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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 o Toronto f

@Copyright by Yi Zhan 1999

m*I

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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, stochastir 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 viscositu 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? = O . with p-Laplacian D2u)

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, w e 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

of free l~oundary 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,

Perron met hod.

cornparison principle

Euler t ransforrnation,

Lipschitz continuity

HJB equation, Black-Scholes equation,

free b o u n d q

Leland equation

p-Laplacian

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 oral esamination. 1 a m iudebted to the Department of Mathematics for providing an excellent environniciit for learuiug aud working.

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

T.S. Abdelrahrnan, for

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

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

0.1

0.2

1

Models and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

5

Revien . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Results

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

0.3

1

17

19

Viscosity Solution Theory of E'ully Nonlinear Degenerate Paraboüc Equations

1.1 Preliminaries

20

..................................

20

1.2 Cornparison principle and maximum principle . . . . . . . . . . . . . . . . 1.3 Estimates of uniformly continuous moduli

27

. . . . . . . . . . . . . . . . . . 35 1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2

Regularity and Convexity-preserving Properties of Viscosity Solutions

of HJB Equation

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

45

...... 2.3 Proof of the lemmas . 2.1 Some Matrix -4lgebra . 2.5 Main Tlieorems . . . .

2.2 Basic Ideas

2.6

3

. . . . Conxrexity Preserving Property .

. . . .

. . . .

. . . .

. . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . . . . . . . . . . . . 51 . . . . . . . . . . . . . . . . . . 5L . . . . . . . . . . . . . . . . . . 53 . . . . . . . . . . . . . . . . . . 54 . . . . . . . . . . . . . . . . . . 60

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

land Equation

3.1

Introduction .

. .. ...... . . . . . . . . . . . ... . . .. . . . . . ..

67

- F o d a t i o n of Leland Equation 3.3 Cornparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Esistence of The Viscosity Solutions . . . . . . . . . . . . . . . . . . . 3.5 Properties of The Pricing Functions . . . . . . . . . . . . . . . . . . . 3-51 ,Monotonicity in time t . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Monotonicity in the Leland Constant . . . . . . . . . . . . . . .

3.2

Delta-hedging with Transaction Cost

4

. . 68 . . 71

..

-/s

. . 76 -. . // . . '78

Existence and Lipschitz Continuity of t h e Fkee Boundary

of Viscosity Solutions for the Equations of p-Laplacian Type

4.1

80

Properties of the support

4.2.1

4.2.2

4.2

. .. .. . . . .. ... ... .. . . .. . .. . . Lipschitz continuity of the interface . . . . . . . . . . . . . . . . . . . . . ,

80

86

Basic lemmas - rnonotonicity and ~~vmmetricity viscosity solutions 86 of Lipschitz continuity in spatial variables and asymptotic symmetric-

ity of the interface . . . .

4.2.3

References

. . . . . . . . . . . . . . . . . . - . . . 88 Lipschitz continuity in time . . . . . . . . . . . . . . . . . . . . . . 91

,

96

A Perron Method

B Ascoli-Arzela Theorem on Unbounded Domain

102

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 priciug 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 equatious 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~iclSn 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

(Fi)

F

satisfies the foUowing assumption:

q, X

F ( z . t. =, q, ,Y) C ( & ) . E F is degenerate elliptic, i.e. F ( x , t , z?q, X + Y )5 F ( x , tt z,

x R x Rn x S n , X ? YE S n

) WM/ O 2

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

.To=Q

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.

iIïc

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 L o ( u ) = o.

aEA

X,

are

TL

x r n real matrix functions in Q, a is sub index,

A is a given set.

2. Leland Equation

u-hrre f i = J1

t + . 4 s i g nf ((Sw ) ,R is called

2

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 n o d n e a r 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

T) c-oridition f (S? = 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 t y p e

(PLE)

of which F talces the form:

ut = div([DulP-*Du).

p

>2

F ( x , t, r:q. X) = -IqlP-2tr{[I

q a + ( p - 2)-]X)q

lq12

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

GW

of whicli F takes:

ut

= div(g([Dul)Du)

n-liere g

C1 ((O, cm)) satisfying certain structure conditions, one of which is:

lVliat's more, w e consider the aaisotropic version:

(APL)

11-hereQ = 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 evolirtiou 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 t o p-Laplacian equations

have free bonndary wit h Lipschitz continuity?

