equation,Leland equation and

equations of p-Laplacian type, which h d a lot of applications in 0Uld mechanics, stochas-

tir control theory and optimal portfolio selection and transaction cost problems in finance.

Further stildies are done on the properties of viscosity solutions of the abot-e models:

1 ). Bernstein estimates ( especially estimates ) and convexity of viscosity solutions

of the H-JB equation: 2). monotonicity in time and in Leland constant of the viscos-

i tu solutions to the Leland equation and the relationship between Leland solutions and

Black-Sclioles solutions; 3). the existence and Lipschitz continuity of the free boundaries

of viscosity solutions for f d y nonlinear equations ut + F (Du? D2u) = O . with p-Laplacian

eqiiation as model. Our study estends the application of viscosity solution theory and

aids in the qualitative analysis and numerical computation of the above models.

To construct continuous viscosity solutions. we m&e use of Perron Method and various

estimates by virtue of viscosity solution theory; we generalize Bernstein estimates and

Iiruzlikov's regularization theorem in time from smooth solutions to viscosity solutions;

our met Lod applies to initial boundary d u e problem.tbougli the estimates of uniformly

coiitinuous ~nociuli near the boundary need to be obtained and suitable viscosity sub- and

siiper-solutions need to be constructed; to study the Leland equation, we transform it into

s t audard form by Euler transformation and linear translation. then study the property of

the visrosity solutions by virtue of comparison principle ; to study the properties of the

free l~oundary of equations of p-Laplacian type: we employ comparison principle, reflection

pririciple. rnoving plane methocl and the construction of sub a d super solutions.

Key words and phrases:

iiorilinear degenerate equation, viscosi ty solution, cornparison principle

Perron met hod. HJB equation, Leland equation

Euler t ransforrnation, Black-Scholes equation, p-Laplacian

Lipschitz continuity free b o u n d q

ACKNOWLEDGMENTS

Tliauks are due to my supervisor, Prof-Luis Seco: his guidance and encouragement

have been invaluable assets.

Tliaukç also go to Professor Gabor Francsics: Professor Robert McCanq Professor C .

Srrlern. Professor &f .D.Choi. Professor Ian Graham and Professor T.S. Abdelrahrnan, for

th& carefully reading and commenting my thesis as well as preparing and attending my

oral esamination.

1 a m iudebted to the Department of Mathematics for providing an excellent environ-

niciit for learuiug aud working.

1 aiii grateful to Ida Bulat for her help during the years of my graduate study.

1 gratefully acknowledge the financial support of the University of Toronto and the

Goverunient of Ontario.

Firially. 1 would like to tLank my family and friends for their constant encouragement.

TO Bin Yu

Contents

O Introduction 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 Models and problems 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 Revien 5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.3 Results 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 .-\ rrangement 19

1 Viscosity Solution Theory of E'ully Nonlinear Degenerate Paraboüc Equa-

tions 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Preliminaries 20

. . . . . . . . . . . . . . . . 1.2 Cornparison principle and maximum principle 27

. . . . . . . . . . . . . . . . . . 1.3 Estimates of uniformly continuous moduli 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Existence 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Applications 45

2 Regularity and Convexity-preserving Properties of Viscosity Solutions

of HJB Equation 48

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Ideas 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Proof of the lemmas 5L

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Some Matrix -4lgebra 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Main Tlieorems 54

. . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conxrexity Preserving Property 60

3 Delta Hedging with Tkansaction Cost-Viscosity Solution Theory of Le-

land Equation

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2 Delta-hedging with Transaction Cost - F o d a t i o n of Leland Equation . . 68 3.3 Cornparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

-- 3.4 Esistence of The Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . / s

3.5 Properties of The Pricing Functions . . . . . . . . . . . . . . . . . . . - . . 76 --

3-51 ,Monotonicity in time t . . . . . . . . . . . . . . . . . . . . .. . . . . / /

3.5.2 Monotonicity in the Leland Constant . . . . . . . . . . . . . . . . . '78

4 Existence and Lipschitz Continuity of the Fkee Boundary

of Viscosity Solutions for the Equations of p-Laplacian Type 80

4.1 Properties of the support . , . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Lipschitz continuity of the interface . . . . . . . . . . . . . . . . . . . . . - 86

4.2.1 Basic lemmas - rnonotonicity and ~~vmmetricity of viscosity solutions 86

4.2.2 Lipschitz continuity in spatial variables and asymptotic symmetric-

ity of the interface . . . . , . . . . . . . . . . . . . . . . . . - . . . 88

4.2.3 Lipschitz continuity in time . . . . . . . . . . . . . . . . . . . . . . 91

References 96

A Perron Method 102

B Ascoli-Arzela Theorem on Unbounded Domain 106

C Notations

Chapter O

Introduction

Some problems of practical interest reduce to nonlinear degenerate evolution equations,

su& as the Hamilton-Jacobi-Behan equation( HJB for short) from stochastic control

theor!; alid the portfolio selection problem in finance, Leland's equation from option pric-

iug tkeory with transaction costs and the p-Laplacian equation from non-Newtonian fluid

clyriarnics. They do not in general have smooth solutions due to the possible degeneracy.

Becaiise of the noniinearity, it is in general difficult to define Sobolev weak solutions using

iritegration by parts formulae. The theory of viscosity solutions applies to certain equa-

tious of the form ut + F ( x , t. u. Du. D2u) = O . where F : Rn x [O. T ] x R x Rn x Sn + R.

a~icl Sn denotes the space of n x n symmetric matrices with the usual ordering. This

tlieory allows merely continuous functions to be solutions of fully noniinear equations of

sc~corirl order and ~rovides very general esistence and uniqueness theorems. and applies

to the above mentioned t h e e types of models.

Tlie purpose of this thesis is to present a new and unifying construction of the esistence

aucl iinicpeness theory of viscosity solutions for the above mentioned models, and to study

tlic properties of their solutions by virtue of viscosity solution method and estimating

terhiclues from Sobolev weak solution tlieory.

0.1 Models and problems

Ll'c d l stiicly the foilotving initiai value problem (often called Cauchy problem):

ut + F ( x , t , u , Du, D2u) = O in Q = Rn x (O, Tl.

LVe cal1 equation (0.1) degenerate parabolic if F satisfies the foUowing assumption:

(Fi) F ( z . t. =, q, ,Y) E C ( & ) .

F is degenerate elliptic, i.e. F ( x , t , z? q, X + Y) 5 F ( x , tt z, q, X ) WM/ 2 O

il-kere Sn denotes the space of n x n symmetric matrices with the usual ordering and

. T o = Q x R x Rn x S n , X ? Y E S n

If tliere are positive constants X and h such that

wliere t7-EP denotes the trace of the matrix Y , then we say that F is uniformly elliptic,

and the equation (0.1) is unzformh~ parabolic.

i I ï c also assume that F is proper, nameli-, F satisfies:

for 1 - 5 s . V ( r . t : r .q .X) , ( z . t . s .q ,X)E .JO-

The above equation includes the follovving three types of models as examples:

1. Hamilton-Jacobi-Bellman equation

HJ B equation appears in optimal control theory of stochastic difierential equations([L] . [Kr] ). and especially in the optimal portfolio selection problem in fina.nce[Du]; the general

forni of the HJB equation is:

ut + sup Lo(u) = o. a E A

X, are T L x r n real matrix functions in Q, a is sub index, A is a given set.

2. Leland Equation

2 f ( S t u-hrre f i = J1 + . 4 s i g n ( w ) , R is called Leland constant. It is noted that the above

equatioii is in backward form, a simple transformation s = T - t wiU change it into

the forward form. Equation (Le) is introduced by Leland [Le] to study the dynamic

liedging portfolios for derivatives in the presence of transaction costs; the formulation

of the mode1 d l be given in Chapter 3. If we just consider convex solutions, then the

ahove equation is reduced to a linear parabolic equation(B1ack-Scholes equation). We

are interested in studying its non-convex solutions; then (Le) is in general nodnear and

-1 5 1 is required such that parabolic condition ( F I ) is satisfied.We are also interested

in the asyniptotic property of the solution as A goes to zero and its relationship with

Black-Scholes solution. -4s h = 1. (Le) is actually degenerate parabolic, and can not

in general have classical solutions, but it is still amenable of being studied under the

frainework of equation (Le) ; as h > 1: for non-convex payoff functions. the equation

( Le) is mathematically ill-posed? i.e.. the evolution of a payoff function under equation

( Le) leads to exponentially large modes. Accordingly, the function f (S. t) develops huge

oscillations or blows up for t arbitrarily close to T. Thus the equation (Le) with terminal

c-oridition f (S? T) = f (S) has no solution for generic, non-convex payoff functions f (S).

To solve this problem, Avellaneda and Paras [A, Pl propose new hedging strategies that

can be used with h > 1 to control effectively hedging risk and transaction costs. The

strategies are associated with the solution of a nonlinear obstacle problem for a diffusion

equation. Although viscosity solution theory also applies to such type of problems. our

attention in tlus thesis is on Cauchy problem and we leave it to the future studies.

3. Equations of p-Laplacian type

( P L E ) ut = div([DulP-* Du). p > 2

of which F talces the form:

q a q F ( x , t , r: q. X) = -IqlP-2tr{[I + ( p - 2)-]X) lq12

n-liere. q ~3 (I is the tensor product of q; and more general form :

GW ut = div(g([Dul)Du)

of whicli F takes:

n-liere g C1 ( ( O , cm)) satisfying certain structure conditions, one of which is:

lVliat's more, we consider the aaisotropic version:

( A P L )

11-here Q = diag[leiqlp-', - . - ? le,qJp-'1.

The above equations describe the motion of fluids with large velocity and non-Newtonian

fluicls.(refer to [AsMa], [EsV], [An] and [PaPh].) W e will study the non-negative solutions

Ircaiise the function u in the equations generaily stands for physical quantities such

as temperature or concentration of fluid. One of the most important properties of p-

Laplacian equations is that its solutions have compact supports if the initial functions

do. i .e.. so-called property o f f i f a i t e propagation , which is caused by the degeneracy of the

equation. and is contrary to the property of infinite propagation speed of the classic heat

eqiiatioii. The t heory of viscosi ty sohtions allows us to seek corresponding properties for

more general equations (GLE) and (APL).

For the above models and equatioris. we will study the following problems:

l.uricler what conditions does the Cauchy problem (0.1) and (0.2) have unique viscosity

solution?

2.Hon- smooth are the viscosity solutions to H.JB equations under suit able structure

co~icli tions'?

Cari the convexity of viscosity solutions to (0.1) and (0.2) be preserved with the evo-

lirtiou of tirne'!

3.Doc-s there exist a unique viscosity solution for Leland equation for non-convex (not

riecessarily piece-wise linear) payoff function :> How does the solutions of Leland equation evolve with time and the Leland constant'?

-4s Leland constant goes to zero (namely, the transaction cost decreases to zero), does the

soliit ion to Leland equation converge to a solution to Baick-Scholes equation ?

4.Cnder what conditions do the solutions to p-Laplacian equations (GLE) and (APL)

have free bonndary wit h Lipschitz continuity?

In tlie following sections of this introduction, we wiU firstly review some basic facts

of viscosi ty solution theory; then introduce the background of the above mentioned three

types of rnodels: after that we present the main resdts and the arrangement of this thesis.

0.2 Review

1 .A brief review of viscosity solution theory

(1). The definition of viscosity solution

The viscosity solution was introduced by Lions and Crandall [CL] in 1983 when

the- studied Haniilton-Jacobi equations ut + H ( z , t , u, Du) = 0' its name nias obtained

fro~ii 'vatzichzng uiscosity rnethod (Le.. approximating the solution of ut + H ( x , t t u , Du) = O

by a secluence of solutions to ut - EAU + H ( x . t . u, D u ) = O as E -+ O.) It was later ex-

tzuded to general second order equations by Lions [LI and quickly found applications in

riiauy fields. The viscosity solution theory is amvng Lions' Fields Medal winning works.

To iiiake tlie notion clear, n-e begin by assuming that u is in C 2 - ' ( Q ) a d

ut (x ' t ) + F ( r . t . u ( x , f ) . D u ( x , t ) , D ~ U ( X . ~ ) ) 5 O

for al1 (s. t ) E Q(i.e. u is a classical subsohtion of ut + F = O. and F is degenerate

paral~olic). Suppose that y E C2v1(Q) and (2.F) is a local maximum of u - y in Q : theu

U , ( X i) = ; J i t t). DU@. t) = D ~ ( F , and D 2 u ( ~ : i ) 5 D 2 i ? ( ~ . t ) : by ( f i ) .

The inequali ty

does not depend on the derivatives of u and so we may consider dcfining an arbitrary

friiiction n to be (some kind of generalized) subsolution of ut + P 5 O if

ivlienever r, E C2*' (Q) and (2. t) is a local maximum of u - p. This is the definition of

\-iscosi ty subsolution, the definition of viscosity supersolution can be given andogously

(just replace 'maximum' with 'minimum' and '5' with '2'). The basic idea of viscosity

sol~itiou is to transfer the 'derivatives ' of the solutions by test functions via maximum

priliriple. namelx to replace the 'derivatives' of any order of u with those of the smooth

function 9 at the local maximum or minimum points of u - y; people familiar with the

cldinition of Sobolev weak solutions will find this idea to be very intuitive. This definition

filids au quivalent which can be described with so-cded sub- and super- difierential . For esample. u is a viscosity subsolution to (0.1) iff

1 ~ ' . + u ( . r . t ) = ( ( r . q . A) E R x Rn x Snlu(x + h. t + s ) 5 u(x. t ) + rs + (p. h ) + - (+4h.h)

2

D2.+ u( r . t ) is called super-digerential of u at (z, t ). The sub-differential is defiued as

R 2 - ~ ( s . t ) = -D2-+(- u(x' t ) ) . Details can be foiind in [CIL] and in the first section of

Chapter 1( Definition 1.1 and 1.2).

Frorii above discussion. we naturally require that the viscosity solutions be continous

to giiarantee the existence of local maximum or minimum of u - 9: however. the continuity

i-au I>e relaxed and "weak viscosity solutions" can be defined, just as Isliii did in [Il] ivhile

studying the existence of the viscosity solutions; we will also notice this in Section 3 of

uc3st chapter.

( 2 ) . Cornparison principle

Given the concepts of sub- and super-solutions, we can study their relationship . We

say conparison principle between viscosity sub-solution u(x , t ) and super-solution v(x, t )

holdq if u (z . O) 5 v(z, O ) implies that u(x, t ) 5 u ( x , t ) in Q. This is actually an extension

of the maximum p h c i p i e . whick says that. if u is a solution of (0.1) and (0.2): then the

rnasimiirn of u can be bounded from above by the maximum of initial value u(x , 0) and

ot her paraaieters depending on F.

The basic idea to derive comparison principle is to estimate the function 9(x, y. t ) =

ii(r. t ) - u ( y , t ) - q; the main technique lies in how to apply the condition (Fi).

In 19SS. Jensen [dl] observed thato after suitably regularizing u and u. we can find

tn-O rnatices ,Y, -Y E Sn, which are respectively the second order super- and sub- dif-

fereutiaI(whose definitions will be given later). so that X O

,Y + Y 5 0. then (Fl) can be used. Using this idea. .Jensen established the cornparison

priuciple of W1sp viscosity solutions for a class of elliptic equations with F indepen-

dent of x : then, .Jensen. Lions and Souganidis [JLSo] studied the cornparison principles

of Boundea uniformly continuous (BUC ) viscosi ty solutions for the eUiptic equations

witli the forni of F ( z , z , q, -Y) = G(X) + H ( x , z, q ) . IshiiEIl] refined Jensen's idea and

s t ildieci the comparison principle of semi-continuous viscosity solutions on bourided and

iinbouncled domains for F of which the coefficient of the second order terni depends on

x. Ishii and Lions[IL] summarized the resiilts and concentrated on studying unifomly

d ip t i c equations: Though the study of [IL] and [Cl]. the crucial idea for comparison

priiiciple finds a ver' explicit and clear description in [CI]. whicli is a lemma on the

st riictuïe of super-differentials of semi-continuous functions, and will be rest ated in $1 of

Chapter 1 (see Lemma 1.9).

Parabolic equations generally can be studied in a way analogous to eUiptic equations,

biit they have their own properties. The method for studying parabolic equations was

iiirlitiouecl in [ILj. Dong and Bian [DBl] studied the initial boundary value prohlem on

I~oiiucled domains: Cauchy problems were studied mainly for the geometry equation ut =

1 DU ldi o(-) : Chen. Giga and Goto [CGG] studied the comparison principle between ID4 sii11- and super solution with compact supports for F independent of x by virtue of the

iiietliod in [IL]: Giga. Goto. Ishii and Sato[GGIS] studied the comparison principle of

viscosity soiutions growing linearly at infinity on unbounded domain, they made many

assiirnptions on F with the geometry equations as models, two of a-hich are:

(F7j implies tha.t F is locally u i i i fody continuous in q? (F4)' describes the continuity of

F iri x. incorporating the basic techique lemma ( rf. Lemma 1.9 in Chapter 1). In this

paper. we will obtain the sarne cornparison principle with milder conditions. In particular,

ive will replace (0.4)' with the following inequality:

V u > O. a F is independent of x. y, t , S ? -Y. Y. p, U. a, a .

To lie clear. w e restate this new condition as following:

-APpIying to H.JB equations, (0.4)' requires tkat b, satisfy: l b , ( x l t ) - b , ( ~ , t ) ] < L [ x - Y I , ivhile . to satisfy (F4). ha only needs to satisfy:

(B) is xiiilder than Lipschitz continuity . e-g. b, (x. t ) = (bixpl , . . . : b,,zcn ), (b; 2 O, ai E

(0.1). i = 1.. . . . r t . ) satisfies (B) but does not sat i s l Lipschitz continuity .

(3). Existence

There are rnainly two methods in studying the existence of viscosity solutions:

S

(1). approximate method via the stability property. (refer to Proposition 1.8 (

chapter 1) later, which says that if {u,) are a series of viscosity solutions to equation

uCt + Fc = O, where Fc + F as E 3 0, and u, + u as E -+ O, where u is a continuous

function. then u is a viscosity solution to ut + F = O.) It is essential to prove that

the set (u.) is compact. From Ascoli-Arzela compactness theorem, we only need

to prove that this series is bounded and unifordy continuous. namel_v: there is a

continuous modulus m independent of E such that ( u , ( x , t ) - u , ( y . s ) 1 < m(lz - y 1 + It - sl).One example of the application of this method is 'vanishing viscositÿ

method' in [CL], where, to construct solution to HJ equations u t + H ( x . t. u. Du) = 0.

approsimate second order equations ut - EAU + H = O are studied and relevant

boundedness and uniform continuity of the viscosi ty solutions are ob t ained and

applied to get the existence of the solutions to HJ equations;

( 2 ) .Perron method. Ishii [I2] reduces the existence of continuous viscosity solution

to the construction of viscosity sub- and super- solutions talcing the same value

at the boundary and initial time(these sub- and super- solutions are often c d e d

baeer functions). By Perron method, Chen. Giga and Sato [CGG] proved the

global existence of viscosity solution for the geometry equation: the crucial point

here is to construct suitable barrier functions. For completeness. we present details

of Perron method for parabolic equations in Appendix B.