(GLE) and (APL)

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

by a secluence of solutions to ut

+ H ( z , t ,u , D u ) = 0'

its name

nias

obtained

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

+

- EAU+ H ( x . t . u , D u ) = O as E -+ O.) It was later extzuded to general second order equations by Lions [LI and quickly found applications in

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

ut(x't )

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

+ F ( r . t . u ( x , f ) D u ( x , t ) , D ~ U ( X . ~ )5 O . )

ut

for al1 (s. ) E Q(i.e. u is a classical subsohtion of t

+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 , ( Xi)=

; J i t t).D U @ .t) = D ~ ( F ,and D 2 u ( ~ : i5 D 2 i ? ( ~ . t ) : ( f i ) . ) by

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*' and (2.t) is a local maximum of (Q)

u

- p. This is the definition of

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

(just replace 'maximum' with 'minimum' and

'5' with '2'). basic idea of viscosity The

u

sol~itiouis to transfer the 'derivatives ' of the solutions by test functions via maximum priliriple. namelx to replace the 'derivatives' of any order of 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

~ ' . + u ( . t ) = ( ( r . . A) E R x Rn x Snlu(x r. q

1 + h. t + s ) 5 u(x. t ) + rs + (p. h ) + -2( + 4 h . h )

D2.+u ( r . t ) is called super-digerential of u at (z,). The sub-differential is defiued as t R 2 - ~ (t s = -D2-+(- t ) ) . Details can be foiind in [CIL] and in the first section of ). u(x'

Chapter 1( Definition 1.1 and 1.2).

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

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

i-au I>erelaxed

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

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

holdq if u ( z . O)

5 v(z, ) implies that O

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; technique lies in how to apply the condition (Fi). the main

u.

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

fereutiaI(whose definitions will be given later). so that priuciple of

we can find

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

X

O

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

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) viscosity solutions for the eUiptic equations witli the forni of F ( z , z , q, -Y) G ( X ) =

s t ildieci

+ H ( x ,z, q ) .

IshiiEIl] refined Jensen's idea and

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 i p t i c equations: T h o u g h 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

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

1 DUldi o(-) ID4

: Chen. Giga and Goto

[CGG] studied the comparison principle between

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 a t infinity on unbounded domain, they made many assiirnptions on F with the geometry equations as models, two of a-hich are:

(F7jimplies tha.t F is locally u i i i f o d y continuous in

q? (F4)' describes

the continuity of

F iri x . incorporating the basic techique lemma ( rf. Lemma 1.9 i Chapter 1 . I this n ) n

paper. w e 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. p, U.a,a . Y.

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 )

ivhile . to satisfy

- b , ( ~ t, ) ]< L [ x - YI,

(F4). only needs ha

to satisfy:

(B)is xiiilder than Lipschitz continuity . e-g. b, (x.t ) = (bixpl ,. . . :b,,zcn), (b; 2 O, a i (0.1). i = 1.. . ..r t . ) satisfies ( B ) but does not s a t i s l Lipschitz continuity .

( 3 ) . Existence

E

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

S

(1). approximate method via the stability property.

uCt Fc = O, where

(refer to Proposition 1 8 ( . where u is a continuous

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

+

Fc + F

as E

3

0, and u,

+ u as E -+O,

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

y1

E

such that ( u , ( x ,t ) - u , ( y . s )1

< m(lz -

+ 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 viscosity solutions are obt 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 above two methods . Our idea is: firstly Ive get the existence of viscosity the by sollitions for initial functions uo in w2*m(Rn) 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 uiiiformly elliptic equations were studied in [IL], mainly Ly virtue of viscosity for solut ion tecliniques: the C'varegulari ty was first obtained by CafFarelli [Cal] for uniformlÿ equarlliptic equations; Wang[W] extended Caffarelli's method to uniformly ~arabolic 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 degenerate equations. Ivanov of solritions

[IV] introduced some results on local and global estimates of gradient

. 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-ositysolutions 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 theor 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 -1is nondecreasing in Iql, X i j is the ijth entry of the matrix

rsists a Y

S: assume that there

> O.

set.

S u i e that (0.5)requires that

F grow in order of 1qIa and

lxlPas

Iql?11goes to infinity x

for certain nuniber a,0, example: for

satisfying ( F s ) ,i t is noted that natural structure conditions (0 = 1: 1

cases in the above inequality.

4 a + 2) are special

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-qiiations 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 d e t e d n i s t i c problem: Hamilton-Jacobi-Behan equation.

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

Consider the following Ito Process

[LI.

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>).LetZ Le the state space, a meastuable subset of Rhc ) . p is a rneasurable

~~-valued function on .4 x Z x [O, T ] , Ais a measurable subset

of Euclidean space

d).a : -4 Z x [O, Tl x

+ M ( K , N) is measurable, where M ( h ; N) is the space of l x N i

Tl

matrices

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

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

V ( x ,Î ) = V c o ( x T ) t/ 7

tlien q would be an opt-lmal control.

(2, ) T

c0 E

C such that

E Z x [O, T ]

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

for arry (x. ) E Z x [O' 7

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-

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

tion is a solution of

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 solutions 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

II xa.bu[[ 5

C, Ca

2O

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. calJrv=.- < CG, in f c, > O. f, bu,

X.