Iri this article. Ive will obtain the existence of viscosity solutions for (0.1) and (0 .2) by

ro~nbiniug the above two methods . Our idea is: firstly Ive get the existence of viscosity

sollitions for initial functions uo in w2*m(Rn) by the Perron method? then for uo E Co(Rn)

arict uo E B U C ( R ) by approximating uo with smooth functions and using the estimates

of iiuiformly continuous moduli as well as the stability property.

(4). Estimates of uniformly continuous moduli

The local and global Holder, Lipschitz continuity of viscosity solution of Dirichlet

prol~leiu for uiiiformly elliptic equations were studied in [IL], mainly Ly virtue of viscosity

solut ion tecliniques: the C'va regulari ty was first ob tained by CafFarelli [Cal] for uniformlÿ

rlliptic equations; Wang[W] extended Caffarelli's method to uniformly ~arabolic equa-

tioiis: folloiving Caffarelli's method, Dong and Bian[DBi2] and Chen[Chl] stuclied Cl*"

rcgularity for a ciass of uniformly elliptic (parabolic) equations under various structure

coudi tions.

The regularity results for degenerate equations are comparatively fewer. For degener-

ate equations. Ivanov [IV] introduced some results on local and global estimates of gradient

of solritions . mainly by virtue of the construction of barrier functioris; For fully nonlin-

r a r degenerate equations. Ishii [Il] got the estimates of uniformly continuous moduli of

\-isc-osity solutions on unbounded domain depending on the continuous moduli of F and

tlie continuity of solutions near boundary. His method uses some ideas in [Bra]. In this

paper. t hc techniques in [Il] combined mith viscosity solution theory will be used to get

the estimates of uniformly continuous modulus depending on rno and the maximum

of the solutions. For a class of equations with F independent of x: as stated before, our

cornparison principle in fonn of maximum principle gives the explicit dependence of the

riio duli.

To obtain the regularity estimates in time, we will extend Kruzhkov's regularity theo-

r e m to viscosity solutions from classical solutions(Theorem 1.3.2)' this theorem discloses

tlie relationship between the regularity of solutions in space variable and the regularity

of solutions in time, namely, if we h o w that sohtion u(s , t ) to equation (0.1) is Holder

(-oiltinuous with respect to x, then we claim that, u is also Holder continuous with respect

to f.

To achieve that Ive will take Kruzhkov's condition about F as foLlowing:

wlirro -1 is nondecreasing in Iql, Xij is the i jth entry of the matrix S: assume that there

rsists a Y > O. s e t .

S u i e that (0.5) requires that F grow in order of 1qIa and lxlP as Iql? 1x1 goes to infinity

for certain nuniber a, 0, for example:

satisfying (Fs ) , i t is noted that natural structure conditions (0 = 1: -1 4 a + 2) are special

cases in the above inequality.

The C l q a estimates for degenerate equations are difficult to study: the classicai method

is Bernstein estimating method: however. to apply this niethod? higher regularities on the

c-qiiat ions are required. which excludes many equations with non-smooth coefficients. It is

ueressary to generalize this method to more general equations. Viscosity solution theory

allows us to achieve this goal for a class of nonlinear degenerate equations-Hamilton-

.Jacohi-Bellman equations, Chapter 2 wili be devoted to this topic.

2.HJB equations

The Control of Ito Process and Hamilton-Jacobi-BeUman Equation

The control of Ito process is the basis for the analysis of portfolio optimization problem.

Li-ben the related parameters are "smoothnt Ito's Lemma. Bellamn's Principle of Dynamic

Programniing, and the Markov property of the Browpian Motion reduce the stochastic

coutrol problem to a de t edn i s t i c problem: Hamilton-Jacobi-Behan equation.

Ilé briefly recall this deduction. Details can be found in [Du] and [LI.

Consider the following Ito Process

siicli that the expectation

V C ( q 7)

ilsists auci

Hcrc E,, denotes expectation under the probability measure governing X for starting

point r and control c.The primitive b c t i o n s p, a: u and r of ( a ' x . t ) E -4 x R" x [O'T]

are to satisfy certain regularity conditions[Du] ; the notations are explainecl as follows:

n).ll;, = (M/-l. - . IVi''-) is a standard Brownian Motion in R"

l>).Let Z Le the state space, a meastuable subset of Rh-

c ) . p is a rneasurable ~ ~ - v a l u e d function on .4 x Z x [O, T ] , A is a measurable subset

of Euclidean space

d).a : -4 x Z x [O, Tl + M ( K , N) is measurable, where M(h; N) is the space of li x N

matrices

e) .u is a measurable real-valued function on -4 x Z x [O, Tl

f ) . r is positive scalar discount rate

g ) . C is a set of predictable control process taking value in A

l i e also assume that

T h above defined function is c d e d the value function: if a control c0 E C such that

V ( x , Î ) = V c o ( x 7 T ) t/ (2, T ) E Z x [O, T ]

tlien q would be an opt-lmal control.

According to the Behan's Principle of Optimality [Du]. under re,darity conditions.

for arry (x. 7 ) E Z x [O' Tl the value function

is a solution of the following Hamilton-J acobi-Beban equation

This is a nonlinear equation. if w e just require the non-negativity of a.the equation

i d 1 also be degenerate and we can not generally expect analytic solutions. then viscosity

solution t heory applies.

The viscosity solution theory of HJB equation

-4s stated before. by Dynamic Programming Principle(DPP for short). the value func-

tion is a solution of HJB equations, however, it requires more regularities of the value

fiiiiction to test DPP(see [LI and [Du]).Before 1979. HJB eqüations were studied maidy

1)'- probabilistic method; after then. Krylov et-al developed some analysis method based

ou PDE theory(refer to [Kr] and references therein) , but they only considered convex so-

lutions or solutions with bounded second order derivatives and assumed that F is convex.

Krylov obtained the existence and uniqueness of concave solutions of Cauchy problem

with al1 coefficients in C2(Q) and I I xa. bu[[ 5 C, Ca 2 O and cm grows Linearly in x at

infinity (see theorems in 57.3. p329 in [Kr]). In [Il, Lions showed that the continuous

value function is a viscosity solution of HJB equations, which filled the regularity pap.

tliiis viscosity solution is a correct definition of the solution of the HJB equations. By his

inetliod. Lions got the existence and uniqueness of viscosity solutions for Dirichlet probleni

1)y assuiiiing that II Ca. bu, calJrv=.- < CG, in f c, > O. f, E BUC(Rn) . The semiconcavity

of viscosity solutions nas got in [IL] by viscosity solution method. Assuming that b,

are uniformly Lipschitz continuous in X . c,. fa E BUC(Rn) and in fc, > O. Ishii[Il] got

the cornparison principle for Dirichlet problem and got the existence by Perron method.

In this paper. our conciitions for uniqueness relaxes the Lipschitz continuity of O, as

3c > O. s-t. < ( b a ( x . t ) + ~ ~ ) - ( b m ( y - t ) + ~ ) > Z 0:no new conditions areneeded

for l i ( r . 0 ) E W2qm or u(x .0 ) E Co. especially no convexity of F is assumed to get the

existence. Yote that the Lipschitz continuity and scmiconcavity of viscosity solutions c m

t ~ e gor under corresponding assump tions on the initial function and the coeficieuts. this

will l>e done in Chapter 2.

3. Viscosity Solution Theory of Leland Equation

The s t itdy of optimal consump t ion and investment in continuous-t irne finacial models

ivas started hy Merton in a series of pioneer papers([Ml] and [M2] ). The application

of coritiniious-time models led to a quite satisfactory arbitrage pricing tLeory for no-

transaction cost, complete market models ( Black 91 Scholes [B,S]. Iireps [Krepj etc).

However in practice, transaction costs can not be overlooked in many cases, people

ueed to study the problem of optimal comsumption and investment in the presence of

truisaction costs to seek a model which has solid empirical support. This motivated Le-

lcud in 1985 [Le] to introduce the Leland equation to incorporate the transaction cost into

Black-Scholes analysis of option pricing theory. In a complete financial market without

t rausact ion cos t , the Black-Scholes equation provides a hedging portfolio t hat replicates

the contingent claim, which, however, requires continuous trading and therefore, in a mar-

ket with proportional transaction costs, it tends to be inhitely expensive. The require-

meut chat replicating the value of option has to be relaxed. Leland [Le] considered a model

that allows transactions only at discrete times. By a forma1 Delta-hedging argument he

derived au option price that is equa! to a Black-Sciioles price with an augmented-volatility

rvlicrc; -1 is Lelaad constant and is equal to fi-& and v is the original volati1ity.k iç the

proportional transaction cost and 6t is the transaction frequency. and both dt and k are

assumed to be s m d while keeping the ratio k / J s t order one. He got the above results

for coiivex payoff functions fa(S) = (S - Ii)+.where I< is the strike price of the assets.

Le also assumed that 11 is small(e.g. < 1). For non-convex payoff functions(e.g. for the

payoff of a portfdio of options-like bull spread and butterîly spread), Leland equation can

uot bc reduced to Black-Scholes equation and Leland equation is nodinear. and generally

n-c cau riot h d analytical solutions.

Hoggard et.al[HWW-] gerieralized Leland's work to non-convex(piece-wise linear) pay-

off fuuction with 21 < 1, -4lbanese and Tompaidis studied smail transaction cost asymp-

totics under several hedging strategies [..\.SI: as A > 1. the coefficient of the second

clerivativc xnay be negative and thus the Leland equation is ill-posed. -4s A = l 1 the Le-

l a d equation is a degenerate parabolic equation and rnay not have classic solutions. so for

-1 2 i.A4vellauda and Paras introduced new model to describe the d p a m i c hedging prob-

lciii [.A .Pl : Zariphopoulou et al considered the preferences of inves tors to incorporate t rans-

action costs into the optimal comsumption problems( see Davis,Panas & Zariphopoulou

[D .P.Z] . Davis & Zariphopoulou [D .Z] and G-Constantinides &- Zariphopoulou [C.Z] etc):

herr we discuss Leland equation for A 5 1 and establish the viscosity solution theory for

rion-convex(not necessarily piece-wise linear) payoff function.

N:î will study the follotving Leland equation

wliere fo is the payoff function which may be non-convex, e.g..the payoff of a portfolio

of options.like bull or buttedy spread. We will derive the existence and uniqueness of

its viscosity solutions for payoff function fo(S) with linear-growth at infinity and for

-1 5 1. We also study the properties of viscosity solutions of Leland equation and their

relationship tvith solutio~s of Black-Scholes eqiiation.

The tradi tiond method to get the existence of solutions is construct value function and

prove that it is a solution of Leland equation. however, strong regularity conditions are

r~cpired. Our method will be of pure PDE analysis; we maidy tvant to apply the results

cstablished in Chapter 1 to tlie Leland equation. However we have two main difficulties:

one is tkat the terminal function is possibly linearly growing at infinity; the other is that

the coefficient of the second order deritative is not lineady growing at iïlfinity. What we

lia\-e done to overcome these two difficulties is to observe that any linear homogeneous

furiction is a solution of Leland equation and use Euler transformation to reduce (Le) to

an equation of the form ut + r u + F(Du, D2u) = 0.

-4fter obtaining cornparison principle,we can easily study some properties of viscosit-

sciltitions to Leland equation, including the monotonicity of option price in time t and in

the Leland constant,and the relationship between the Leland solution and Black-Scholes

soliitiori gives us some knowledge about the role of the transaction cost, and also provides

a iisefril estimates of solutions to nonlinear Leland equation by Black-Scholes analytic

solutions.

4.Some properties of Mscosity solutions for equations of p-Laplace type

The p-Laplacian equation(PLE for short) was first studied in [BI: where Barenblatt

comtriicted for p > 2 a class of self-similar solutions with finite propagation velocity . The existence and uniqueness of Sobolev weak solutions for (PLE) can be found in

[LSC]. aiid in [dBH] with measures as initial functions. The study of (PLE) concentrates

ou tlie local and global Ca, C'va regularities, some of whicli are extended to quasi-linear

~quations of divergence type with second natural structure conditions(see [Ch21 and ref-

rreiices t herein).

and Esteban([EsV]) studied the properties of strong solutions for (PLE) of

1-diniensional. the estimate ( t; IVr l~-*), 2 - f plays the crucial role. In high-dimensional

case. the finite propagation nras got by Diaz & Herrero in [DA] for (PLE) and (APL),

Zhao and h a n [ZY] got the Lipschitz continuity of free b o u n d q for (PLE).Their method

follows that in [ C W ] : employing the special structure, the self-similar solutions and some

Lasic estimates of solutions for (PLE). Many techniques developed by Caffarelli, Vazquez

L LVolanski in [ C m ] for studying the regularity of free boundary of the solutions for

Poroiis Medium Equations. can be applied to more general equations.

In this paper, we will stucly the following equations using viscosity solution theory:

for wliicli. the existence and uniqueness of viscosity solutions are the results in Chapter

1. To proceed . we assume that:

/(SI(, = llS+II+IJX-II, X = X i + X - , X + > O , X - g l - g satisfies (G) in section 2:

Cnder (F6). Ive get the properties of finite propagation speed and positivity of viscosity

soliitiom by constructing suitable sub- and super solutions.

To stiidy the regularity of the free boundary, we introduce the condition:

wliîrr T = I - 2 n W n o n ~ Rn and In1 = 1.

This coudition guarantees that the viscosity solution remains to be a sub-solution

imder reflection transformation I'.

ilTe also introduce:

(Es ) F ( p 7 AX) 2 A F ( p . X) VA 2 O ( p , X ) E Rn x Sn

this condition actually requires that F is quasi-linear; whch will be used to derive the

Lipschitz of the free bouiidary in time.

Under (F; ) . applying cornparison principle and reflection principle. we get the mono-

tonicitl- of viscosity solutions and the regularity of free boundary with respect to spa-

tial tariables(Proposition 4.2.2 and Theorem 4.2.4)? we also get the asymptotic spher-

ical sy~mnetricity (Proposition 4.2.5). This geometric method follows Caffaralli et-al's

idea in studying the regularity of free boundary of solutions to porous medium equa-

tious( [CVW] .[GXNi]).

To get the Lipschitz continuity of free boundary in tirne: we require that uo E W2@.

aiid get the estimates of viscosity solutions . a h k h plays the role of (KI I/;IP-~), 2 -1 t in Vazquez and Esteban's work([EsV]).

I t is easy to test that the above conditions are satisfied by (PLE) and (GLE). (F6) is

oiily usecl to derive the properties of finite propagation and posi t ivi t~ other conditions

are comparatively general.

Sote that . the regularity results of free boundw hold after the support of viscosity

solution move outside a sphere containing the support of the initial function. This involves

t ilc s t udy of the wait ing t ime, while we leave it open due to the generality of the equations.

Filially it is pointed out that the following idem apply to any weak solutions:

( 1 ) .cornparison ~rinciple +reflection principle-+ the asymp totic spherical symmetric-

ity

(2).comparison principle +reflection ~rinciple + the existence of the free boundary

-i the Lipschitz continuity of free boundary in spatial variables

(3).coinparisori ~rinciple +reflection principle + the existence of the free boundary

+local 1V17' .- estimates-+ the Lipschitz continuity of free boundary in time.

0.3 Results

Tlierc are four results:

1.Couiparison principle estimates of unifordy continuous moduli and existence of

viscoçity solutions for Cauchy problem of (0.1) under geueral structure conditions for

l~oiiiidrd uniformly continuous initial functions (see Theorem 1.2.1, 1.3.1 and 1.4.1) : ap-

plication of these results to HJB equations (see Theorem 1.5.1); equations of p-Laplacian

type( (PLE), (GLE) and (APE))(see Theorem 1.5.2), and Leland equation (Le) in Chap-

ter 3(Theorem 3.4.1 and Theorern 3.4.3)

2.Extension of Ishii and Lions' techniques [IL] for studying serniconcavity of viscos-

ity solutions of static HJB equations to the Bernstein estimates of viscosity solutions of

parabolic HJB equations, especially, the CL*" regulari ty of solutions (Theorem 2.1 . The-

oreni 2.3 and Theorem 2.4): finally, generalization of convexity-preserving property to

uonliuear non-homogeneous equations(Theorem P. 7) . from homogeneous ones ( [GGIS] ) . 3.-Application of the techniques and results in Chapter 1 to Leland equation (Le), by

t rausforrning t ke Leland equation into the "standard' form, for get t ing the cornparison

priuciple and regularity (Theorem 3.3. l ) , then the existence of a class of non-convex con-

tinuous viscosi ty solution (Theorem 3.4.1); relaxiation of the constraints of non-convexity

and piece-wise-linearity on payoff functions; finally, the properties of the viscosity so-

liitions and tkeir relationship with solutions of Blacli-Scholes equation(Theorem 3.5. l:

Theoreln 3.5.2 and Theorem 3.5.3)

4.Esistence and Lipschitz continuity and the âsymptotic sphericd symmetry of free

hoiindary of viscosity solutions for Cauchy problem of equations of p-Laplacian type under

assumptions (G) and some structure conditions on F. (seeTheorem 4.1.3, 4.2.4, 4.2.6 and

Proposition 4.2.5).

Coxnpared wi th the we& solution theory, there are several characteristics for viscosity

solut ion theory:

( I ).viscosity solution theory is simple, insight and elegant. It consists of only one defi-

~ i i tion. one property (stability). one lemma (Jensen-Ishii- Crandall-Lions) and one method

(Perron). it provides an efficient way to study PDEs without too many techniques and

provides a complete theory for H J equations and uniformly elliptic equations. Cornparison

principle is one of its most important results, it enables us to study the properties of solu-

tious wi t hout construting special solutions for general nonlinear equations. Many results

cari 1)e estended to viscosity solutions from classical solut ions under milder conditions.

(2 ) .The main disadvantages of viscosity solution lie in that: the test function 4 con-

riccts the solutions u only at the 'match points'(the maximum of u - 4): taking relatively

less ififormation from solutions; it is in general required that test functions are in C2*',

wliicli liniits people to construct functions matching the regularity of the solutions; On

t ke otlier Iiand, it is difficult to employ integration operation to viscosity solutions and it

is hard to use the established estimating techniques in Sobolev solution theory to study

the properties of the solutions in detaii. It is noted that Caffarelli and Trudinger etc

have made some refinements for the defini t ion of viscosity solutions and introducecl some

original ideas(see [Cal], [Tl and [ES]).