E B U C ( R n ) .The semiconcavity

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

are uniformly Lipschitz continuous in

c,.

b,

fa E B U C ( R n )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>edone in Chapter 2.

3. Viscosity Solution Theory of Leland Equation

The st itdy of optimal consumpt 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 notransaction 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 Lelcud 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 cost , the Black-Scholes equation provides a hedging portfolio t hat replicates

the contingent claim, which, however, requires continuous trading and therefore, in a market with proportional transaction costs, it tends to be i n h i t e l y expensive. The requiremeut 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; -1is Lelaad constant and is equal to

fi-& v is the original volati1ity.k iç the and

k are

proportional transaction cost and 6t is the transaction frequency. and both dt and for coiivex payoff functions fa(S) =

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

Le also assumed that 1 is small(e.g. 1

(S- Ii)+.where I< is the strike price of the assets. < 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) payoff 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

-12 i.A4vellauda and Paras introduced new model to describe the d p a m i c hedging problciii [.A .Pl : Zariphopoulou et al considered the preferences of investors to incorporate t ransaction 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 b u t t e d y spread. We will derive the existence and uniqueness of its viscosity solutions for payoff function f o ( 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 Black-Scholes eqiiation. of 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 w e 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 u t

+ r u + F ( D u , 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 comtriicted for p

[BI: where Barenblatt > 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, regularities, some of whicli are extended to quasi-linear C'va

~quations divergence type with second natural structure conditions(see [Ch21 and refof

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] (PLE) and (APL), for

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

. w e assume that:

/(SI(,llS+II+IJX-II, =

X=Xi+X-, X + > O , X - g l g satisfies ( G ) in section 2: Cnder (F6). get the properties of finite propagation speed and positivity of viscosity Ive

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

soliitiom by constructing suitable sub- and super solutions.

wliîrr

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

I ' .

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

imder reflection transformation

ilTealso introduce:

(Es)

F ( p 7AX) 2 A F ( p . X ) VA

2O

( p , X ) E Rn x

Sn

this condition actually requires that F is quasi-linear; w h c h will be used to derive the Lipschitz of the free bouiidary in time.

Under ( F ; ) . applying cornparison principle and reflection principle. we get the monotonicitl- of viscosity solutions and the regularity of free boundary with respect to spatial tariables(Proposition 4.2.2 and Theorem 4.2.4)? we also get the asymptotic spherical sy~mnetricity(Proposition 4.2.5). tious( [CVW].[GXNi]). To get the Lipschitz continuity of free boundary in tirne: we require that uo E [email protected] aiid get the estimates of viscosity solutions . a h k h plays the role of

T i geometric method follows Caffaralli et-al's hs

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

(KI I/;IP-~), 2

-1t in Vazquez and Esteban's work([EsV]).

It 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 p o s i t i v i t ~ other conditions

are comparatively general. Sote that

. the regularity

results of free b o u n d w hold after the support of viscosity

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

t ilc st 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 ~ r i n c i p l e

-i the

+ the existence of

the free boundary

Lipschitz continuity of free boundary in spatial variables

(3).coinparisori ~ r i n c i p l e +reflection principle

+ the existence of the free boundary

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

0.3

Results

estimates of u n i f o r d y continuous moduli and existence of

Tlierc are four results: 1.Couiparison principle viscoçity solutions for Cauchy problem of (0.1) under geueral structure conditions for l~oiiiidrduniformly continuous initial functions (see Theorem 1.2.1, 1.3.1 and 1.4.1) : application 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 Chapter 3(Theorem 3.4.1 and Theorern 3.4.3)

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

~ii 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 HJ equations and uniformly elliptic equations. Cornparison principle is one of its most important results, it enables us to study the properties of solutious 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 conriccts the solutions u only at the 'match points'(the maximum of u

less

- 4 ) : taking relatively 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 definit ion of viscosity solutions and introducecl some

T original ideas(see [Cal], [ land [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~isforniingLeland 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 Solution 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.. u!Du,D ~ U )= O t

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 is

F ( x , t . i , q , X ) E C ( J o ) . J o= Q x R x Rn x S n , X IE Sn Y degenerate elliptic.i.e. F(xlt, . q , X Y ) F(x. z . q,.ir') V Y z 5 t.

+

2O

Son- we state the definition:

Definition 1.1 Let u be an upper-semicontinuous (USC for short) (resp. lower semico-otciirluo-us(MC for short)) function i Q. u is said to be a viscosity subsolution of (0.1) n

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

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

-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'.

N e s t we recall an equivalent definition given by 'super(sub)d2flerentiaP, where superdifferential i n domain Q:

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

the closure of the ~ u ~ e t d i f l e r e n t iis: al

(-U(X,

t))

the closure of subdiflerential0~-u ( r t ) can be defined analogously. We also use B2.+u( 5 .t )

aiid

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

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

Proposition 1.2 Assume that F E C ( Q x

tiorz (resp. supersolution ) of e-q. (0.1)

Î

R

x

Rn x S n ) , Sn

i the space of n x n s

symmetric matrices, then u E U S C ( Q ) (resp. LSC(Q)) i said to be a viscosity subsohs

. if

and o d y if the following statement holds.

(x, t ) E Q ?( T ?q, -4) E D Z 1 - u ( z ,) ) t Remark If u is a viscosity subsolution of ut F 5 O and F is continuous, then ut

T

+ F ( x . t. u . q. A) 5 O for (resp. + F ( x . t ,u , q, A ) 2 O

(3:

t ) E Q , (T'q?.l E D 2 b ( xt, i) )

for

+

+

F ( r . t :u(s:t ) , q7A) 5 O for (x, ) E t

t u supersolutions and solutions.

Q and

(T,q, A) E D**+u(x, Similar rernarks apply t).

Xow w e give the definition of viscosity solution of (0.1) undcr the initial value condit ion:

U

= ~ J ( tx),

072

d,Q

(l-l)

{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 ) o n Rn. Definition 1.3 Let u E U S C ( Q ) ( ~ ~ ~ ~ . L S C ( is said to be a viscosity subsolution u Q)), ( r e s p . supersolutzon) of (O.1 : (1.1):if u ï.s a VLScoSity ~ubsolution(resp. supersolution ) )

wlirre dpQ = C? x

of (0.1) on Q. and u $ <i> o n apQ (resp. u

3$

o n 3,Q.J

E C ( Q ) is said to be a viscosity solutior~of (0.1) and (1.1);i u is a uiscosity f subsolutzon and supersolution of (0.1)and u = on a, . &

Th.en

u

+

\'iscosity solution is weak sollition.

solution^ it

is closely related to strong solution and classical

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

and

DO^]).

Proposition 1.4 Let F satisfy (Ft),ifu E

IL,

wE1"+' )n C ( Q )satisfies (Q

a.e.in

+ F ( x , t . u 7Du, D*U)= O

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 ( F i ) .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, Z u ) D

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

Proposition 1.6 Let

2 1

= e-Ctu.u be a viscosity solution of (0.1), then v i a viscosity s

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

i a viscosity solution of s

Q,then v(y,t) = u(r-ly,t).

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 ? - y & F) (

-7 =

r-1Y?

O ) ,(y, f ) E

Qr.tlien a t (y. f).with

+ ~ ( yt;. U. rrDyy. rrD2ypr) 5 o.

L e t y = T x . and set $(x: t ) = p(l?x:t ) , then

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

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 + O.

E

The follorving

proposition claims that this met hod works for viscosity solutions.

Proposition.l.S.(stability) Assume that

u, E

U S C ( Q , ) (resp.LSC) i a viscosity subs

solution (resp. super solution)^f the following

where Q , z tioninc~emingurith respect to s

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

and

- Y that

converges respectively

zf

II

E LrSC(Q) r e s p .LSC(Q)):then, u i a viscosity subsolution (resp.supersolution ) of ( s

Proof of Proposition 1 8 .

For any

;j E

C2,1(Q)7such that sup(u - 9) ( u - ii)(c . = f)

Q

and

(2. f ) E Q

(2. i f ive replace 9 with f)

11-e GLU assume that

c7=$+

u

- 9 attains its local strict maximum at

we assume that

I r - i IJ+lt-1l2

S o w ( 3 .f) E

Qcfor e small enough and

B(3.i)

sup ( u - y ) = (u - y ) ( z , f )

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-Sincethat u, converges uniformly to u on

B

aiid (x,.t,) kas a iimit point (xo,to) as E

neccssary ) , t hen

+ O.(we can choose a convergent subsequence if

sup(u - $9) = ( u - P ) ( X O , to)

B

by let ting

E i

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

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) 0 5

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

if n-e let

5

go to

O and use the assumptions on FcQ .E.D.

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

Lolds.

3.Basic lemma

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

Y .G.Crandall and H-ISHII ([CIL]): I. Letnrna 1 9 Let ui E U S C ( ( 0 , T )x R') for i .

P.L.Lions.

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

= 1,. . . ,k with u ,

<

m. let m be a

for r = (r,. ( 7 .p.

-O-

.

rk) E

R ~ where . = NI + - - . N

+ Nk.For ( z ' s ) E RiV x

any g i v e n ibf

R: suppose that

-4) E LIZ.+ tu(=. .s)

cR

Assume that there i s

R x SN" an w > O such that for

x

> O , we have a; < C for

s o m e C=C(M), whenever the followzng condîton i satisjied: s

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 s a t i s e ( 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

-( -u)- for super solution.)