0.4 Arrangement

1 .Chapter 1 is devoted to the cornparison principle ? the estimates of uniformly con-

tiriiious niodulus and the existence of viscosity solutions for (0.1) and (0.2). and the

application of t hese results to HJB equations and equations of p-Laplace type:

2.Cliapter 2 studies the regularity of viscosity solutions of Cauchy problem of H.JB

eqiiat ions: similar techniques are used to study the convexity-preserving property of t ke

viscosi ty solut ions of nonlinear nonhomogeneous degenerate equations:

3.Viscosi ty solution theory established in Chapter 1 is applied to Leland equation

1 ) ~ tra~isforniing Leland equation to the standard form: the monotonicity of the pricing

fuuction in time and Leland constant is studied. also the convergence of Leland solution

to Black-Scholes solution as Leland constant goes to zero is proved (Ckapter 3):

3.Ckaptcr 4 studies the existence and Lipschitz continuity of the free boundary of

viscosity solutions with (PLE) and (GLE) as models.

Chapter 1

Viscosity Solut ion Theory of Fully Nonlinear Degenerate Parabolic Equat ions

1.1 Preliminaries

1 .Definition of viscosity solution

We first recall the definition of the viscosi ty solution for equation (0 .1)

ut + F(x. t . u! Du, D ~ U ) = O

oii domain Q = R x (0' TI, R C Rn is open(maybe unbounded).

Tlirougliout w e assume that F satisfies the following degenerate elliptic condition:

( F, ) F ( x , t . i , q , X ) E C ( J o ) . J o = Q x R x Rn x Sn,XIY E Sn

F is degenerate elliptic.i.e. F(xlt, z . q , X + Y) 5 F(x. t . z . q,.ir') V Y 2 O

Son- we state the definition:

Definition 1.1 Let u be an upper-semicontinuous (USC for short) (resp. lower semi-

co-otciirluo-us (MC for short)) function in Q. u is said to be a viscosity subsolution of (0.1)

(re.~p.s~upersolutiot~) if for all y E C 2 - ' ( Q ) , the fullouring inequalzty holds ut each local

rn.nzimum (resp.minimum) point (xo, t o ) E Q of u - 9

Then u E C(Q) *> said to be a viscosity solution of (0 - l ) , if u is a viscosity subsolution

and supersolution of (0.1).

Remark It is possible to replace 'local? by 'global' or 'Local strict' or 'global strict'.

Nes t we recall an equivalent definition given by 'super(sub)d2flerentiaP, where superdif-

ferential i n domain Q:

and sabdifferential ~ : - u ( z ? t ) = - D:+ ( - U ( X , t ) )

the closure of the ~u~e td i f l eren t ia l is:

the closure of subdif lerential0~- u ( r t ) can be defined analogously. We also use B2.+u (5. t )

aiid h2-- u ( x . t ) to denote the closure of super- and subdifferential.

Remark The definition of sub(super)differentiaJ is closely related to the domain of the

h l i c t ion. Ive can check without difficulty the foilowing conclusions:

90~1- we state an equivalent definition as following(refer to [CL], [El] and [Dol] ):

Proposition 1.2 Assume that F E C(Q x R x Rn x S n ) , Sn is the space of n x n

symmetric matrices, then u E USC(Q) (resp. LSC(Q)) is said to be a viscosity subsoh-

tiorz (resp. supersolution ) of e-q. (0.1) . if and o d y if the following statement holds.

Î + F ( x . t. u . q. A) 5 O for (3: t ) E Q , (T' q? .il) E D2b(x, t )

(resp. T + F ( x . t , u, q, A) 2 O for (x, t) E Q? (T? q, -4) E DZ1-u(z , t ) )

Remark If u is a viscosity subsolution of ut + F 5 O and F is continuous, then ut + F ( r . t : u(s: t ) , q7 A) 5 O for (x, t ) E Q and (T, q, A) E D**+u(x, t). Similar rernarks apply

tu supersolutions and solutions.

Xow w e give the definition of viscosity solution of (0.1) undcr the initial value condi-

t ion:

U = ~ J ( x , t ) 0 7 2 d,Q ( l - l )

wlirre dpQ = C? x {O) U a R x [O , T] . R is an open set in Rn: if R = Rn, ( 1 . 1 ) becomes the

initial \ d u e condition u(z, 0) = u o ( x ) on Rn.

Definition 1.3 Let u E U S C ( Q ) ( ~ ~ ~ ~ . L S C ( Q ) ) , u is said to be a viscosity subsolution

( r e sp . supersolutzon) of (O. 1): (1.1): if u ï.s a VLScoSity ~ubsolution (resp. supersolution )

of (0.1) on Q. and u $ <i> on apQ (resp. u 3 $ on 3,Q.J

Th.en u E C ( Q ) is said to be a viscosity solutior~ of (0.1) and (1.1); if u is a uiscosity

subsolutzon and supersolution of (0.1) and u = + on a,& . \'iscosity solution is weak solution^ it is closely related to strong solution and classical

sollition.

The following proposition declares t hat strong solutions are viscosi ty solutions ([LI

and DO^]).

Proposition 1.4 Let F satisfy (Ft),if u E wE1"+' ( Q ) n C ( Q ) satisfies

IL, + F ( x , t . u7 Du, D*U) = O a.e.in Q

thcn u 2,s a viscosity solution of (0.1).

By the definition of viscosity solution , it is easy to prove that :

Proposition 1.5 Let F satisfy (Fi) . then a classical solution of (0.1) i s a viscosity

solution of (0.1)

2. Changes of variables

In proving cornparison principle, it is often required that the coefficient of u in F ( x , t . u, Du, DZu)

lx . positive: the following proposition reduces this requirement to the condition ( F 2 ) .

Proposition 1.6 Let 21 = e-Ctu.u be a viscosity solution of (0.1), then v is a viscosity

solution of the following equation:

Ou the transformation of self variables x 1 let i' be a n x n invertible matrix: Qr =

Or x (O. Tl. ivhere Rr is the image of domain R under the transformation y = rz, then

have:

Proposition 1.7 Let u be a viscosity solution of (0.1) o n Q, then v(y,t) = u(r-ly,t).

is a viscosity solution of the following equation:

Proof of Proposition 1.7: W e only need to prove the case of subsolution, the case of

supersolution is analogous.

By the definition of viscosity subsolution? we only need to prove that for d p(g, t ) E

C2.'(Qr). if marg,(v(y, t ) - p(y, t ) ) = ( v ( i ? F) - y(& O ) , (y, f ) E Qr.tlien at (y. f).with -7 = r-1-

Y?

+ ~ ( y . t; U. rrDyy. rrD2ypr) 5 o. Le t y = T x . and set $(x: t ) = p(l?x: t ) , then

max(u(z. t ) - $(z. t ) ) = ( ~ ( 2 . f) - ll.(z'f)) QT

Silice if is the viscosity solution of (0.1). we have at (2. f)

Son- let r = r - 'y and by virtue of transformations (') and (""), we get at ( t j . f ) :

To construct a viscosity solution. approsiarnatc approach is often used. For example.

Ive cari use a series of solutions to the uniformly parabolic equations ut -EAU + H(Du) = O

to obtain a solution to the equation ut + H ( D u ) = O by letting E + O. The follorving

proposition claims that this met hod works for viscosity solutions.

Proposition.l.S.(stability) Assume that u, E USC(Q,) (resp.LSC) is a viscosity sub-

solution (resp. super solution)^ f the following

where Q , zs tioninc~eming urith respect to E , and u , > ~ Q , = Q, u, converges uniformly

to a furzction u on any compact subsets of Q . About F, , we assume that there eztsts a

fr~rtctiorz F, such that for al1 sequences x,, t,, z,, p, and -Y that converges respectively

to points x. t . z, p and X7 we have that

zf II E LrSC(Q) ( re sp . LSC(Q)): then, u is a viscosity subsolution (resp.supersolution ) of

Proof of Proposition 1.8

For any ;j E C2,1(Q)7such that

sup(u - 9) = ( u - ii)(c f) . and (2. f ) E Q Q

11-e GLU assume that u - 9 attains its local strict maximum at (2. f) i f ive replace 9 with

c 7 = $ + I r - i I J + l t - 1 l 2

S o w ( 3 . f) E Qc for e small enough and we assume that

sup (u - y) = (u - y)(z , f ) B(3.i)

wlit.re B is a bal1 centered at (2. F) in &. on which u - y attains its strict maxinium. Now

sirppsc that

wr claini that (z, t z ) € B for E smaU enough-Since that u, converges uniformly to u on B aiid (x,. t,) kas a iimit point (xo, to) as E + O.(we can choose a convergent subsequence if

neccssary ) , t hen

sup(u - $9) = (u - P)(XO, to) B

by let ting E i O in (*) ,thus (xo, to) = (5, t) E B from the assumption in the beginning.

Hence ( x , : t , ) E B for small E.

Xow ue is the viscosity subsolution of the equation (C,),then at (x,: t,),

p, + F&, t,: u,, Dy. D2y) 5 0

froin the definition. Then we get at (5,t):

if n-e let 5 go to O and use the assumptions on Fc-

Q. E.D.

Remark If FE converges to F on any compact subsets of Q. the above conclusion also

Lolds.

3.Basic lemma

Sow n-e state the fundamental iemma of viscosity solution theory, wliich is given by

P.L.Lions. Y .G.Crandall and H-ISHII ([CIL]):

Letnrna 1.9 Let ui E U S C ( ( 0 , T ) x RI'.) for i = 1,. . . , k with u , < m. let m be a

functior~ in ( O . T ) x R ' ~ gzuen by

for r = (r,. - - O - . rk) E R ~ . where .N = NI + - - - + Nk. For ( z ' s ) E RiV x R: suppose that

( 7 . p. -4) E LIZ.+ tu(=. .s) c R x R" x SN- Assume that there is an w > O such that for any g i v e n ibf > O , we have a; < C for

some C=C(M), whenever the followzng condîton is satisjied:

Thert for each A > O? there exists (ri, -Yi) E R x SN' svch that

and

7 = T* +...+ Tk

Remark The above lemma holds for locally compact space.

4. Perron method

Ishii(Il] extended the classical Perron method to a class of weak viscosity solutions.

WP cari define viscosity ( sub-, super-) solutions which do not sa t i se ( semi) continuous

properties by requiring in the subsolution case that the USC envelope of u , narnely:

is fiiiite and a viscosity (sub-, super) solution.( similady , one uses LSC envelope u. =

-( -u)- for super solution.) We denote such a viscosity sub(super) solution by WV sub-

(super- j solution.

Perron method of viscosity sohtions has been done for first-order equations by Ishii

[IZ] ancl for elliptic equations by Chen et.al[CGG]; for parabolic equations, we give the

proof of this method in -4ppendis -4 for cornpleteness.

Son- ive state the Perron metliod as following:

Theorem 1.10. Let F sabisfy ( F I ). f , g : Q i R are respectively WV snb and saper

solutioiz of problem (C). and f 5 g on Q. Then there exists a W V solutiorr u satisfying

f L u s g i n Q .

Remark: This theorem can be used to obtain a viscosity solution by incorporating the

cornparison principle: if u is a WV-solution, then. by cornparison principle, u' < u.. then

1>y the definition of u* and u,. u' 2 u,. so u' = u, = u, so u is continuous and a viscosity

solution.

5. Theorem of cornpactness

The following compact t heorem is the basis of proving the existence of viscoslity solutions.

Theorem 1.11 (Ascoli-Arzela theorem on unbounded domain) If E C Rn is separable

space. f,, E C( E ) ( n = 1,2? - -), there exists a continuow rnodufus m independent of n,

.go that jf,(z) - fn(y)I 5 m(lz - y[), {fn) are bounded pointwisely on E, then {f,,) has

locally uni fomly convergent subsequence.

The proof of this theoren; is enclosed in Appendix B.

The following Dini theorem gives the (local ) uniform convergence and continuity of

t lie solutions mit hout estimating uniformly continuous moduli.

Theorem 1.12. ([Ru])Assurne that K is a compact set in Rn and {f,) is continuow

s e p e n c e on K satishing

a). fil E C ( K ) Vn

6 ) . f,, converges to f pointwisely , V n? where f E C ( K )

c) . f,(x) 2 f,+i (x) V x E n = l , 2 , - - -- then f,, converges to f uniformly on K.

1.2 Cornparison principle and maximum principle

This section establishes comparison principle for problem (0.1) and (0.2). W e mainly

eiiiploy the techniques in [GGIS] and the basic lemma 1.9. Our results easily yield the

rrgiilarity of solutions. Finally ive give an estimate of the maximum. which is actually a

gcueralization for classical solutions.

1 .Cornparison Principle

Llè firstly recall the condition ( F2) :

X).

for I - 5 S. V (2, t , r, q? ,Y), (x, t , S. Q: ,Y) E JO.

Shen we state the comparison principle:

Theorem 1.2.1 (cornparison principle ) Let u E U S C ( Q ) , v E LSC(Q) be respectively a

tiiscosity sub and a super solution of problem (0. I ) , (0.2) and limlZl,, u < m. limlZl,, L. >

-os. and let F saticfi (Fi ) - (F4) ,u(z, O ) or v ( x , 0) be uniformly contznuou.s with moduli

o f cotttinuity nz ( .) . Then

vhere Q is a constant from the condition (F2)

Proof: By ( FZ) and Proposition.l.6. for convenience we can assume that (0.1) owes the

for 111

ut + u + F(x. t, uo Du' D ~ U ) = 0-

F is nondecreasing in u and satisfies other conditions ( Fl ), ( F3) and (F4). then we only

ueed to prove that

( 1.3) is easily got by the transformation u = w . 1. I\;c d l prove (1.3)' by contradiction. If (1.3)' is fdse, then

Herc B plais a role of barrier for space variable z at infinity and t = T .

3. u E LrSC(Q), v E LSC(Q) and the assumptions on the behavior of u . v a t infinity

imply that 5 2 M for sorne constant M > 0, and

n-lirre (P. y. t) E and U = Rn x Rn x (O. Tl.

4. Denote sup(u - v ) ? N and sup(u - v)+!~v,~ by the contradiction assumption in Q

- - k 0

> - çupw(x, s, t ) = N > Nl Q

\Vc claini that sup O(z? y , t ) 2 k a > NI for all 5 < k < 1 and O < 6 < 60, O < y < 70 for u

do. 70 small and for al1 E > 0.

Below we prove this claim.

Denote s = y. t h n for E , = SCI

sup{w :

,3 B 0 > O a s 8 < 8 0 , s . t .

lx - y1 < 6) > (1 - s ) a

a i l c l 3 (xo. yol t o ) E U. (10 - y01 < B. s.t.w(zo, go, to) > (1 - 2s)a . kj$ < J and J(l2ol2 + 1 ~ 0 1 ~ ) < J if we choose 6 < JO,& s m d enough and < 3 if we choose

-1 < d e r e small enough.

Thus

@(zo. yo, to) > (1 - 3 . s ) ~ ~ = lia > A-, 3 O

5 . Sow sup @ > O implies that

Claini tkat

Brcause. from 4. for a < a - Ni y take k = y > %, then there exist & ( E ) . 70(s) , set. as

O < 5 <&.O < 7 < Aio. and E > O.

Let 5 -+ O. w e get the result by noting that lim,o a(&) = a.

6 . We claim that 3 eo > O. s- t . (3: y. F) E C: V E < By the definition of B, we have that f # T: nom- ive claim that f # O. Othemise?

3 a, > 0: 6, E (0 . JO), 7, E ( O 0 70): s.t.O attaics its maximum at (5 , . ij,, 0) for E = E ~ < S =

5,. 7 = - , j . then from 4.

Son- by 5. IIj - 3 0, as j -+ O O ( E ~ + O), then we get a contradiction if we let j -+ m. - I . Es~anding iE' at (2. y, f ) yields (@,, I ,,,, A) (Z, y, f ) E D2-+u(z, F) , D2 !P(T . t) 5

-4 E Sn -

S o w apply Lemma 1.9 with K = 2. ul = u, u2 = - v . s = t , z = (2. y), it is easy to see

tliat assumption (1.2) is satisfied. Since (2. tj, t) E Cr. by the remarlc after propersition

1.2. auci Lemnia1.9. weconcludethat V X > 0 . 3 ( Î - ~ . X ) . ( T Z ~ Y ) E R X Sn set.

Then by virtue of the definition of viscosity solution,

0 2 ~ , + ~ ( r . t ) - ~ ( y , t ) + I ~ > k a + &

ivliere I I = F ( r J . u(Z.r), @,.X) - F(y,i. U ( Z , q, -Gy, -Y) S. 'c'est we take a special -4

-4fter letting 7 + 0. then

if wr uote that [al : !$ -+ 0: as i + O. thus ( 1 . 4 ) leads to O < ka 5 O' a contradiction.

Remark

1.If u(,. O ) - ~ ( y . O) 5 m ( l x -y[). other assumptions are as Theorem 1.2.1, then tliere

rsists a continuous modulus rn', s.t. u ( x : t ) - v(y't) 5 m'(lx - yl), V ( x ' t ) , ( y , t ) E Q. 2.If Q = R x ( O ? Tl , R is an open set in Rn( may be unbounded ),

&ere BC = a R x R x (O, TI U R x aR x (O, Tl U R x 0 x {O) . Then there exists a continuous

nioclulus nz' S. t .

u ( x . t ) - u ( y . t ) < m f ( l x - y ( ) for ( x . y , t ) ~ U

3. The results above hold for Wv-solutions.

4. If F(r . t lOIO,O) = 0 , u is a bounded viscosity solution of (0 .1) . then suplu[ 5 Q

f ( c ~ + i ' T ~ ~ P Rn

For F independent of x, tl we can get explicit dependence of the continuous rnoduli

for 1-iscosity solutions of (0 .1 ) .

Proposition 1.2.2 I f F does not depend on z and t . i.e., F is ofthe fonn F ( u - Du. D2u).

wh,crc F sativfies ( F I ) and ( F r ) (not necessady satisjies (F3) or ( F 4 ) ) . then comparison

principle holds for any bounded USC subsoiution and LSC supersolution ; if u E C(Q)

is a uiscosity solution of (0.1): liml,l,,lul < rn? u(x .0 ) - u(y ,O) 5 m ( l x - y[)- then

and

fol- (1. t ) . ( . r . t + r ) E Q.