, one uses LSC envelope

u = .

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.

solutioiz of problem

Let F sabisfy ( F I). f , g : Q

i

R are respectively WV snb and saper

(C). f 5 g and

o n Q. Then there exists a

W V solutiorr u satisfying

fLusginQ.

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'

1>y the definition of u* and u , . u'

< u..

then

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

space.

.go

(Ascoli-Arzela theorem on unbounded domain) If E C Rn i separable s

f,,E C (E )( n = 1,2? - -), there exists a continuow rnodufus m independent of n, that jf,(z) - fn(y)I 5 m(lz - y[),{fn) are bounded pointwisely o n E, then {f,,)has

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

locally u n i f o m l y convergent subsequence.

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

t l e solutions mit hout estimating uniformly continuous moduli. i

Theorem 1.12.

sepence

([Ru])Assurne that K is a compact set in Rn and

{f,) is continuow

on K satishing

a. )

fil E C ( K )

Vn

n ? where f E C ( K )

6 ) . f,,converges to f pointwisely , V

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

L è firstly recall the condition (F2) l :

X).

for

I-

5 S. V

(2,

t , r, q?,Y), t , S. Q: ,Y) E JO. (x,

Shen w e 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,,

-os. and let F saticfi

u

< m. limlZl,,

L.

>

(Fi- (F4) O ) ) ,u(z,

or v ( x , 0) be uniformly contznuou.s with moduli

of 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

for1 1 1

ut

+ u + F(x. t, u oDu' D ~ U= 0)

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

Q

and

U

= Rn x

Rn x

k0

(O. Tl.

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

>

çupw(x, s,t ) = N > Nl

Q

\Vc claini that sup O ( z ?y , t )

do. 70 small and for al1 E > 0.

u

2 k a > NI for all

5 < k < 1 and O < 6 < 60, O < y < 70 for

Below we prove this claim.

Denote s =

y. n for th

E,

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

sup{w : lx

ailcl

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

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

1

<

dere

small enough.

@(zo. yo, to)

Thus

> ( 1 - 3 . s ) ~ ~lia > A-, 3 O =

5 . Sow sup @ > O implies that

Claini tkat

Brcause. from 4. for a O

< 5 <&.O

< a - Ni take k = y > %, then there exist < 7 < Aio. and E > O.

y

&(E).

7 0 ( s ) ,set. as

Let 5

-+ O.

w e get the result by noting that lim,o a(&) a. =

eo

6 . We claim that 3

> O.

we

s-t.

(3:y. F)

E C: V

E

<

By the definition of B,

have that f

# T:

nom- ive claim that f

# O.

Othemise?

s.t.O attaics its maximum at ( 5 , . ij,, 0) f r E = E ~ S< = o 3 a, > 0: 6, E ( 0 . JO),7, E ( O 0 70): 5,. 7 = - , j . then from 4.

Son- by

I

-. E s ~ a n d i n giE'

5. IIj -

3 0,

a j s

-+

OO(E~

+ O),

then we get a contradiction if we let j

-+

m.

at (2. f ) yields (@,, I, A) (Z, f ) E D2-+u(z, y, , y,

F) ,D 2 !P(T.

t) 5

-4 E Sn

S o w apply Lemma 1.9 with

K

= 2. u l = u, u2 = - v . s = t , z = (2. it is easy to see y),

-

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

1.2. auci Lemnia1.9. weconcludethat V

X>0.

3 ( Î - ~ . X ) . ( T Z ~ YX Sn ER )

set.

Then by virtue of the definition of viscosity solution,

0 2

~ , + ~ ( r . t ) - ~ ( y , t ) + Ik~a + & >

- F(y,i. ( Z , q, -Gy, U -Y)

ivliere I I = F ( r J . u(Z.r),@,.X)

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

- ~ ( 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 ? T l , R is an open set in Rn( may be unbounded ),

1.If

u(,. O )

&ere BC = a R x R x

(O,T U R x a R x (O, Tl U R x 0 x {O). Then there exists a continuous I

nioclulus n ' z

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 l O I O , 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 i ofthe fonn F ( u -Du. D2u). s

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

i a uiscosity solution of (0.1): liml,l,,lul s

; if

u E C(Q)

< rn? ( x . 0 ) - u ( y , O ) 5 m ( l x - y[)- then u

and

fol- (1.) .( . r . t 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 ~ step 5 and step S. the

talc(. siibseqilence if necessary, aud

fruiii (1.5) of

the step S'take subsequence if necessary. and

X,I satisfy:

t)+ ü ( take subsequence if necessary!) by virtue of the boundedness we compute (1.4) &ter letting d + 0,

and

IL (i.

of u. XOW

1 9 - (1.6) and the condition

(Fi)? from (1.4),O < k a 5 O. a contradiction. So thus

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 i p ( u ( r . t )- u ( z

Q

+ h, t ) ) 5 e

(W+I)T

sup(u(z, O)

Rn

- u(z + h, O' ) )

5 e(Ca")Tm(lhl)

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

principle sup(u(t. t

Q

+ r ) for

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

+ T )-

U(Z?

t ) ) 5 e ( ~' I + s u p ( u ( q r ) - u(x, O ) ) + T

Rn

2.Maximum Estimate

To give the mauinium estimate. we assume that:

(fa

V

u ( z . t ) F ( x , t. ~ ( xt ) , 0,O) .

2 -pluZ - ~

2

1

~

1

~

( s , t ) E Q, for certain pi442

L O+ E ( 0 2 ) -

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

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 ( x o l E Q, so to)

lvliere .\:(:II. T) is a constant dependent on M , T and (xo, E to)

3.We cclaim that M = l i m c + o (v(~.f)l. since Ivl

6-4

Q.for S. E < 1. - ~ ( r l- - - Iv(i.F)l 5 M. w e get ' T-t <

11y lettirig

E

-+

0.6

-t

O

t akiug niasinium on the above formula we have:

M

= lim .-O

&+O

1 v ( 2 .F) 1

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

5 .IC7edisci~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'

11r oved-

5 e-(Z-a)cip2~(~oqa + L. then 1>y(Fia). and let ; 07weget ( c - pi)i\.12 5 p 2 M a 7so i l $ (A)&, -+ k c-PI

< O then

(1.7) is

2 ) . if i:(.E.t)

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

conditions.

sati*

the above

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 [GGIS] that in

d ( g )

=

. s ~ ~ { ~ ( i l l . U ( Y >t ) l l z

t )-

-

<

O,

(z. t) E U } is a continuous modulus of the y.

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

tlir depeuclence of the w ( a ) is unknown.

In this section, we will construct a uniformiy

roiitiniious modulus n i t h explicit dependence by virtue of the constructing techniques of [Il] and the nietliods in Theorem 1.2.2.

F' will use a lemrna in le

[Il] to construct

our

test function:

Lernma 1.3.1. VE >

O , [ > O, m(-) c o n t i n u o w m o d u l w , thete exists a function $ E a

C'((O. C G ) ) . dy' > 0.~5''

< 0.s.t.

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

fiirictiou.

Sest we prove:

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

of (0.I)and u

<

1Vl: tT>

-hf: M i.s certain constant, F satisfies (Fi) (F4),( X . O ) U

mo and OF, s.t.

r ( ~ . 5 m g (l - y 1) ,mo is a continuous modulus, then there ezists a contznuoz~s 0) x rnodulus

W.

depending only o n

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

*(x: y. t) = d ( x :y )

+ B ( x . y, t )

_ t

B ( z .y, t) = 6(lz12

+ IyI2) + ~Y

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

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

3 6 E C 2 ( ( 0 ,m)): .

( O )

qi;

> O, P:( < O.

satis f y i n g

5E (

1

2

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

36

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

L.wbere

A = ((5.y) E

II7e prove it by contradiction. If not,

~ ~ " -y1 1 5 ~1)1

1

-

6 -t O (take subsequence if necessary.) 3. Claim that ( s o t M~t o ) E U A i la(

QI

as

Clearly. E

# Tt # 0: f

If 1 - J I = 1. then s

a contradiction to the assumption. 4. Espanding 9 at (i.ij?C)yields

(*t.

l

4 ) ( ~ . i j ' t ) D ~ ~ + w ( z D ~ + ( f .y'f) 5 E . 2.~

s = f. z = (3. fi) it is easy to

-4 E S"

Sow apply Lemnia 1.9 witli k = 2. ul = u . uz =

svr

-W.

that assuniption (1.2) is satisfied. Since (Z.j f) E C, by the equivdent definition of i.

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.

(-72:

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

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

5. A direct calculation yields that:

then 1 . 1 1 [ 4

5 2$

+ 26

5&

(yi

r o t e that

fi < s <_ / ,

5 l), from 1 and the properties of 0. :

we

take X =

+.

0, (9)

(1.8) becomes

where 11 = 3 O'!s'

+ 16.w = 26 + 4

~ O:( 2): 6 ~

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

tLen

TIL(I-)

rLO

OF-

is a continuous modulus? and depends only on M ,rno and

Q.E.D.

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

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

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

dere

-1 is nondecreasing in IqI, -Yij is the i j t h entry of the matrix X:and there exists a

S-t.