Proof of Prop 1.2.2

1 .we firs t prove the comparison principle

Following the proof of Theorem 1.2.1 we can reack the end of step 8 without rnaking

auy changes,now

Step 9. let b -+ 0 . b ~ the step 5 and step S.

talc(. siibseqilence if necessary, aud

fruiii (1.5) of the step S'take subsequence if necessary. and X, I satisfy:

and IL (i. t) + ü ( take subsequence if necessary!) by virtue of the boundedness of u. XOW

we compute (1.4) &ter letting d + 0,

1 9 - (1.6) and the condition (Fi)? thus from (1.4), O < ka 5 O. a contradiction. So

c-orriparison principle holds.

2. Because F does not depend on x, so for viscosity solution u ( x , t). u(x. t ) = u(x+h, t )

for any h E Rn is also a viscosity solution of the equation, so by the comparison principle

ç i ip(u( r . t ) - u ( z + h, t ) ) 5 e ( W + I ) T sup(u(z, O) - u(z + h, O))' 5 e(Ca")Tm(lhl) Q Rn

Replacing h with y - x.we get

3. Since F does not depend on t , for viscosity solution u(z, t ) ? u(x,t) = u ( x , t + r) for

an- t E ( O . T - T), r > O is also a viscosity solution of the equation, so by the comparison

principle

sup(u(t. t + T ) - U(Z? t ) ) 5 e ( ~ + ' I T sup(u(q r) - u(x, O ) ) + Q Rn

2.Maximum Estimate

To give the mauinium estimate. we assume that:

(fa u ( z . t )F(x , t. ~ ( x . t ) , 0,O) 2 -pluZ - ~ 2 1 ~ 1 ~

V ( s , t ) E Q, for certain pi442 L O+ E (02) -

Proposition 1.2.3 u E C(Q) is a viscosily solution of (0.1), limIzl+, lu 1 < ca? F sathfies

Proof of Proposition 1.2.3

1.Let u = eCtu. c > pi then t. is viscosity solution of the following equation:

2. Denote M = supq (ul. and consider

for a fised point (xol to) E Q, so

lvliere .\:(:II. T) is a constant dependent on M, T and (xo, to) E Q. for S. E < 1. 3.We cclaim that M = l i m c + o (v(~.f)l. since Ivl - ~(rl' - -

6-4 < Iv(i.F)l 5 M. we get T-t -

11y lettirig E -+ 0.6 -t O

t akiug niasinium on the above formula we have: M = lim .-O 1 v ( 2 . F) 1 &+O

4.If supQ Io1 > supaq Ivl 2 O. then ( i , t ) E int(Q)?

5 .IC7e disci~ss two cases:

thcr! by the definition of viscosity solution

riiiiltiply the above inequality with v(I, f).we have

Froiii condition ( F9), ( c - pi )v(+, 9' 5 e-(Z-a)cip2~(~o qa + L. then 1>y (Fia). and let ; -+ 07we get ( c - pi)i\.12 5 p 2 M a 7 so ikl $ (A)&, c-PI (1.7) is

11 r oved -

2 ) . if i:(.E.t) < O then

i l f cari get (1.7) by the definition of viscosity solution and the siniilar discussion in 1).

Remark

1.If Q is bounded, then (Fia) is not needed.

Z.It is easy to test that ( H J B ) and equations of p- Laplacian t-e sati* the above

conditions.

1.3 Est imates of uniformly continuous moduli

T Lis section establishes the estimates of uniformly continuous modulus with respect to

spatial variables (r) depending only on the maximum. the continuous modulus of the

iuitial fiinction and op. then ivith respect to time t. and ive get the estimate of Lipschitz

coritiriuity of viscosity solutions for F independent of x: t .

1.The Estimates of Uniformly Continuous Modulus with Respect to Spatial

Variables

Cucler conditions of Theorem 1.2.1 in last section. it is actually pro~red in [GGIS] that

d ( g ) = . s ~ ~ { ~ ( i l l . t ) - U ( Y > t ) l lz - < O, (z. y. t) E U } is a continuous modulus of the

\risrosity solution u. where it is shown by contradiction that l i ~ , ~ w ( o ) = 0. however.

tlir depeuclence of the w ( a ) is unknown. In this section, we will construct a uniformiy

roiitiniious modulus n i th explicit dependence by virtue of the constructing techniques of

[Il] and the nietliods in Theorem 1.2.2.

Fl'e will use a lemrna in [Il] to construct our test function:

Lernma 1.3.1. VE > O , [ > O, m(-) a continuow modulw, thete exists a function $ E

C'((O. CG)) . dy' > 0. ~5' ' < 0 . s . t .

Remark rn(-) is defined as: m ( - ) : [O. m) -+ [O? o;) is nondecreasing concave continuous

fiirictiou.

Sest we prove:

Theorem 1.3.2 Let u E brSC(g), v E LSC(Q) be respectively the su6 and super solutions

of (0.I) and u < 1Vl: tT > -hf: M i.s certain constant, F satisfies (Fi) - (F4), U ( X . O ) - r ( ~ . 0) 5 mg( lx - y 1 ) , mo is a continuous modulus, then there ezists a contznuoz~s rnodulus

W . depending only on mo and OF, s.t.

u(x. y . t) = U(Z, t ) - u(y. t )

*(x: y. t) = d ( x : y ) + B ( x . y, t )

Y B(z . y, t) = 6(lz12 + IyI2) + ~ _ t

~ ( z . y ) = Q . ( ( 1 ~ - ~ l ~ + ~ ~ ) f + y , ' O < y l <1.

hme, oc is defined by Lemma 1.3.1, namely, for

3 6. E C 2 ( ( 0 , m)): qi; > O, P:( < O. satis f y i n g

( O ) 5 E ( 1 2 & ( r ) 2 m(r) , O 5 r 5 S.

36

2. Hope to prove that w(x: y. t ) 5 9 ( x , y. t ) . VE. 6 7. -yl > O. (xl y. t ) E 9 x ( O . T ) =

L. wbere

A = ((5.y) E ~ ~ " 1 1 ~ -y1 5 1)-

II7e prove it by contradiction. If not,

1 1 - Q I i la( as 6 -t O (take subsequence if necessary.)

3. Claim that ( s o t M~ to) E UA-

Clearly. E # Tt f # 0:

If 1s - J I = 1. then

a contradiction to the assumption.

4. Espanding 9 at (i.ij?C) yields ( * t . l 4)(~. i j ' t ) E D ~ ~ + w ( z . ~ . ~ . D 2 + ( f . y ' f ) 5

-4 E S"

Sow apply Lemnia 1.9 witli k = 2. ul = u. uz = -W. s = f. z = (3. fi) it is easy to

s v r that assuniption (1.2) is satisfied. Since (Z. ij. f) E C, by the equivdent definition of

t h visrosity solution and Lemma 1.9, ive conclude that V A > O 3 (71. X ) . ( 7 2 . Y ) E

R x S" s.t.

(TI. Gr, X ) E D2'+u(Z7 f)

( - 7 2 : -\fiy. -Y) E D2'-o(y. f )

5. A direct calculation yields that:

then 11.41[ 5 2$ + 26

r o t e that fi < - s <_ ,/- 5 & (yi 5 l), from 1 and the properties of 0:. we

take X = +. (1.8) becomes 0, (9)

where 11 = 3 O'!s' + 16. w = 26 + 4 ~ 6 ~ O:( 2):

6. otlier discussions are analogous to 7 and 9 of Theorem 1.2.1. ive have that

lct 7 , 1 . 6 . ~ i O. then

the above inequality holds for x? y E Rn from +(1) 2 211.1 and 4: > 0.

7. Define

r n ( r ) = i n f { & ( r ) , ~ > O ) for r L O

tLen TIL(I-) is a continuous modulus? and depends only on M , rno and OF-

Q.E.D.

2. The Extension of Kruzhkov's Regularity Theorem in Time

Lié will estend Kruzhkov's regularity theorern ([Kr]) to viscosity solution from classical

soliitions. We do not require the smoothness of F: but only require the following:

d e r e -1 is nondecreasing in IqI, -Yij is the i j th entry of the matrix X:and there exists a

-, > O. S-t.

iim p(p)p7 = O, p(p) = h(2Mp-' . 2Mp-' ) ~ 4 0

Let R b r a domain in Rn and Q = R x ( O . Tj: we will consider the viscosity solution

ii E C ( Q ) . 1 u 15 &f of parabolic equation of the form

L ( u ) = ut + F ( z 7 t . u. Du, D ~ U ) = 0.

lierrafter u i ( . s ) . w ( s , t ) will denote functions which are moduli of continuity type, and are

definecl and continuous for nonnegative values of t heir arguments7 are nondecreasing wi t h

respect to each argument, and w(O), # ( O , O) = O

Theorem.l.3.3 Let (xo, to), (xo + Ax,to + At) E Q: At > O . d = dist(Xl, ro). u is a

uiscosity subsolutiort (resp. supersolution ) of (0.1).

If u(x. to) - ~ ( 2 0 ~ to ) I w ( I 5 - xo 1 ) (?.c.qp- - u(x, to) + U ( X O ? ~ O ) 5 w(l x - z o 1))

then,.

1) . I f d > O and I 4 x I < d,

u ( z o + As, to + At) - u(zo, to) 5 min [ w ( d + p(p)At + 2M 1 AX l Z 1 l 4 4 l ~ l d P'

( r c s p . - u ( x o + ~ x , t ~ + ~ t ) + u ( x ~ , t ~ ) 5 min [ u ( p ) + p ( p ) ~ t + 2 ~ I 1 2 ] ) ~ a ~ l s ~ s d pz

and dij is the Kronecker symbol. in particular

( r e s p . - u ( x ~ , t o + A t ) + u ( x o . t o ) <wd(Ljf)= m i n [ w ( p ) + p ( p ) A t ] ) O<p<d

2). If d=O (xo E 30) and if (2, t ) E Q\Q,

I L ! J O + Ar. t o + At) - u(x0, to) 4 min At) + u ( ~ ) + p ( p ) A t + 2M 1 Ax 1'

PLI^^ p2 1

To prove the theorem. we first give two lemmas. They c m be checked directly by the

dcfiuition of viscosity solution .

Lemma 1.3.4 Let u E C ( Q ) be a viscosity su6 (resp. super) solution of L(u)=O. Then u

mil1 be a sub ( resp. super) solution of L f ( v ) = O , where L f ( v ) = v t + F ( z . t _ u (z , t ) . Dv, D'L.).

aucl the cornparison principle between a viscosity subsolution and a classical super solution

on bounded domain.

Lemma 1.3.5 Let u E CrSC(Q) be a vtscosity subsohtion of

N ( w ) = wt + G ( x . t , Dw, D Z w ) = O.

i1 bc a classical s u p e ~ s o h t i o n of N(v)=O. G sathfies (FI )! Q is bounded. Then

Son- let us prove the main theorem.

Proof of Theorem 1.3.1:

Define a new operator L 1 ( v ) as in Lemma 1.3.4.

Let d > 0. 1 Ar 15 d.Let us take an arbitrary nurnber p E (1 At 1, 4 in the cylinder

consider the functions

Ily virtile of condition (F5)

It is uot difficult to verifv that u 5 vf laq,: then by the basic cornparison principle stated

i n Lemma 1.3.5, we have that u < vf whence it foilows that

thiis the result in 1) is got.

To prove estimate 2) for the case d=O. i t is not difficult to consider. entirely analogously

t lic fuuc t ion

u ( x 0 . to )+ [u(p . At) + W ( P ) + p ( p ) ( t - to) + 2.q z - xo 1 2 P - 2 ]

i n the cyliuder Q" = Q' f~ Q , p 21 & 1 Q . E . D

Remark

1.If tliere exists /3 > 0:s.t. l imp ,~p (p )pa = C 2 O, then W(p) in 1) can take the forrn:

" ( p ) = c(+*) +Ph). C depends on I [ U O ( ( ~ . Particularly. if w ( p ) = C'pe ,a . > O , then

i ( p ) = cp*. C clepends on C' and IIuollm.

Y. For HJB equation. if the coefficients are bounded. then p ( p ) = C(p-' + p-' + 1). theu it satisfies conditions in 1 if we take ,O 2 2.

For the equation of p-Laplacian type with (&). p ( p ) = Cg(l)(p-61-2 + p-6<- ' ) . then

n-e c w take ,b' >_ SI + 2 to satisfy the condition in 1.

3. Lipschitz continuity

For equation

ut + F ( D u , D Z u ) = O in Q

tve study the Lipschitz contiuuity of the viscosity solution .The results are:

Theorem 1.3.6. Let F satisjij ( F I ) and u E C(Q) be a viscosity solution of (1.7), (1).

if u ( s . O) - u ( y . O) < L1x - y[, then u(z7 t ) - u(y. t) 5 Llz - y[; 2 - If u&) = 4 2 . O) E 1.1 -2.93 n C ( Q ) then u(x,t) - u(y. r) 5 C ( l x - y1 + (t - T I ) , C depends on I ( U ~ ~ ( ~ Z - = . Proof:

Ouly '2) need to be proved.

BI- Proposition 1.2.2, Ive only need to estimate

sup ( u ( x . t) - u(x, O)). t>O.zERn

Drfitie o* = &d + uoo c = sup 1 F( Duo. D 2 u o ) Io then ci are respectively the super and sub

çulrrtions of ( l.ï).By virtue of the cornparison principle Theorem 1.2.1. we have that

1.4 Existence

11; will construct a bounded continuous viscosity solution for Prob.(C) by virtue of Per-

rou's niethod and approximate method.Perron method of viscosity solution is developed

11y H-ISHII (15 [Il ). Our result is as following:

Theorem 1.4.1. Let F satisfy (Fi ) - (Fs ) , uo E W'~m(R*) n C ( Q ) , then there exists for

(O. 1). (0.2) a unique viscosity so~ut ion u E BUC(Q).

Proof:

1. Let u = eQtvt then u is the solution of the following

2 . Define

v* = f C f + g(z) rvliere C = sup l e - C ~ t [ ~ g e c ~ t + F ( r ? t , e C ~ ' g ( x ) 7 eCot D g ( x ) : eC0' D Z g ( x ) ) ] 17co is the constant in

Q (F ' ) . g(x) = ~ ( x ) .

Obviously. r!+ and u- are respectively a super and a sub solution of L ( v ) = O . by ( F z )

aucl froni Proposition 1.4, and

aud u* = cC0' ( i ~ t + g) are respectively a super and a çub solution of (0.1) (0.2).

3. Tlieorern 1.10 implies that there exists a W V solution u so that :

t h s ~ ' ( x . O) E W'." and

1irn uœ < 00: lim .t~, > -CG l4+- 14-+-

4. Yow Theorem 1.2.1(Remark 3) is applied to deduce that

so [ L E C ( Q )

5 . Froni Theoreru 1.3.2 and Theorem 1 .3 .3 (Remark l), we have that

wliere C depends on I l g l l w2.w mi and r n z are continuous moduli depending only on

I t ! / j /L1.2.s aud aF.

Sext we study the case uo E BUC(Rn).

Tlieorem 1.4.2 Let F satisfy (Fi) - (Fs), to get maximum estimate. assume i n addition

that ( F 9 ) . (Flo) hold. g(z) E BUC(Rn). Then. there e z i s b a unique viscosity soliltion

11 E BC:C(Q).

Proof :

Firstl~: Ive disscuss the case uo E Co(Rn) :

1. Define g,(x) = g * p,(x) where p, is a mollifier. Then

and

1t~J 5 1 + maxlgl, gL(x) + g(x) uni formly in Rn

2. Replace g(z) with g-(z), and denote this problem as (0.1) and (0.2)-;

3.11 follorvs from Theorern 1 .41 that there exist a sequence of functions uc E C ( Q ) . u,

is the viscosity solution of (0.1)(0.2),:

Bi: Proposition 1.2.3.

iuc1 5 cm I ~ I J :

.Analogous to 4 in the proof of Theorem 1.4.1. there exist two continuous rnoduli ml

aiid m2 tlcpnding only on T l lgl,,so that

By Tlieorem 1.11. there exists a function u E c(Q). s.t. u, + u locally uniford- and

4. By the stahility property Proposition 1.8' u is the viscosity solution of (0.1) in Q.

5.By 3.

lu.(.. t ) - u&, O ) l 5 Cm&)

from 1 and 3, if we let E + O. thus u(x,O) = g ( x ) .

For the general case u o E BCIC(Rn). we approximate u o with U O , = uo x en. where

z ira(x) is a "cutting function" defined as &,(z) = 1 for 1x1 5 n and O for 1x1 2 n + 1, and

nierges linearly in n < 1x1 < n + 1: then:

1 s t ~ ~ ~ f C o ( R n ) i

2.Iz10,J 5 Iuol: 3.uon(x) - Uon(y ) 5 m'(lx - y(), where m'(-) is a nodecreasing unifonnly continuous

~iiodulus depending on the uniformly continuous modulus of uo and maximum of uo.

4 . t ~ ~ ~ converges to uo 10cdy uniformly

With the above properties and uniformly continueous es timates as well as the stability

property. and use the similar discussion as above, we can prove the theorem.

Q.E.D.

Remark

1 .If O is a solution of (0.1): then the assumptions ( F9), (FIo) can be abandoned.

2. It is possible to get existence result with uo satiseing other conditions. e.g.

IF(Duoc. D2uo,)l 5 Cc for certain constant Cc dependent on E .

1.5 Applications

LVr apply Theorem 1.4.1 and 1.4.2 to HJB equations and equations of p-Laplacian type.

1 .The existence and uniqueness of viscosity solutions for HJB equations

ii'e assume the coefficients of the HJB equations satisfy:

( 4 ) sup IlZallwl.- (Rn). IIh-: ca: f,ll, 5 c:c is independent of a; W o , r l

cout iiiuous modulus.

Our result is as following:

Theorem 1.5.1 Let ( A i ) - (&) hold, the initial fvnction u o E W2@(Rn) n C(Rn) or

Bl-C( Rra). then (XJB) has a unique wiscosity solution u E BUC(Q).

This is the result of the former sections, only (F4) need to bc tested.

Recall that in ( F4), the inequality (0.3) is :

Sliiltiply (0.3) from two sides with

and take the trace, we have

5 vL21z - y \2 + WC.

wlirre Cr(s) = 2 n ( s ) i L2s + CS? ai(s) = 5s.

Remark Linder the corresponding conditions on the coefficients, w e can prove that

tr f I.IG-'*". Cl7" and u is semicoiicave, this will be done in the next chapter.

2. The Existence and Uniqueness of viscosity solutions for Equations of p-

Laplacian Type

\f.7e apply the former results to equations of p-Laplacian type.

Theorem 1.5.2 F satisfies ( F I ) , (Fs) and (F9). uo E BUC(Rn), then the above problem

lias a unique viscosity solution u E BUC(Q). If uo E IV2.-, then (F5) and (F9) are not

urressar- and in addition,^ E CV1?'.*(Q).

Remark The above result Lolds for (-4PL).

In cliapter 3. w-e will apply the techniques and results obtained here to study Leland

ccpation.

Chapter 2

Regularity and Convexity-preserving Properties of Viscosity Solutions of HJB Equation

In Chapter l.we establish the cornparison principie and the existence of the viscos-

i ty solutions of the Cauchy problems for Hamilton-.Jacobi-Bellman(H.JB) equation.This

chapter is concerned with the regularity of viscosity solutions. The techniques of vis-

rosit' solution method given by H. ISHII and P. L. Lions in [IL] allow us to deduce

tlre estimates without differentiating the equation. which is in a completely different way

froiii traditional one. We mainly get the estimate of < Du >:A under the corresponding

assliinptions on the smootlmess of the h o w n functions in the equation.which general-

izcs Ishii and Lions' semiconcavity estimate results for viscosity solutions of ellip tic HJB

eqiiat ions. Finally. we extend this met hod to st udy the convexi ty-preserving property of

~ioiiliuear non-liomogeneous equations.

2.1 Introduction

Tlic classical Bernstein's method ~resented a way for estimating the mauiniums of the

iiiodiili of derivatives of any order of solutions for linear parabolic equations rinder the

asstiriiption that the solution itself with dl of the known functions in the equation are

siifficieritly sniooth. The basic idea of this methocl is to linearize the equatioil by clifferen-

tiation. However, this technique can not be used if the solutions are we& or the known

fiiuctions in the equation are not smooth enough. In [IL], H. Ishii and P. L. Lions studiecl

the semiconcavity of viscosity solutions of HJB equations. The idea and techniques in

[IL] motivate u s to seek the estimates of Bernstein type for viscosity solutions. In this

paper. w e will deal with the following Cauchy problem:

{ ut + F ( x . t . u . Du, D2u) = O in Q = Rn x (O; Tj u ( x . 0 ) = u g ( x ) in Rn

wliere F ( x . t . u' Du. D2u) = supo,, Lo with

L . ~ ( J - . t . u . Du. D2u) = - t r ( C ( x ' t)' C ( x o t ) ~ ~ u ) + ( b g ( x . t)' Du) + c~(r' t ) u - fd(x. t ) B /3

CSj is 7~ x nt matrix, t r r i is the trace of n x n matrix A, b E Rn, ut . Du and D2u denote

respectively the time derivative of u. the gradient of u and the Hessian matris of u in

spatial variables: for x, y E Rn, (r: Y) denotes the usual scalar product on Rn. ,3 is subindex

in a family B.

Ii7e first List assumptions on F and uo. The following assumptions hold for 3 E B

iiuiformly.where li is a set:

(HI) 0 5 Er(CB X a ) E V< E Rn

( H z ) 3Co > -m. s.t C&, t ) 1 Co on Q

i ~? ) x , . ~ ~ . c ~ . E L = ( ( o . TI n c 1 . m ( ~ n ) ) n C ( Q ) : ~ -

tLc space T*V is defined as:

I=-*'llPo

The initial fiinction satisfies:

(C;) U O ( X ) E C 1 > a ( R n ) =

l = - = ' I s ~ o Sow w e state our resdts.

Theorem 2.1 Let u E C ( Q ) be a viscosity solut ion of (2-1), under the assumpt ions of

(HI ) - (&) and (UO), t h e n

I u I d C

for some constant C depending o n the W modul i o f 6- c. f and I ~ ~ l ~ l . a ( ~ n ) and lu/^=(^) Theorem 2.2 Let u E C(Q) n L m ( Q ) be a wkcosity solutions of (2.1): under the

a..wmzptions of (Hi ) - ( H 3 ) and (Uo). i f Co > O . t hen u satisfies the foilowing inequalzty.

V.r. y. z . .F E Rn. t E [O. Tl C depends o n the Mi modul i of x; b? c , f and o n ( uo I c l . a ( R n )

and l u IL=(^) To get theorern 2.1 sve should first get the Lipschitz continuity of u in x.

Theorem 2.3 Let u E C ( Q ) f~ L w ( Q ) be a viscosity s o h t i o n of (2.1). Co b. c. f E

L X ( [ O . Tl. WA(Rn)) n c(Q). 1 uo(x) - u, (y ) 15 L 1 x - y 1 then

I 4 x . t ) - u ( y 3 ) I i L' I x - Y I

L' depends o n L and the corresponding rnoduli of Cy b, c. f in space Lm([OO Tl. W , ( R n ) ) .

Remark.lt is obvious t ha t W c LOD([O. Tl , 6V&(Rn)) fi C ( Q ) and i f uo sat+s (Co) . then

The uest theorern is actually a corollary of the above theorems. it gives the w?iQ, cstimate of the solutions.

Theorem 2.4. Let u E CCi?'(Q) n c ( Q ) . ( H ~ ) - ( H z ) and (Lio) hold for cr = 1. then

C depends o n the 6V modul i of Cl b. c. f for a = 1 a n d l ~ ~ l , i . i ( ~ n ) ,

l w2.l = sup 1 u 1 +sup 1 ut 1 +supID,ul +sup 1 D==u 1 - ( Q I Q Q Q Q

2.2 Basic Ideas

W c first clarify the relationship between Theorem 2.1 and Theorern 2.2.

Lemma 2.5 u E Lm ([O; Tl' CIa(Rn)) and (2.2) holds for certain constant C. then

Therefore. Theorem 2.1 is a corollary of Theorem 2.2 and Theorem 2.3.

i\-e d l concentrate on the proof of Theorem 2.2 because Theorem 2.3 can be proved

in an analogous argument-

The basic ideas of the method of viscosity solutions are contained in Lernma 1.9 of

Chapter 1.

To prove Theorem 2.2, the following lemma is needed.

Lemma 2.6. I f g ( x , t ) E Lm([O, Tl, CIqa(Rn)), then

g ( s . t ) + g(y t ) - g(x,t) - g(z, t ) 5 Cao Vx, y, z . s E Rn t E [O . Tl.

is d e f i e d in (2.2).

111 the nest section, we will prove Lemma 2.5 and Lemma 2.6: in section 4, Ive study

soine niatris algebra needed for the proof of Theorem 2.2: in section 5.we will prove

Theorem 2.2, Theorern 2.3 and Theorem 2.4 ,finally,in section 6,we will extend this method

to study t lie convexity-preserving property of viscosi ty solutions for general nonlinear

riorilioniongeiioiis equations.

2.3 Proof of the lemmas

Proof of Lemma 2.5: In (2.2): we set s = x + h z = y + 12 E Rn then

( ~ ( x + h, t ) - u(xl t ) ) - ( ~ ( y + h, t ) - ~ ( y , t ) )

< &C(I x- y I P I il 1 + 1 h I l+") -

with hiei, where h; E R', e; denotes the i-th unit vector o i Rn , then

( ~ ( 2 + hie,, t ) - U(Z, t)) - ( u ( y + hie,, t ) - u ( y 7 t))

51

replace h efor

5 AC([ r - y 1" hi + h f f " )

Divide the above inequality on the both sides by hi and let hi i 0: we get

By the symmetricity of xt y, we have

1 Diu(x) - Diu(y) 15 &C 1 x - y l a

Proof of Lemma 2.6. g(x. t ) E LOD([07 Tlt C ' @ ( R n ) ) so

Sote that ( = 9s + (1 - 8)z' O 5 9 5 1,so

2.4 Some Matrix Algebra

In this section we recall some martix algebra to be used in the proof of the main theorem.

1.Let -4 be a real symetric mariz,then all egenualues of -4 are real

Proof: Let A be an egenvalue of A and f be its corresponding egenvector,then by the

defiui t ion of egenvalue:

-4< = A<

~Iultiplying on the both sides of above equa1ity:we get:

where 'I-" over J means conjugate. Now take conjugate aad transpose in the above

so X = X and X is real.

Q.E.D.

2 . Let -4 be a real symetric ma*, then A2 is semi-definite positive

Proof: ActuallyJet X i ( l 5 i 4 n ) be egenvalues of the matrix A. then Xf (1 $ i 5 n)

arc all egruvalues of .42.and -4' is semi-definite positive. Q.E.D.

3. Let =li( l 5 i $ n ) be real symetric man'ces,then (xLl -4i)2 $ 2"-' -4;). Proof: We only prove for the case of rr = 2.

Since --I1 - -a2 is real symetric,so by the above Proposition 2.(-41 - -A2)' 2 O.and

T h nest few propositions are about the computation of tensor product and derivatives

of I S - yl. we oxnit the proof because it c a a be checked directly.

4.(s ::: s)' = Ix('x c3 2

t r [ ( x :I: x)A] = xrrlx - T - Y g-&l r - Y I - I=-,,,

D,lr - yl" = alz - yla-2(x - y) = -DYIx - y (a

2.5 Main Theorems

Before starting proving the main theorem.we make some simplifications: we only prove

oiir theorems for linear parabolic equations.namely,uve will &op all sub indices ,B of all

coefficients : the proof of H.JB case is completely similar.if we notice that for any small

~. t l iere is ,o. such that for Q < Bo,

wlirre F (x. t . u. p, X) = supass Ld.

Proof of Theorem 2.2: We assume that u is Lipschitz continuous in space variable.

wliirh will be proved in Theorem 2.3. From Cauchy's inequality, we see that, to prove

iricquality (2.2). we only need to prove that

for al1 > O . s , y ? z ? x ~ R", t~ [O,T]where

To prove (2.3) , we fix any 6, 61 > O, and -kI > 0, and set

,-(.S. y. z . .c. t ) = Mik(s, y. 2. r) + r ( I2 +*. Where [ = (s? y. 2. x).

L\-e only need to prove that (w - y ) ( s , y:--, x.t) 5 O on U = R4" x [O. T].for r., ri > O siiiall eiiougli and M > O big enough.wlere LM depends on norm of initial value uo

aud 1.t' uornl of al1 coefficients of the equation.

' rote tliat r 1 C 1 2 . play respectively the role of a barrier at infinity and t = T.

aricl tllat is bounded and p is nonnegative, so the function (u! - p)(s, y,z. x , t ) on

R"" x [O. Tl achieves a maximum. We assume that this maximum value is positive. and

will grt a contradiction for C , M large enough and r s m d enough. Let ( ~ . y , ? . f ? f ) be

one of its maximum ~ o i n t s . Then 5 # y for !Li large enough. This can be proved by

contradiction. for if 5 = y. by Theorem 2.3 and Cauchy's inequality.

for -11 > L' .

This is a contradiction to our assumption. Hence i # ij. It is obvious that F # T.

nest. f # 0. otherwise, by Lemnia 2.6. using the initial condition? we get

for JI > C .

Soir ive prove r 1 f [+ O as r -t O. since that (w - p)(~,ïj,r.à.f) > O and w is

l~otlridecl from certain constant .say B that does not depend on r,then

Espaiidiug y at (5, F) yields

ivitli ~ z ~ ( F . f ) < E 7 D É = (D .?D, ,D , .D , ) , where E is to be chosenlater.

Sow by virtue of Lemma 1.9, here. b = 4. U I = u(s , t ) , uz = u ( ~ . t ) . ug = -.U(Z. t ) : u4 =

-~ l ( s . t ) . ( ( .F ) E Ci. then V A > 0, 3 (T~.X)~(F~~Y),(T~,Z).(~~,S) E R x Sn such that

aud

< E + X E ? -

By the definition of viscosity solution,

72 + F(y.5. u(y, t), D , v ( i , t), Y ) 5 O

-73 + f (5, f: ~ ( 5 : f ) . -l);v(f , f ) : -2) 2 0

-T4 + F(l? f. U(X. t). - D = ~ ( z ~ f), -X) 3 O

Siibtracting the last two inequalities from the surn of the first two inequalities . we have

1 I -1 -I 1 0 0 - I I I -1 -1

s - r I 2 +2(1 +a) 1 s - 1 12" ) O O 0 O O O 0 O

- 1 0 0 I

S + (- + 4a(l + a)) 1 s - x 1 2 a - 2 ) 6 1

,<otite t hat each matrix above is semi-defin ite positive.only the coefficient of the second

2 / S G term is negativejf we denote G = D2$ subtracting the second term.then D v

wr then choose E = Y G + 2 r l . and it's not difficult to check that E2 < 2(gp + 4r2 I ) .

aud G" 5 CdG. where Cd is a constant dependent on a. &bi7 b2 and Ir - yl. Then.by

clioosing X = r n i n { L 2hiCd J-):we 4r have that

1 2 3

aud in t lie following,for simplici ty,we denote the corresponding coefficients of the HJB

cqiiatiolis by 6;: C;,and we also &op the "-l' sign over x and t .

Miiltiplyirig (2.5) by the nonnegative matrix ~3 Co taking the trace and using the

above results. We get

To show that the right Land side of (2.7) 5 Cf& for certain constant Cf depending on

14- tiioduli of the coefficients?we just check the third temi in the right hand side of the

fortnida of tr(C t3 C)G,

t h retiiaiiied arguments are analogous.

.Uso. \ve observe that

Sotice that

iisiiig the similar discussion as above,we daim that expressions in Nght hand side of (2.8)

5 C,\l\k. Xow. analogously,

Son- from (2.1)-(2.10) and Theorem 2.3. we then obtain

\I7lirre o(1) -i O as r* + O and C is a positive constant depending only on the W moduli

of C. 6. c. f and 1 uo IC~.a(Rn) and u I L m ( q i

Thus Ive arrive

aiid 1wnce.noting Proposition 1.6, a e can always make Co > C by suitable transformation.

and and r are srnall enough, this inequality leads to a coutradiction. theu -11 >

Proof of Theorem 2.3. To prove u ( x , t ) - u ( y ' t ) 5 L' 1 x - y 1 we need only to prove

t ha t

U(Z, t ) - U ( Y . t ) 5 M ( S + I X - ~ l 2 ) Y 6 > 0 S

Set P(Z' y. t ) = ~ ( 6 + w) + r 1 z I2 +* the following argument is completely analogous to that of Theorem 2.2. M;e won't restate

here.

Proof of Theorem 2.4. Set a = 1 in Theorem 2.2. then let s = s+h. y = x-h, z = z.

by (2.2). we have

u ( r + h. t ) + u(z - h o t ) - 2u(r,t) 5 2 ~ &

l e t s = ! / . . r = y + h , z = y - h w e h a v e

2 u ( y ) - u ( y + h ) - u ( y - h ) 2 ~ \ / 2 1 h

thlis 1 D2u 1,s C. By Theorem 2.3. 1 Du 1,s C. Now by the equation

2.6 Convexity Preserving Property

Corivesity is an important property of the value function of HJB equation. In this sec-

tiori.we will seek how a concave initial function evolves in time: we hope to study the

s trilctiire of the equations such that the concavity is preserved dong time by the viscosity

soliitions. W e will not constraint ourselves to HJB equation, however. we wi11 deal with

gerieral uonlinear nonhomogeneous equations.

In [GGIS]. Ishii et.d proved the convexity preserving property for linearly growing

x-iscosity solution of equation:

They sliowed that the concavity of u in x is preserved as time evolves provided that

F (q. -47) is convex in X. However. th& method does not apply when F depends on time

t or x. The main difficulty is that they have to get estimates for growing property of

\-iscosi ty solutions before proving the convexity presenring property. But w e often see

siick type of equation:

ut + ru + F(D,u, D2,u) = O

where r is a real n ~ m b e r ~ t o apply viscosity solution method,we u s u d y make a transforma-

t ion u = cC'u to guarantee the coefficient of u is positive or big enough,then unavoidably.

tiine variable t may appear in P. So it is necessary to study a more general type of

eqiiat ion. In t his section. we consider the following Cauchy problem:

Brcause ~ v e study bounded viscosity solutions, rve can apply s im~le r test functions to

&rive the convexity preserving property under the following conditions:

( 1 ). F is degenerate parabolic

(2 ) . F is continuous

(3).5 i F ( t . q. -Y) is convex on Sn for al1 t E (0:TI.q E Rn

(4) .r i G(r. t ) is concave on Rn for a l l t E (O. Tl. G is also globally Lipschitz with

coustarit Lc in Rrz

( 5 ) . r . is any real constant.

The theoreni is stated as following:

Theorem 2.7 Assume that above conditions ( 1 ) - ( 5 ) are satisfied. let u be a bounded

cotrtit~uous viscosity solution of (2.1 1 ) and (2.12). If the initial fvnction rro is concave

and globally Lzpschitz with constant L in Rn? then

holds for x. y . r E Rn, t E [O' Tl. In particafar. x + u ( x : t ) is concave for t E [O? Tl'where

li = rnax{L. Lc).

To prove Theorem 2.7 we need the following two lemmas,

Lemma 2.8 Suppose that function v (z) is concave and glo bally Lipschitz with constant

L in Rra. then

v(x) + .(y) - 27-44 5 LIx + y - 2 4

for ail x, y. 2 E Rn.

Proof: Since v is concavejt follows that

tlir last inequality uses the global Lipschitz of v .

Q.E.D.

Lemma 2.9 Let u(x) be continuous in Rn and satisfy

then r q is concave.

Proof: W e only need to prûve that

for al1 X E (0.1). W'e prove it in three steps:

S t r p 1. for X = & ? B is an integer

By iucliiction.

1. 7 1 = 1 (') is the assumption;

2. Assume that (') holds for n,now we prove tliat it liolds for n + 1

The second las t inequality uses the induction assumption.

Step 2. Using similar method as above. we can prove that ('1 holds for X = &, k 5 2". T L . k are positive integers.

Step 3. For a.ny4 real number A E (O, l), and for any positive integer n.there is an

integer k > O. satisfying k + 1 5 2", such that

and

n-ith h(rr. k) = $. So if we let n + m.then k -t m.so

By s t e l~ 2. (') h l d s for X = X(n?k) , and v is continuous.so by letting n -t m. we get

Proof of Theorem 2.7: We will prove that

for al1 ( = (x. y. z ) E R3".t E [O. Tl. Without loss of generality. we assume that r > 1 For

-!. 6. F > O and Ii > 1 we set

with 1 1 7'

b ( { ) = %lx + y - 2~1' + F E , B(c.t) = s1(1' + - - T - t To prove (2.14) we only need to prove that for every 6: y > O : there e'cists =

&(F.-,: A-) > O such that

if O < 6 < &. By virtue of Cauchy's inequality,

and the equality holds by letting E = lx + y - 2-1. Taking this c and letting 7: d -+ O in

('2.15): we got (2.14).