~

-, > O.

iim p(p)p7 = O , 4 0

p(p) = h(2Mp-' . 2Mp-' )

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 1 &f of parabolic equation of the form 5

L ( u ) = ut

lierrafter

u i ( . s ) . w ( s , t ) will

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

denote functions which are moduli of continuity type, and are

definecl and continuous for nonnegative values of t heir arguments7are nondecreasing wi t h

respect to each argument, and w(O), # ( O , O) = O Theorem.l.3.3 Let (xo, to), (xo

+ Ax,to + A t )

E

Q: At >

O . d = dist(Xl, ro). i a u s

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

If

u(x. to) - ~ ( 2 t o ) ~ w(I 5 - xo 1) 0 I (?.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(zo

+ A s , to + A t ) - u(zo,to) 5

l 4 4 l ~ l d

min [ w ( d

+ p(p)At + 2M 1 AX lZ1 P '

(rcsp.

- u ( x o + ~ x , t ~ + ~ t ) + u ( x 5 , tmin [ u ( p ) + p ( p ) ~ t + 2 ~ 12]) ~ ~ ) I ~a~ls~sd pz

and dij i the Kronecker symbol. i particular s n

(resp.

- u ( x ~ , t o + A t ) + u ( x o . t o )< w d ( L j f ) =

(2, ) t

min[w(p)+p(p)At])

O<p<d

2).

If d=O (xo E 30) and if

E Q\Q,

IL!JO

+ Ar. t o + A t ) - u(x0,to) 4

PLI^^

min

At)

+ u ( ~ )p ( p ) A t + 2M 1 Ax 1' 1 + p2

To prove the theorem. we first give two lemmas. They c

dcfiuition of viscosity solution .

m

be checked directly by the

Lemma 1 3 4 Let ..

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.).

u

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

i1

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

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

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 1 d.Let us take an arbitrary nurnber p 5

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(x0.to)+[u(p. At)

+

W(P)

+ p ( p ) ( t - to) + 2 . q

z

- xo 12P-2]

i n the cyliuder

Q" = Q' f~ , p 2 & 1 Q 1

Q.E.D

Remark

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

"(p) =

c(+*) +Ph). C depends on I [ U O ( ( ~ . i ( p ) = cp*. C clepends on C' and IIuollm.

Particularly. if w ( p ) = C ' p e , a .

> O , then

Y. For HJB equation. if the coefficients are bounded. then p ( p ) = C(p-'

theu it satisfies conditions in 1 if we take ,O2 2.

+ p-' + 1).

then

For the equation of p-Laplacian type with (&). p ( p ) = Cg(l)(p-61-2

n-e c w take

+ p-6<-').

,b'>_ SI + 2 to satisfy the condition in 1.

3. Lipschitz continuity

For equation

ut

tve study

+ F ( D u , D Z u )= O

in

Q

t h e 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 ) - u(y. t) 5 Llz - y[; 2 - If u&) = 4 2 .O) E t 1.1-2.93 n C ( Q ) then u(x,t)- u(y. r) 5 C ( l x - y1 (t - TI), 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 + uooc = sup 1 F ( Duo. D 2 u o ) then ci are respectively the super and sub Io

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

1.4 Existence

1; will construct a bounded continuous viscosity solution for Prob.(C) by virtue of Per1

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 ) - ( F s ) ,uo E W'~m(R*)C ( Q ) ,then there exists for n

(O.1). (0.2) a unique viscosity s o ~ u t i o n E BUC(Q). u

Proof:

1. Let u = eQtvtthen u is the solution of the following

2 .Define

v* = fC f

Q

+ g(z)

rvliere C = sup l e - C ~ t [ ~ g e cF (t r ?t , e C ~ ' g ( x eCot g ( x ) :eC0'D Z g ( x ) )17cois the constant in ~ )7 D ]

+

( 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 ~ tg) 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 ~ ' ( xO) E .

W . and '"

l4+-

1irn u oe < 00:

14-+-

lim

.t~,

> -CG

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 ( R e m a r k l ) ,we have that

wliere C depends on

It!/j/L1.2.s

Ilgll

w2.w

mi and r n z are continuous moduli depending only on

aud aF.

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

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

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).

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

1. Define g,(x) = g

Proof :

* p,(x)

where p, is a mollifier. Then

and

1t~J

5 1 + maxlgl,

gL(x) + g(x) uniformly

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 . 4 1 that there exist a sequence of functions uc E C ( Q ) . u,

i the viscosity solution of (0.1)(0.2),: s

B :Proposition 1.2.3. i

iuc15

cmI ~ I J :

ml

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

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 ) .