Sow we prove (2.15) by contradiction, if it is false,there would exist €0: > O such

that

sup @(f: t ) > O with r = €0: = 70. IC = K0 (2.16) O

Lolds for a subsequence 6, + O.By the boundedness of u and (2.15): we have O < O for

siifficiently large f, clearly @(J, t ) = -oo at t = T and t ) 5 O at t = O by Lemma

2.S and (2.15).s0 a((. t ) attains its maximum inside U. we assume the maximum point is

( f . t ) witii < E R3":f E ( O : T ) .

Sotv ive prove 6 1 ( 1- O as 6 = S, -t O. Since O( [ : i ) > O and w ( f . t) is bounded

from certain constant (say, B) dependent on the bound of u,

Siuce <P attains its maximum over (i at ((: 5). so

with D ; Q ( ( ~ ~ ) $ A, Dt = (D,: D,:D=).

';on- by virtue of Lemma 1.9, here,

k = 3, u, = U(I. t ) . U2 = u ( y . t ) , u 3 = - 2 4 ~ : t): ((.t) E O.

By the definition of viscosity solution.

Actding the first two inequalities and subtracting the last one twice yields

-1pplying (2.16) and Lemma 2.8, we have

Sow ive compute the derivatives of \k,we denote r , ~ = 2 + ij - 22.

witli c = and 6 = 4. II'

~ * \ k = -S + 261 E

I - 2 1 . We take A = DZY. since S2 = 6S, so llSJl = 6 and

- 2 1 -21 41

Takiug X = 1 and (2.18) now becomes

Xolv w e let 6 + O, then q + a for a subsequence of {d,)(still denoted {6,) as 6, + 0. By (-.ZO).tliere is further a subsequence of (6,) and X, Y. S E Sn such that

witL -Y, = X(6,) and so on. So after letting 6 + 0,(2.20) becomes:

wit h

thcri ive have -Y + F- + Z 5 0:so by the parabolic condition of F.we have

aucl (2.19) becomes:

Siuce F is îontinuous,and rIi 3 Lc,then w e get a contradiction if we use the couvexity

assiiiiiptioii to get yoT-2 < O. Thus ive prove (2.14) and complete the proof. Xow by

~.irtiie of Lemma 2.9, u is concave.

Q.E.D.

Remark

Tliroreni 2.7 applies to HJB equation of the following fonn:

Chapter 3

Delta Hedging with Transaction Cost-Viscosity Solution Theory of Leland Equation

3.1 Introduction

Lplancl rquation was first introduced by Leland in [Le] to incorporate the transaction

rost into Black-Scholes analysis of option pricing theory. In a complete financial market

withoiit transaction cost, the Black-Scholes equation provides a hedging portfolio that

replicates the contingent claim, which. Iiowever. requires continuous trading and therefore.

iu a market with proportional transaction costs. it tends to be infinitely expensive. The

rquirement of replicating the value of option h a . to be relaued. Leland [Le] considers a

rncjdel that allows transactions only at discrete times. B y a formal Delta-hedging argument

lie derives an option price that is equal to a Black-Scholes price with an augmented-

volatility

f i = v , / n

wlicrr .\ is Leland constant and is equal to E-& and v is the original volatility?k is

tlie proportional transaction cost and 6t is the transaction frequency, and both d t and k

arc assuiiied to bc srnall while keeping the ratio L/& order one. He obtained the above

resiilts for couvex payoff function fa(S) = (S - K)+ ?where 11- is the strike price of the

assets. he also assumed that A is sma.ll(e.g. A < 1). For non-convex payoff function(e.g.

for a portfolio of options,like b d spread and buttedy spread), Leland equation can not

Iw reduced to Black-Scholes equation and Leland equation is a nonlinear equation, and

gcnerally we can not find analytical solution.

Hoggard et.al([HWW]) generalized Leland's work to non-convex(piece-wise linear)

popoff function with A < 1: for -4 > 1- the coefficient of the second derivative m a -

1,e uegative and thus the Lelaud equation is ill-posed: for A = l. Leland equation is a

degaerate parabolic equation and may not have classic solutions. so for h 2 1.Hoggard

et.aI iiitroduced new model to describe the d ~ a m i c hedging problem. Here we study the

transaction cost problem under the fiame of the Leland equation for A 5 1 and apply

viscosity solution theory to this problem for non-convex(not necessarily piece-mise linear)

payoff fiirictioris.

Iu this cliapter: Ive will study the following Leland equation

wlierr fo is the payoff fuiiction which may be non-convex' e.g.?the payoff of a portfolio

of options.like bull or buttedy spread. We will derive the emstence,uniqueness of its

viscosity solutions for non-convex payoff function f o (S ) with linear-growth at infinity and

for -1 5 1. iVe also study the properties of the viscosity solutions of the Leland equation

aud their relationship with solutions of Black-Scholes equation.

This chapter is arranged as following: we f i s t r e c d the formulation of Leland equa-

t ioii(tj2 ). t hen prove the comparison principle of the equation by transforming it into the

forni to wliicli our results in Chapter 1 apply( $3); then in $4 we establish the existence of

thta viscosity solution: finally, in $5 we study some properties of the solution, in particular?

n-e stucly the relationship between the Leland solutions and the Black-Scholes solutions.

3.2 Delta- hedging with Transaction Cost - Formula- tion of Leland Equation

ni. first recall the formulation of Leland model. We are interested in constructing hedging

strategies to replicate Europeau-style derivative securities with a payoff function fo (S)

drpendiug only on the value of the underlying assets at the expiration time T. We will

(-oriibinr al1 techniques in [Le]: [WDH] and [&Pl to derive the model. We mmalie the

following assumptions:

1.Consider a market in which a security is traded with a bid-ask spread - =

k S t . where St is the average of the bid and ask prices and k is a constant percentage; it

is fair to assume that:

we also assume that lending and borrowing at the riskless rate does not involve significant

costs

2.The portfolio is revised every S t , where dt is a non-infinitesimal fixed time-step and

does not goes to O.

3.The random process for the stock price is given in discrete time by

wùere W; is a Brownian motion,ECti< = etJ6i,and et is standard normal distribution: v is

the annualized volatility and p is the drift.

4.Ttie value f t of any portfolio consisting of shares St and risk-less discount bond Bt

n-ith interest rate r,only depends on St and time t. Le. f t = f(St.t)

.Assume that an investor sells an option with payoff f o ( S ) and t d e s a position con-

sisting of At shares of the security and of risk-less bonds wi th value Bt. Subsequently the

portfolio is dynamicdy adjusted in a self-financing manner. Its value at time t is

Theu the change in the value of the portfolio from t to t + St is

n-herc. 6Bt = rBt6Bt and r is the risk-less interest rate. The first term on the RHS is

tlic profit/loss due to the change in the value of the underlying security, the second is the

interest paid or received from the bond: and the third is the transaction cost of rehedging.

i x . of changiug arnount of units of security from At to At+at-

B y assumption 4.

f t = f (Sttt)

ive expand f (St , t ) using Ito's lemma

w-liere ive c m uot replace E: with its e.xpectation E(E: ) = 1 because we can not let dt -+ 0.

Son- ive use delta-hedging. following the same hedging strategy as Black-Scholes' ar-

giiiiieut and noticing that 6ft = 6 f (S,. t ) 'we have

II depends only on St , t,but not the past history of prices, so Ito's lemma applies:

6At = a2f(St't)dSi + terms of order 6t or higher as2 Keeping the first term and plugging 6St i vStGMr, into above formula, we have

Lsiug the relation

B, = fi - &St = f ( S t . t ) - af (St' t ) st as

wcA have that f (S. t ) satisfies the equation

for S E ( O , m), t E ( O , T ) and

The Leland constant A = fi--& plays an important role in this equation. If -4 >> 1.

t h ~ u the, transaction costs term dominates the basic tariance, this implies that transaction

costs are too high and the rehedging frequency is too big( 6t is too small).

If A << I then transaction costs tenn has little effect on the basic tariance. This

iiiiplies very s m d transaction costs, and 6t is too large. the portfolio is being rehedged

too seldoni.

Compared ~ ~ i t h Black-SchoIes equation, Leland equation has one more term

where IwI is the Gamma. a measure of the degree of mishedging of the hedged

portfolio due to that bt can not Le infinitesimdy small. Intuitively, the bigger the Leland

constant ;\, the more vaulable the option is. This relationshi~ will be studied in detail in

55.2.

3.3 Cornparison Principle

This section is devoted to the cornparison principle of a class of viscosity solutions for

Lelantl equation; w e will relax the requirement that the paoff function is convex or

picce-wise linear.

The Leland equation derived above is back-ward form. For convenience we trsnsform

tiriic variable t into T - t and still use f (x, t ) to represent f ( x , T - t ) . then ( L e ) becomes

f (S' O ) =

Frorii now on. we mean Leland equation by this new form (Le)' . We will seek linearly-

growiug continuous viscosi ty solutions for the Leland equation:

To guarantee that the equation is parabolic,we require that O 5 h 5 1.

T h e are two difficdties that prevent us from directly applying Theorem 1.2.1: (1).

The çoefficierits of the second order derivatives are not uniformly continuous and not

liiiearly growing at the infinity of the space, so conditions ( F3). (F4) axe not satided; (2).

The solution is linearly growing instead of bounded unifonnly.

To overconie these two difficulties we need to make some transformations. Observing

that any linear function h ( S , t ) = C * S for constant C satisfies Leland equation,and we

iet g(S. t ) = f (Sr t ) - CS,for f a Leland solution, it is not difficult to check that g satisfies

Leland equation. and Ig(S, t)l 5 h: as S 2 So,C is the constant from (3.3): so g is a

II ouiided continuous function.

To overconie the first difficulty, noticing that Euler transformation can simpii& the

rocfficients of the equation. namely, make S = ex. then

If ive write h(x, t ) = ertg(er, t ) = ert( f ( e Z l t ) - C e r ) . the Leland equation (Le)' becomes:

1 ht - ;.h, - -üZ(h , - h,) = - rh

2

a2s a29 V' = v2(l + A.sgn( - ) ) = u2(1 + 12sgn(s2 - ) ) = v2(1 + Asgn(h,, - h,)) as2 d S 2

i1i.i te the above equation into the general form:

ILl + F(?z,. h,,) = O in Q' = R x (O'T) ( L c h )

h ( x , O ) = fo(ez) - C e z > s E R

wlicrc F ( z . q: X ) = -rq - $v'(x - q) . and i2 = u 2 ( l + A.sgn(X - q ) )

Xow WC claim t hat F is continuous and satisfies ( F I ) , ( f i ) required by Theorem 1.2.1

ancl Proposition 1.2.2.

F is coiitiuuous. if we note that

Scst w e check the condition ( F , ) ,

F ( t , q , X + Y ) 5 F ( = , q , X ) . for Y 2 0

Son* w e discuss the sign of X - q,(notice that Y 2 0,O 5 R 5 1)

1.If ,Y - q > O.then 6

1,ccaiise each term in the brackets is nonnegative.

Sow- we can use Theorem 1.2.1 and Proposition 1.2.2, and get the follotving cornparison

principle for viscosity solutions of the equation (Le)' .

Theorem 3.3.1 Let u E c(Q). v E C ( Q ) be respectzveLy a viscosity sub- and super-

solution of the equation ( L e ) ' irr Q = [Ol +oo) x [O. T ) and satisfy (3.3). Then

sup(u(S: t ) - u(S, t ) ) 5 e (1tr)T SUD (u($ O ) - V(S. O))+ Q SE[O.+=)

(3-4)

If tr zs a viscosity solution satisfying above conditions. then

Iii(S + AS. t ) - u(S, t)l 5 e(lfrlT sup (u(S + AS, O ) - u(S. O ) ) + + 2C'IASI (3.5) s'€[O,+-)

for A S E R such that A S + S E (0, +m), C' is a constant depending on C'T and r .

and

for d l S E (O. =x;).t,t + T € (O:T),r )_ O . In particular, ij

for certain vniformly-continuous module rn(-)? then

where na'(-) i s a unifonily-continuozrs module depending o n the continuous module m ( - )

and the constant C

Proof: .Ifter malüng the transformation h(x. t ) = f ( e x . t ) -Cez, we have that u h ( z . t ) =

ri(cr. t ) - Cer. ut l (x. t ) = v ( e r . t ) - C e r are respectively the viscosity sub- and super-

solution of the equation (Leh) . and uh E c(Q'), vh € c(Q') are bounded in Q' = R x ( O . Tl.

Bj- virtue of Theorem 1.2.1 and Proposition 1.2.2, we have

sup(u(eZ . t ) - u(ez . t ) ) 5 e('+'IT sup(u(eZ- 0 ) - v(eX? O ) ) + Q ' ZER

theu let S = er.we get

To prove ( 3 . 5 ) . notice that u h ( x + y ) for any fked number y is a viscosity solution of

(Lcli).because F does uot depend on x. So by (3.7)

Soi\- let S = ex. A S = S ( e Y - 1 ) . Ive get

s i ip( i l (S + AS. f ) - u(S.t) - C A S ) 5 e(l+r)T sup (u(S + h S , O ) - u(S.0) - C A S ) + Q SE[O.oo)

so IT get ( 3 . 5 ) by simple calculation of the above inequality.

As to (3.6) notice that ut,(x, t + r ) is a viscosity solution of (Leli) . and the remaining

dis(-iission is similar to the proof of (3.5).

3.4 Existence of The Viscosity Solutions

In this section.we d l give the existence of the viscosity solutions for Leland equation

for two classes of payoff functions. W e first consider the problem for piece-wise linear

functions.

';ow we state the existence theorem as following:

Theorem 3.4.1 Let puy08 function fo(S) satisfy conditions ( I l ) , (12), then there is a

vrrique cor~tinuoz~s u ~ c o s i t y sohtion f (S. t ) satisfying l i n e a ~ l ~ - ~ ~ o w i n g condition:

1 f (S. t ) - CS] 5 K' in Q

.where Ii' is a constant depending on the parameters C. h' of payofl function fo(S) and r.

Proof:

If ive write h (z. t ) = crzg(er. t ) = ert( f (el ' t ) - Ce'). the Leland equation (Le)' be-

romes: 1

hl - rh, - -i2(h,, - h,) = -rh 2

Likite the above equation into the general form:

wharr F ( z . q7 X ) = -rq - ?ü2(X - q). and f i2 = v2(1 + Asgn(X - g ) )

Then h ( x ? O) E BG'C(R) and Theorem 1.5.2 applies to claim that there is a unique

solution h (z . t ) E BUC(R) for (Leh),then f (S , t ) = e-rth(ln(S)? t ) + CS is the unique

solution to (Le)'. and f (S. t) is linearly grotving.

Q.E.D.

Remark:

For Bi111 spread. which consists of longing a c d with strike XI and shorting a c d

witk strike -Y2 and XI 5 ,%, the payoff function is (S - XI)+ - (S - &Y2)+. it obvioudy

sat isfies above conditions, t hus relevant results hold.

Sin?_ilarly,above theorems hold for Butterfly spread, of which the payoff b c t i o n of

Butterfly spread is (S - X i ) + + ( S - &)+ - 2(S - .Y2)+ with X; = XI +x3 2 -

Other examples are: Straddle combination involves buying a call and put with the

same strike price aiid expiration date, has a payoff IS - XI; Strip consists of a long

position in one c d and two puts with the same strike price and expiration date? the

payoff is (S - -Y)+ + 2(X - S)+: Strangle involves buying a put and a call with the same

expiration date and different strike prices. it has payoff (S - Xi ) + ( X - S2 )+.

Corollary 3.4.2 Assume that fo E W2*OD satipfies (12), then the unique vtscosity solution

f is a h global Lipschitz.

Remark: This is the result of Theorem 1.5.2.

3.5 Properties of The Pricing Functions

i i-e know that Black-Scholes equation has an analytic solution for c d payoff function

( S - -Y)+.

c = S N ( d l ) - ~ e - ' ( * - ~ ) N(d2)

-Y( - ) is the cumulative normal distribution,and it is not difficult to conipute its Gamma.

Tlicta and Veea:

The above formulas show us that the pricing function of Black-Scholes equation is

couves. non-decreasing with respect to the time to maturity T - t and non-decreasing

in vwiance a. In this section, we will prove that the eo Vega of Leland equation have

siiuilar properties: and they hold for more general payoff functions.

3.5.1 Monotonicity in time t

In tliic section rve study hovv the d u e of the option evolves witli respect to tinie. Uë

claini that if the p - o f f function fo(S) is a viscosity subsolution of the Leland equation

and linearly growing at infinity, then the pricing function has monotonicity property.

Theorem 3.5.1 Assuming that the payoff function fo(S) is a viscosity asbsolution of

( L E ) ' and satzsfies (3.3): then the value function f ( S , t ) of (Le)' is nondecreasing with

respect to the time t ,

proof:

By cornparison principle Theorern 3.3.1, we have

fo(S) - f ( S . t ) 5 e('+ 'IT S ~ P ( f o ( S ) - f ( S . O ) ) + = O SE(O,P)

Sest. by ( 3 . 6 ) . we have

f ( s . t ) - f ( S - t + ~ ) S ~ ( l + ' ) ~ SUP ( f ( S . 0 ) - f ( s . ~ ) ) + = O for r > O Q.E.D. s ' € [ O , c ~ )

Reniark The ~ayof f function fo(S) = ( S - K ) + is a viscosity subsolution of (Le)' but

uot a s~iper-solution of (Le)', we can directly check this by noting the following results:

{ O ) x { O ) x [ O 1 c c ) S c K .21+j&q = { 4 s = rc

{ O } x { 1 } x [o. 00) s > r - {O) x ( 0 ) x ( - ~ . O I S < K

D2,-fo(.S) = { { O ) x (Ol 1) x R u { O ) x {O, 1) x [O' os) S = Ic {O) x (1) x (-00.01 s > ri-

3.5.2 Monotonicity in the Leland Constant

Leland constant is an important parameter,it measures the transaction cost. In vierv of

finance. the bigger the Leland constant. the more valuable the option is. That is to Say:

Theorem 3.5.2 Assume that the comparisorz principle holdc for equation (Le) ' , and let

f-1,. i = 1.2 respectively be a viscosity solution of the Leland equation v i th Leland constant

-1;. i = 1,2. then

111 5 i l 2 * fA1 5 f ~ *

Proof: Let LeA4,. be the Leiand operator with Leland constant ili, i = 1.2, then we only

rieed to prove that fAI l is a viscosity subsolution of Len2 = O 9 actually

in viscosity solution sense,where LeA, ( fA, ) = O and Al 5 A2

TLeii by cornparison principle Theorem 3.3.1, we have that

Leland equation (Le) and

Remark:

1 .In particulas. for A = 0,we get Black-Scholes equation for

f ~ . s 5 f., for any A 2 O wliere fBS is Black-Scholes solution.

2. By virtue of the fact that - 1 fssl 5 fss 5 1 fssl and simi

lia\*c that

lar argument as above, ~ t . e

f ( s . t . (1 - A)) I f.\(S.t) 6 f(S,t? (1 + 11)) where f (S. t. (1 + A ) ) is the Black-Scholes solution for the BS ecpation with volatility

G = v and f(S,t;(i - Li)) is the BlacBScholes solution for the BS equation

wi th volatility I / = ud=, f., (S. t ) is the solution to the Leland equation with Leland

coustarit -1.

Sest ive derive the relationship between the Leland solution and the Black-Scholes

soliition as following:

Theorem 3.5.3

fBs(S, t ) = lim fA(S.t) locally uni fol.mly in Q A 4 0

78

Proof:

1 .By Theorem 3-52, for any (S, t ) E QI fA (S, t ) is nondecreasing with respect to A:so

there is a function f(S?t) such that

point-wisely in Q.

2.By Theorem 3.3.1, we have uniform estimates for the continuous module of fi\

aucl the continuous module m does not depend on A.

3. Let -1 + O in the above inequality, we have from 1 that

so f is uniformly continuous in Q

4.By the nionotonicity of fa, in A and Rudin's theorem 1-12! we have that

f (S? t) = lim fA(S, t) locally uni formly i r t Q A-O

5-By the stabilit- Theorem 1.8. we have that f is the viscosity solution of the Black-

Srlioles equation. and by the uniqueness of the solution of BS equation.we have that

f ( S . t ) = fss(S, t ) . so (3.12) holds.

Q.E.D

Chapter 4

Existence and Lipschitz Continuity of the Free Boundary of Viscosity Solutions for the Equations of p-Laplacian Type

In $5 of Chapter 1 ive have obtained the existence and uniqueness of the viscosity solution

for the follon-ing problem( see Theorem 1-52) :

wLcre Q = Rn x (O: Tl.

In this chapter.we will study some properties of the viscosity sohtions by virtue of the

coinparisou principle obtained in Chapter 1. We mainly study the existence and regularity

of the frcr bounclary. We always assume that uo satisfies:

4.1 Properties of the support

Iii tliis section we study the support of the viscosity solution of problem (4.1). We first

prove that. the support of u is compact if the support of the initial fiinction isl that is the

property of finite propagation speed? then we establish that the support of nonnegative

solution is non-contractible.

U'e assume that F satisfies ( F I ) and (&)> i.e.

( FI ) F(q , X ) E C(*fo),

F is degenerate elliptico i.e. F(q , X + Y ) 5 F ( q , X ) VY > O

n-here Sn denotes the space of n x n symmetric matrices with the usual ordering and

.Io = Rn x Sn. Ar, Y' E Sn

and

where & and sl are constants. Next, we study the condition (G), i t is not hard to prove:

Lemma 4.1.1 Let g sattsfy ( G ) , then

Thus ( F6) together with ( G ) can be replaced by the following condition:

Hcre -4 = --l(al. p). B = B ( n l , p ) : p > 2 .

Remark. Here we define IlXII = t 7 f X ) for X 3 O or - t r ( X ) for X 5 O , which is actually

cq~~ivaletrt to the general definition with the maximum of absolute value of eigenuahes.

We first construct a classical supersolution with compact support.

Lemma 4.1.2 Let F satisFJ (FI ) and (Fs)', uo satisfy ( C b ) , then there ezists a fitnction

ii(r. t ) E c2.'(Q) s- t . L' iS a classical supersolution of problem (4.1) and there ezists a

trnrnber C = C(A. B , p , no 1, T ) > O s - t . u = O as alx12 - t > - aR2 + 1 for all n 5 C .

Proof:

We o d y prove the conclusion for B = 0, the case of B # O can be proved analogously.

and

a . 7- and k are positive constants to be chosen.

3. by virtue of ( F I ) and (Fs)'

41: I Soting that the last inequaiity above employs the fact p > = 2 + for large

4 . ~ ( x . O) 2 O on Rn: hence v(xt O ) > g(x) as 1x1 2 R.

For 1x1 5 R,alx12 - t 5 aR2

Let 7' = a R 2 + 1. then

Thus ~ v e can easil'; get the following property of finite propagation velocity.

Theorem 4.1.3 Let u be a viscosity solution of problem (4.1): then under the assumptionq

of Lemma 4.1.2. there exists a number a = a(,& B,p,n, l .T) > O s-t . u ( z . t ) = O as

nl.r12 - t >_ aR" 1.

Xest ive study the positivity. We first construct a class of subsolutions as following:

Lemma 4.1.4 F satisfies (Fi) and (Fs)', then 3 u E C2.'(Q) is a classical subsolution

of problem (4.1) and sati.$es

Proof:

Let

mhre s = n,p P 2 > 2 + A, k large enough and a > &: r is a constant to be chosen.

Tlieii &) E C2*'([0, m)), o E C2*'(Q).

-4 direct calculation shows

Shen O 5 g 5 1.-k < - d < - 0 0 5 g" 5 k(k - 1) and g(0) =1. Non-

Sou-

whvre we take s = 2. If we substitute g , g' and g" with their representations,then ive P - 2

L(v) 5 - QP ( t + T)"+' dl-

Xon- choose k so large that p > 2 + & then

so ttiere exists a ro > O set. L ( v ) < O if we take a > 5 and choose r > TO.

Xow we give the theorem of positivity propagation.

Theorem 4.1.5 Let u E LSC(Q) be a v iscosi ty subsok t ion of (4.1) and let F satisfy (Fi)

and (F6)'. u ( z o . t o ) > O : u ( z . t o ) ts continuous at x = xo' then l l (xo l t ) > O for t > t o .

Proof:

u ( t o ? t o ) > O and the continuity of u(x,to) at xo imply that there exists a positive

nuniber po s.t. u(z, t o ) > co > O for x E B ( x O , pO) for some constants Q. Now define

.S. O and g are chosen as above in Lemma 4- 1.4, T > ro is to be chosen. Then v is a

viscosiry subsolution of problem (4.1).

Blit

Sow ive clmose 7 so large that $ 5 Q. Then by g' 4 O we obtain that c(z . to) 5

r*(.ro. t o ) 5 CQ < u ( x . t o ) in B ( s o . p o ) . then v(x. t o ) 5 u( .c . to) on Rn. Now by virtue of

<-oiiilxuison principle . ive have that o(z. t) 5 u ( x , t ) in Rn x [to. T] then

O < v ( x o . t ) _< ~ ( " 0 , t ) for t 2 to Q-E .D

It is riot hard to prove the following:

Corollary 4.1.6 Let u E USC(Q) Le a viscosity solution of problem (4.1) and R ( t ) =

{.L. E Rn l u ( s . t ) > 0: 0 < t < T } . then

4.2 Lipschitz continuity of the interface

Iu last sertion? w e prove that the viscosity solution of problem (4.1) has a fiee b o ~ d ~

if the initial function has. In this section, we fist study the monotonicity and syametric-

ity of the viscosity solution by the moving plane method( [GNNi] and [Li].) Then the

iiioliotonicity resdt dl be applied to study the regularity and the as-mptotics of the

interface.

4.2.1 Basic lemmas - monotonicity and symmetricity of viscos- ity soiutions

first prove the so-cded reflexion pincipleo it describes the relationship between the

d u e s of the solutions at two points s p e t r i c d y locating on the either side of a plme-

We i d 1 use a new condition (f i):

witli î = 1 - 2n @ n.n E Rnlnl = 1

Lemma 4.2.1 Let F satisfy (Fi) and (FT). u E C(Q) be a uiscosity solution of problem

(4.1). deirote D = suppu(x, O ) , a compact subset in Rn. and D C S = {x E Rn( <

.r - : O . n > < O ) . for certain 20 E R then

Remark: x. y are symetric with respect to the boundary of the set S.

Proof:

Set U ( S . t ) = u ( y , t ) , y is defined as above.

Obviously.t9S = {s E RnI < x - zoo n >= O) is a plane in Rn and E x - y is

parallel to n.

Thus y = x and v ( x , t ) = u(x?t) on as. Wliile D c S,then y E R I S if x E S. so y $ D as x 6 D, therefore v ( x ~ 0 ) = O 5

u(.l- . O ) .

Sow we consider prob.(4.1) in S. -4 simple calcdation shows that

D,u = rrDYu D:V = I'rDy~I?,

then by virtue of (FI), v ( z 7 t ) is a viscosity subsolution of prob.(4.1) in S.

By Proposition 1.2.2

u(x, t) - U(Z. t ) 5 m ( l x - z()

aud the cornparison principle is applied to our case,we have

By virtue of this lemrna,we can prove the following:

Proposition 4.2.2 Let u ( x , t ) be a viscosity solution of prob. (4.1) and F satisb ( f i ) ' (F;):

11. x2 E Rn.wzth 1x1 1.1221 > &,where & = inf{R > O : suppu(x, O) C BR(0)) , l e t u ( x , O )

satisjij (CUo): and

Proof:

Let H be a plane passing the rnidpoint of x i , x2 and being orthogonal to the segment

-1.~1 x 2 . Thus H satisfies the equation

Son- consider the distance from the origin point O to H

By the condition

n-e have that

Then D and zl are at the same side of plane H ,while $2 is at the other side. We thus

claini froni Lemma 4.2.1 that

Cuder the following assumption on uo.

wc cari get the gIobal monotonicity of the viscosity solutions :

Proposition 4.2.3. Let uo E BUC(Rn)?and satiqfy (MUo)7u is a u ~ c o s z t y solution of

problem (4.1). F satisfij ( f i ) and (F;) , then u is nondecreasing in xl for x 1 < 0: and

[ T L particî~lar.if u o ( - z l : y ) = uO(x19 y ) . then u ( -XI , y, t ) = u(xl, y . t ) .

The proof is similar to that of Lemma 4.2.1.

4.2.2 Lipschitz continuity in spat i d variables and asympt otic symmetricity of the interface

Let

S o = { t > ~ I ~ h ( ~ ) cQ( t ) ) . f l ( t ) = { x ~ R " l u ( ~ . t ) ~ O )

aiid assi.iriie tliat So is not empty,let To = inf So, then we have:

Theorem 4.2.4 Let t > To!then the boundary r ( t ) of n( t ) is Lipschitz continuous in Rn

r e p r e s e n t a b l e in spherical coordinates in the fonn

Obvioiisly APc is not empty for E small enough.

LI-e concentrate on proving that x E Rn/(& U Ii:) for 2 E ï(q. lx -il < aR,because

i t clearly implies that the free boundaq r can be represented as r = f(0, t ) . mhere

s = ( O . 7.) and f is l o c d y Lipschitz continuous with respect to t9 . 2.We claim that u(z,t) > O V x E Ii,'

Let x o E &.if w e can prove that

Ro cos(2 - X o , z o ) > -

1x0 1 wliicli deduces that

R 3 Xo E [o. 1) s.t cos(X0l - .q,. xo) 1 -

1x0 l Howver.by the definition of To and Proposition 4 - 2 2

theri rl(ro. t) 2 i r ( X o I . f ) > 0.

Below we prove (")

inzplies

3. Let r o E Ki .then Proposition 4.2.2 is used to derive that u(xo: f ) = O for x0 E h-L.

Q . E . D

Nes t we study the asyrnptotics of the free boundaq.

Deuote R,~ f ( t ) = sup(IxI: x E R ( f ) ) R,(t) = inf (1x1; x E 8fi(t))

Theu we daim that

Proposition 4.2.5

Proof:

LVe only need to prove that

Dcuote .rt - x xf + x

n = z 0 = - 2

: S = { ~ E RnI < y - z ~ ~ n > c O ) = { y ~ R " I I x - y l < I x - ~ ' ~ ) lxt - z[

tlieu ive only need to test that D c S. D is defined in Proposition 4.2.2.

Artually. if Jyl < & then

x - y < z - y and y E S.

Remark

1. Rat, ( t ). R , ( t ) are nondecreasing in t by their definitions and Corollary 4.1.6. Thus

l h , , R:o ( t ). limt3m &( t ) exist( may be infinity).

2. If lirn,,, Rltl(t) = ca then from Proposition 4.2.5, we have that limt,, 2 = 1 :

tv i i i c l i nieans that O ( t ) becomes more and more like a sphere as they expand to infinity:

If l i ~ n ~ + ~ R,&) = RI < w, hmt+, &(t) = r < oo ,then RI < + 2& and n(t) tcud to lie in the area BR, /Br as t goes to infinity.

4.2.3 Lipschitz continuity in t h e

11-e turn to study the Lipschitz continuity of the free boundary in t . W e need one more

condition ( F8). which guarantees that the equation keeps unchanged under scaling trans-

formation. To be clear,we repeat (&) as following:

The inah result is following:

Theorem 4.2.6 Let F satisfy (Fi). ( F i ) and (&). uo E W2-OD(Q) fl C ( Q ) . u is a u i scos i t~

.wL~~tZ'on of problem (4.2). then the fiee boundary of u can be represented in the form

r = f ( O . t ) for 9 E Sn-' . t > To. To is defined as before. f i s locally Lipschitz continuous

2'71 t and urzzformly for 8 .

To prove this theorem, we first give a lemma:

Lemma 4.2.7 Let assumptions be as aboue, then

It < ho srnall enough, and Iro depends on T. &. t and Iluo 11 rv2.a .

Proof : 1 1. Define u , ( r , t) = -u((l + a)x, (1 + ~ ) ~ t + to), then uI is a viscosity solution of

Prol~lern (4.1) from (F8)-

2. Consider Problem (4.1) in the dornain

wc ni11 employ maximum principle to prove that , u ~ ( L c , t ) 5 u,=o(z, t ) for z # BRo (O) and

to large eriougii. and O < t < hl h is small enough.

(1) For t = 0. 1x1 > &.

'5 u , ( J : . O ) - U ~ = ~ ( Z , O ) = -- u((1 + E)X, to) + u((1 + c ) ~ , t o ) - u ( q t o ) 5 O

l + &

froni Proposition 4.2.2.

E u , ( z . t ) - U(X' t + fol = [-- ~ ( ( 1 + i ) ~ , (1 + ~ ) ~ f + to)]

1+,2 + [u ( ( l + E ) I . (1 + ~ ) ' t + f o ) - U(X. (1 + ~ ) ~ t + to)]

+ [u(x' (1 + ~ ) ~ t + to) - U(X. t + to)] A - II + 1 2 + 13 -

i ) Proposition 4.2.2 implies that

~ ( ( 1 + t)x. (1 + ~ ) ~ l + tO) 5 U(X, (1 + ~ ) ~ t + to) f u r 1x1 > &, thus for 1x1 = &:

so - < O .

ii) Let to 2 t l > To: then there is a ci = ci(ti, &:T) , such that u 2 ci > O on

for 6 smaU enough, thus u((1 + E)X. (1 + a jZt + to ) 3 inf u 2 ci > 0. D d

iii).Tlieorem 1.52 implies that u E W I J * m ( ~ ) . thus I3 = e(2 + e)till 5 E ~ C ~ (Q =

3 Il 1~~ 11- j - Then for c < 1 on 1x1 = & , O < t < h:

for. Ir 5 -CO

From t h we conchde that

3. From Theorem 1.5.2. u E W ~ ~ ' . ~ ( Q ) , and

for E < 1. C depends on IIuoll w2.= . 4. The cornparison principle is now used to foUow that

for t a > t i > To7h 5 E.

sud f ro~n Rademancher's Theorem,

Sow ive derive the Lipschitz continuity of the free boundary :

Proof of Theorem 4.2.6

Soting Theoreru 4.2.4, we o d y need to prove that f is Lipschitz continuous in t.

1. Let ïû E T ( i ) , f > To. t h e n ? > &. and 3 a E ( O , l ) , ~ 5 ~ > O t t l > To. s.t.

i > t , + 0 1 2 + 6,.

2.Lemma 4.2.5 implies that

for a.e.1~1 2 &.To < t i 5 to , to < t < t o + h.h 5

we can choose to = t - then

ah c u h u t ( ~ . t ) + x - D u - u < O for t l + - C t a.e . 1z1>&

2 -

3. hlake spherical transformation x = rû. where 9 is a unit spheral coordinate vector.

then we \mite u(z , t ) after transformation as u(7-? 8, t ).

Sxnoothen this function:

Fis r > R,. t 2 f. choose 6 < min{& ( i i - & ) e 3 } so that

Theu by 2. for t > t' > <

Let S i O. then u is non-increasing in t for t > f and fixed 8.

4. For u(? . B. f) = O, i.e. 1:8 E I'(f): then u ( ~ e h ( ' - O . 8. t) = O L - t

thiis r = f(0, t ) 5 Fe=

theri

for To < ? < t < T. where s E (& t ) ; nom* by virtue of Corollary 4.1.6. f ( B o i) - f (8' 9 1 0. so f ( B . t ) is Lipschitz in t .

Q.E.D

References

[.Au] Antonsev7N..-l>ÿ'a11y symrnetric problems of gas dynamics with free boundaries.

Doldady A b d . Na& SSSR, 216(1974),473-476,

[-A. Pl M. Aveuanda 95 Antonio Paras, Dynamic Hedging Portfolios for Deriva tive

Securi ty in the Presence of Large Transaction Costs, worlcing paper, Courant

Ins t i t u te of Mat hematicd Sciences, New York University, 1994

jh.S] C-Albanese k Stathis T. Small Transaction Cost Asymptotics for the Black and

Schoies Model. FVorking paper, 1997

[.AsMa] -1starita.G. Si G.Marrucci,Principles of Non- Newtonian fluid mecliaoics.

Mcgraw-Hill Book Co. ltd. 1974.

[BI Barenblatt.G.I..On self-similar motions of compressible ff uids in porous media.

Prikl. Mat. Mech. 16(1952). 679-698.(Russian).

[Ba] Barles. G.. Uniq ueness and regularity results for first order Hamil ton-Jaco bi

quations. Ind. u'. Math., Vol 39: 2(1990), 443-466.

[Ba.So] G.Barles Ss H.M. Soner. Option Pricing with Transaction Costs and a

.170niinear Black-Scboles Equation, working papers. 1996

[BC]SI.Bardi 5i 1.Capi.zzo-Dolcetta, Optimal Control and Viscosity Solutions of HJB

Equations. Birkhauser. 1997

[Be] Bear..J.. Dynamics of fluidc in porous media,American Elserier Publisliing Co.

Inc..1972.

[B ra] Brandt ..A.. ln terior es ti mates for second-order elliptic differen tial(or

finite-difference) equations via the maximum principle, Israel 3. Math. 7,

95-121(1969).

[B .SI F. Black & M-Scholes, The Pricing of Options and Corporate Liabilities. J .Pol.

Economics.Sl. 63'7-659 (1973)

[Cal] CaffareUi.L. ,Interior a priori estima tes for solutions of fully nonlinear eq uations,

.4rinals Math.. 130(1989). 150-213-

[Cd]-.A Harnack inequality approach to the regdarity of free boundaries:Part

1:Lipscliitz free boundaries are C'va, Rev-Mat-Ibero-Americana 3(1987) 139-162.

[CL] Crandall,M.G. 9i P.L. Lions, Viscosity solution of Hamilton-.lacobi equations.

Tralis. AMS. 277 (1983) 1-42

[CEL] Crandd,M.G..L.C.Evans & P.L.Lions,Some properties of viscosity solutions of

Hamilton-.Jacobi equations, Trans. Amer.Math.Soc. 2S2(l984) ,4Sï-aO2.

[CHU] Crandal1.M.G. Sr Huang Z.D. .A non-uniq ueness example of viscosi ty solutions of

secoud order elliptic equations, preprint.

[C 11 Cra~dal1,M.G.~ Quadratic forms,semi-differen t ids and viscosi ty solutions of fully

nonlinear ellip tic equations,-4nn.I.H.PPP4nal~Xon.Lin. 6(1959).419-435.

[C2] -. Giscosity solutions of PDEs. ANS Progress in Math. Lecture Series. ..\MS.

Providence. RL. l ( l 9 9 l ) . (MR. 92k:30OOl. p6090)

[CIL] Crandal1.M.G. .H.lshii & P. L. Lions. User S guide to viscosity solutions of second

order PDEs. Bull. of Amer.Math.Society vol 27,1(1992).

[Clil] Clieii.Y -2. . C ' v a regulanty of viscosity solutions of fu& nonlinear ellip tic PDE

under n a t ural structure condition~~preprint .

[CW] Clien. Y. Z Holder continuity of gradient of the solutions of certain degenerate

parabolic equations. Chin. Ann. Math.: SB (3) (1987): 343-356.

[ C h ] Clien \\Tm-Fang: .iV-on-New-tonian fluid mechanics, ( C hinese) Science Publis hing

C0111paXly. 1984.

[CGG] Chen,Y-G.,Y.G.Giga 9i S.Goto, Uniqueness and existence of viscosity solutions

of generalized tnean curtnture flow equations ..J.D.Geometr3.,33(1991) ,749-786

[CVW] CafFareilioL ..4 ..J .L.Vazquez & W.I. Woiolanski Lipschitz continuity of solutions

a d interfaces of n- dimensional PME. hd. Univ. Math..J. ,~01.36~N0.2(198'7)

[C.Z] G.M.Constantinides 8i T.Za.riphopouloul Bounds on prices o f contingent claimes

in an intertemporal economy with proportional transaction costs and generd

preferences, Fin. 9r Stochastics, Vol 3 Issue 3? 345-369(1999)

[Do l] D0ng.G. C .? Viscosity solutions o f the firs t boundary value pro blem for the second

order fully nonlinear parabolic equations under natural structure conditions,

Cornm. in PDE116(6&7)(1991),1033-1056.

[Do21 Doug,G. C. 7xonlinear second order PDEso Tsinghua Univ.Publishing

Conipanyt 1988.

[DB 11 Dong G.C. & Bian B ..J: The cornparison principle and uniqueness o f viscosity

solutions o f fully nonlinear second order parabolic equations. Chiu. Ann of Math.

11B:2 (1990). 156-170.

[DBZI-. The regularity o f viscosity solutions for a class of fufl-y nonlinear equations.

Sci. in Chin(A), 34(199l). 1448-1431.

[c lBH]DiBenndettoE M A . Herrero.On the Cauchy problem and initial traces for a

degenerate parabolic equation. Tram. A.M.S.. 314(1)(1989). 187-224.

[ D H ] Dia2.J. S; M.r\.Hemero,Properties de support compact pour certaines equations

elliptiques et paraboliques nodineaires, C.R. Acad.Sc.Paris,t. 286( 19'iS). $1581 7.

[D.P.Z] 1I.H.A.Davi.s. V.G.Panas &- T.Zariphopoulou, European option pricing

trausaction costs. S I A M .J. Control Optim. 31, 4'70-493(1993)

[Dn] D . Diiffie. Dynamic Asset Pricing Tlieory. Princeton, 1992

[D. Z] SI. H.f\.Davis k T. Zariphopoulou. American options and transactions and

transaction fees, Math. Fin. The IMA Vol in Math and Its .4ppl. Vol

6547-62(1995)

[El j Evans,L.C. ?A convergence theorem for solutions o f nonlinear second order elliptic

equations. Ind. Univ. .J. 27(1918), 8734387.

[E2] -. On sol ving certain nonlinear differen tiai eq ua tions by accretive operator

metkods. Israel J . Math. 36(1980), 225-247.

[ES] Evans.L.C.& .J.Spruck, Motion of level sets by mean curvature 1, IIJII. resp-in J .

D . Geometry: 33 (1991): 633 -681,Trans. .\MS. Vol 330,No.l(1992),321-332. and .J.

Geoxii. Anal. 2 (1992). 121-150.

[Esll] Esteban,J.R.k P.Marcati.Appro>iimate solutions to first and second order

cl uasilinear evolu tion equations via nonlinear viscosi ty Tans. AMS, vol N'&No. 2.

April 1994

[EsVI EsteLan.J.R. Sr J . L . Vazquez.Homogeneous diffusion in R with power-like

nonlinear diffusivity. Arch.Rat.~Iech.~4nalal,103(1988)~40-80.

[F.S] Fleming. W.H. & P.E.Souganidiso On the existence of value functions of

two-player.zero-sum stochastic differentid games. Ind. U. Math. J.; 35(1989).

293-314.

[GGIS] Giga.Y.. S.Goto. H.ISHI1 & M-H.Sato. Cornparison principle and coni-exity

presewing properties for singular degenerate parabolic equations on unbounded

domains Ind. Univ. Math..J. .vol.40, No.2(1991) 443-469

[Gi] Gi1tling.B .H. Holder continuity of solutions of parabolic equations. J . London.

Math. Soc. (2).13 (1976). 103-106

[GXSi] Gic1as.B.. W.M. .Ni 6i L.Nirenberg.S~-mmetriy and related properties via the

maximum principle. Com.hIath.Phys.68(1979) ,209-243.

[GT] Gilbarg,D. & N-S-Trudinger: Elliptic PDEs of Second order. 2nd Edition,

Springer-Verlag, New York, 1983.

[Hl .J .Hull, Options? Futures and Other &rivativesl 3rd edition, Prentice-Hd,

Eriglewood Cliffs, N.J. (1997)

[H. W. W] T. Hoggard, A.E. Whalley & P. Wilmot t , Hedging Option Portfolios in the

Presence of transaction Cos ts,working papers , Advances in Futures and Options

Research(l993 )

[Il] Ishii .H. On uniqueness and existence o f viscosity solutions o f fu& nonlinear second

order elliptic PDE's, Comm.Pure Appl-Math. 42(1989),1445

[12] -.Perron S method for Hamilton-.Jaco& eq uations,D&e Math-J .Zi(l9Sï),369-3S4.

[IL] Ishii .H. & P. L. Lions: Viscosity solutions o f fully nonlinear second order ellip tic

PDEs' .J. Diff. Eq., S3(1990), 26-78.

[IV] Ivanov.-A .V. : Quasihem degenerate and non-uniforndy elIiptic equations of second

order. Proc. of Steklov.Inst. l(l984), 1-387.

[ J 11 Jeuseu,R. The maximum principle for viscos&- solution of fuily nonlinear second

order PDEs. Arch. Rat. Mech. Anal. 101 (1988 ) 1-27.

[J2] -. Uniq ueness criteria for viscosity solutions of fully nonlinear elIiptic PD&, Ind.

L'. Math. J . . 38(1989), 629-667.

[J LS] .Jensen.R. oP.L.Lions & P.E.Souganidis.A uniqueness result for viscosity solutions

o f second order fully nonlinear PDEs. Proc.AMS1102(19SS).975-978.

[Iij Iiruzhliov7S.N.Results conceming the nature o f the continuity of solutions o f

parabolic eq iiations and some o f their applications. Translated as

Math.Xotes.6(1969).517-523

[Iir] 1irylov.X.V.. h'onhear elliptic and parabolic equations o f second order. D. Reidel

Publishing Co. Boston. 1987.

[Lj Li0iis.P. L. Optimal control of diffusion processes and H.JB equations. Part .2:

Viscosity solutions and uniqueness. Comrn.PDE(1983),1229-1246.

[Le] H.E. Leland. Option Pricing and Replication with Transaction Costs. J . of Finance.

30. 1'283-1301 (1985)

[Li] Li.C. ..Mono tonicity and symmetry o f solutions of fully nonlinear equations II.

CT~lboitnded domain? Com.PDE.,16(1991),5S5-615.

[LSU] Ladyzhenskaj a,O.A.,V.A.Solonnikov & N.N. Ural'ceva Linear and quasi-linear

eq uations o f parabolic type, AMS .Transi. ,Providence (1968)

[SI 11 R. C . .Mer t on Lifetirne portfolio selection under uncertainty: the con tinuous case?

Rev.Econ.Statist. 51.247-251(1969)

[JI21 R.C.,MertonOptimum consumption and portfolio mies in a continous-time case,

.J. Economic Theory 3, 373-413 (1971)

[PaPh] Padyne.L.E.k G.A.Philippin,Some applications of the maximum principle in the

problem of torsional creep, SIAM J . Math. Anal: 33(1977) 446-455

[Prive] Proter.3I.H. & H.F. Weinberger. ~ l a x i m u m principle in differen tial equations . Prentice-hall, Englewood Cliffs, 1967.

[RI Riidin. W . ,Principles o f mathematical analysis. (third edit ion ). McGraw Hill

International Book Company, 1985.

[S.So] S.E.Shreve & H.M.Soner, Optimal Investment and Consumption with

Transaction Costs. -4nn. -4ppl. Prob.. 4: 609-692. 1994.

[Tl Trtidinger.X.S .: The Dirichlet problem for the prescn'bed curvature equations, .Ar&.

Rat. Mech. Anal. 111(1990), 153-1'19.

[IV] Wang. L. H.. On the regularity tleorq- of fully nonlinear parabolic equations. 1. I I Ss

III. Conun. on. P. A. M., 43 (1992): 27-76, 141-178 and 255-262.

[\\'.D .H] P. Wilmott .J.Dewynne k S. Hom-ison' Mathematical Finance. Oxford Financial

Press. Odord (1993).

[l-] Yarnada.3.. The HJB equation wi t h a gradient constant.

.J.Diff.Eci..ïl(19SS),185-199.

[ZY] 2liao.J.X Si Yuan, H.J., Lipschitz continuity o f solutions and interfaces o f

erolution p-Laplacian equation. Kortheastern iMatli. .J- S(I) (1992),21-37

Appendix A

Perron Method

In this section w e study the Perron method of viscosity solutions for parabolic equation.

wliicli lias been done for first-order equation by Ishii [I2] and for elliptic equation by

Chen et .al[CGG]. For completeness we give the proof of this method because the proof

for parabolic equation is not seen in the literature.

Tliroughout we assume that F satisfies conditions ( F I ) and ( f i ) .

Proposition 1 Let S be a nonempty family of a subsolution of (0.1) and

U(X? t) = sup{u(x. t)lv E S ) for (r, t ) E Q

suppose u R ( x . t ) < r; for (x, t ) E Q.then u is a WV-subsolution of (0.1)

Proof: By the definition of weak viscosity solution. we need to prove that for al1

fiiiiction 6 E C2*'(Q). if

ma--(.* - 4 ) = (u' - 4)(5? t) Q

1 .IfTithout the loss of generality. assume that (u* - d) (5, t) = O.we can replace 6 with

o( .r . t ) + (u* - 9)(5, f ) to achieve this;

2.St.t Cl(r?t) = o(z. t ) + Ir - 11" + It - il2, then u* - ?J attains its strict maximum in Q

at (2.0 so

3.By the definition of u- , there

with an: = ( u ~ - O ) ( x k ? tk)

is a sequence of (xk, t k ) E Q? ( x ~ , tk) -+ ( 2 , t) such that

iim at = (uœ - o)(Z: t) k400

6.Q is locally compact. there is a compact neighborhood B of (2. t),such that

i*; - O E CiSC(Q) aiid kas upper bound, then it attains its ma-ximurn on B at

(fi. sk) E B. so

and ire have if we note that lim,, ak = O

ï.Siiire that LI^ is subsolution of (O.l)'we have at (xr, lk)

wr tlien get hy letting k -t m.so at (1.. t)

Lemma 2: Let g : Q + R be a supersolution of (0.1),

S, = { V ~ U 5 g. u is subs~lution of (0-l)) , i f v E S, and v ( x , t ) 3 w(x; t )? (x. t ) E Q for all u1 E S,then u is a supe~solution of (0.1)

Proof:

1.If 1. is not supersolution of (0.1), then there is a function 6 E C2**(Q). and a point

(.E. f ) E Q s.t.

m..u(v- - 6 ) = (z', - +)(Z, f ) = O Q

since the function 6 can be modified as d + lx - 212 + It - f12 if necessary

2.Clearly c. 5 g. in Q. so v. (f t) = Q(Z: f ) < g-(5 . 0. otherwise it would contradict the

fact tliat g is a supersolution of (0.1 1. 3.F is coutinuous and O E CZvl(Q). for 6 > O small enough Ive have

d(x. t ) + J2/2 5 g. (z. t )

for y E B2& = B n B((5. f). 26). mhere B i s a compact neighborliood of ( r f ) and

B((.r. t ) . 6 ) = {(y. s) E Q [ I x - + It - siï < 6).

4. ("') inclicates tliat the function d(x, t ) + d2/2 is a suLsolution in B2&? furthermore we

liaw

V ( X _ t ) 3 v.(x. t ) - d2/2 2 +(z. t ) + J2/2 on BÎs/Bs

5.Sow define w ( x , t ) b y

Jiccording to Proposition 1: w is a subsolution of (0.1) over Q and thus w E S;

6.Siuce

0 = ( u . - o)(z.~) = liminf{(v - 4)(x,t) l(x.t) E Q and lx - 31 + It - f[ 5 1 ) t-bo

IL-hich implies that there is a point (z, s ) E Ba such that v(z, s) - Q(Z? s) < S2/2 and

r T ( = . s ) < m ( z . s), a contradiction to the assumption. So v is a supersolution of (0.1)

Proposition 3 Suppose that F is degenerate paraboiic and continuous, let f and

g : Q -> R be respectiuely a sub- and ~upersoZution of (0.1). If f 5 g in Q ,then there

e z i s t s a soiution u of (0.1) satisfying f 5 u < g in Q .

Proof: We will use Perron method. As in Iemma 2,we set

S, = {alti is n subsolution of (0.1) and v $ g ) . Since f E S, so Sg # a. Dcfine

U ( X , t) = sup{v(x. t)lv E Sg)

B y Proposition 1, u is a subsolution of (O.l),so u E S since u 5 g. Then by Leuima 2 u

is a supersolution of (0.1) and w e have

Appendix B

Ascoli- Arzela Theorem on Unbounded Domain

Theorem 1.1 1 (Ascoli-Arzela theorem on unbounded domain) If E C Rn is separable

space.f,, E C( E ) ( n = 1, 2, - - -), there exists a continuous modulus rn independent of n,so

that If,,(x) - fn(y) 1 4 m ( l x - y[). { fn} are bounded pointwise on E.then { fn) has Iocally

~unifomly convergent subsequence.

11'~ will use -Ascoli-Arzela compactness theorem and the following lemrila to prove this

t heorerii .

Lemma: If { f,,) is a sequence of functions on the countable set E,. and for any

.r E Ec. {f,z(x)} is bounded. Then there is a subsequence {f,,) such that (f, , ,(r)}

converges for a11 rr E E,.

Tlie proof of the lernma can be found in [RI.

Proof of Theorem 1.11: W e prove it in two steps l.Show that {f,) lias a

stil~sequence converges locally iinifonnly to this function

1. E is separable, so tliere is a countable dense subset E, of E; now by Lemma. there is

a subsequence { f,a,) such that {f,,(x)) converges to a function, Say. f (x)?for al1 z E Ec-

Soir define function f(z) : E -t R

f (2) is well-defined.because

1 ). f ( z,, ) is convergent as zn + z.actually. from

WC> get after letting rt + m

XOW for z,' z,, E Ec7 (2,) is a Cauchy sequence, we have

so { f ( z , ~ ) } is also a Cauchy sequence? so lim =.-= f ( z , ) is well defined. =nEEc

2). The value of f at z E E / E c does not depend on the choice of the sequence {in}

coiiverging to 2, naniely, for al1 zn + 2, x, + r. we should have

Sou- let 772 + s we have

f ( z ) = Lm fnk(z) = lim fnk (2) k+ao k+oo

Xbove al1 Ive get

2 . For any comapct

Lm f&)= f(z). Z E E &-+Cu

subset I< c E, considering the subsequence (f,,) got in 1. by

-Ascoli--4rzela t heorem [RI, we have that there exists a subsequence { f,,, } uniformly

couverges to certain function g on K; now by 1. g ( z ) = f(z) for z E K. so {f,,,) iiuiforrnly converges to f on K.

Q.E.D.

Appendix C

Notations

1. L'ector and set

n-dimensional real Euclidean space

( O ? - - . : 0 , l : 0: - - - 0) (1 is the ith entry)

a point in Rn

K is compact in V

Space of n x m real matrices

2. Fririctions and function spaces

Let Q be nu open set in R+'! and v ( x ? t ) be a function on Q

the derivative in time t of function u(x. t )

the derivative in spatial variables of function u(z , t )

Hessian matrix of function u

the space of upper-sernicontinuous functions in Q

the space of lower-semicontinuous functions in Q

the space of continuous functions in Q

the space of bounded uniformly continuous functions in Q

the space of continuous functions with compact support in Q

the support set of u

r,1.-?,l -"(Q)

C2.' (Q)

the space of essentially bounded functions

the space of Lipschitz continuous functions

{u(zJ) E LODII+:t) - u(y,s)l 5 C(lx - Y[ + I f - 4)) the space of bounded functions

with bounded first and second order derivatives

{u E ~"+'(li)(u~. Du.D2u E Lnf l (V) ,V V CC Q }

{ U E C(Q)Iut? Du,D2u E C(QH

the transpose of the matrix A

the trace of the matrix A

the norm of the matrix -4 and defined as:

sup ,ER. 1 < - 4 ~ . > 1 = max{lXl : X is an eigenvalue of -4) Isl= 1

the unit matrix

the zero matrix

4. Operation and relation marks

the inner product of vectors .E and y in R

the tensor product of vectors x and y in Rn

for ail

there exist (s)

deuote ... as or is denoted by

respect ively

such that