= uo x

For the general case u o E BCIC(Rn).we approximate u o with

z i r a ( xis a "cutting function" defined as & , ( z ) = 1 for 1x1 )

nierges linearly in n

UO,

en. where

+ 1, and

5 n and O for 1x1 2 n

< 1x1 < n

+ 1: then:

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

2.Iz10,J

5

Iuol:

3.uon(x) - U o n ( y ) 5 m'(lx - y(), where m'(-)is a nodecreasing unifonnly continuous

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

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

With the above properties and uniformly continueous estimates 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),F I o ) ( 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 Ccdependent 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,l , Wo,rl

5 c:c is independent of

a;

cout iiiuous modulus.

Our result is as following:

Theorem 1 5 1 Let ( A i ) - (&) ..

hold, the initial fvnction u o E [email protected](Rn)n C(Rn)or

B l - 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), inequality (0.3) is the

:

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

tr

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

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) (F9). E BUC(Rn),then the above problem .. and uo

lias a unique viscosity solution u E BUC(Q).If

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

uo

E IV2.-, then (F5) (F9)are not and

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 viscosi 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 visrosit' 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 generalizcs 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 convexity-preserving property of ~ioiiliuearnon-liomogeneous equations.

2.1

Introduction

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

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

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

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

us

n to seek the estimates of Bernstein type for viscosity solutions. I this

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

{

CSj is

7~

ut + F ( x . t . u . Du, D2u) = O u ( x . 0 )= u g ( x )

in in

Q = Rn x (O; T j

Rn

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

Lo with

L . ~ ( Jt-..u . Du. D2u) = - t r ( C ( x ' t)'

B

C ( x ot

/3

) ~ ~ u b)g ( x .t)' Du) (

+

+ c ~ ( r ') u - fd(x. t ) t

x nt matrix, t r r i is the trace of n x n matrix A, b E

Rn, u t . 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: denotes the usual scalar product on Rn. ,3 is subindex Y) 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:

5 Er(CBX a ) E V< E Rn 3Co > -m. s.t C&, t ) 1 Co on Q (Hz) . L~ n ) i ~ ? x , . ~ ~ E c = ( (.o . T I c 1 . m ( ~ n ) n C ( Q ) : ~ )

(HI)

0

tLc space T*V is defined as:

I=-*'llPo

The initial fiinction satisfies:

(C;)

UO(X)

E C1>a(Rn) =

l=-='Is~o

Sow w e state our resdts.

Theorem 2.1

Let u E C ( Q ) be a viscosity solution of (2-1), n d e r the a s s u m p t i o n s of u

(HI) - (&)

and (UO),e n th

IuI

for s o m e constant

dC

Theorem 2.2

C depending o n the W m o d u l i o f 6- c. f and I ~ ~ l ~ l . a a( n~d n lu/^=(^) ) Let u E C(Q)n L m ( Q ) be a wkcosity solutions of (2.1): u n d e r the

a..wmzptions of (Hi) - ( H 3 ) and (Uo). i f Co > O. t h e n u satisfies the foilowing inequalzty.

V.r. y . z . .F E Rn. t E [O. Tl C depends o n the Mi moduli of

and

x;b? c , f and o n ( uo

Icl.a(Rn)

l u IL=(^)

To get theorern 2.1 sve should first get the Lipschitz continuity of u in x.

u E

Theorem 2.3 Let

C ( Q ) f~ w ( Q ) be a viscosity s o h t i o n of (2.1). Co c. f E L b.

L X ( [ O .Tl. WA(Rn))n c(Q).1 uo(x) - u , ( y ) 1 L 1 x - y 1 then 5

I4x.t)- u ( y 3 )I

i L' I x - Y

I

(Co). t h e n

L' depends o n L and the corresponding rnoduli of Cy c. f in space Lm([OO W , ( R n ) ) . b, Tl.

Remark.lt i obvious t h a t W s

c LOD([O. l , 6V&(Rn))fi C ( Q ) and i f uo sat+s T

The uest theorern is actually a corollary of the above theorems. it gives the

cstimate of the solutions.

w?iQ,

Theorem 2.4. Let u E CCi?'(Q) n c ( Q ) . ( H ~ - ( H z ) and (Lio) hold for cr = 1. t h e n )

C depends o n the 6 m o d u l i of V

Clb. c. f

for a = 1 a n d

l ~ ~ l , i . i ( ~ n ) ,

l

w2.l

-(QI

= sup 1 u 1 +sup

Q

Q

1 ut 1 +supID,ul

Q

+sup

Q

1 D==u 1

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; T 'CIa(Rn)) l 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, CIqa(Rn)), then Tl,

g ( s .t)

+ g ( y t ) - g(x,t) - g(z,t ) 5 Cao

Vx,y, z . s E Rn t E [O. Tl.

i d e f i e d i (2.2). s n

1 1 the nest section, we will prove Lemma 2.5 and Lemma 2.6: in section 4, Ive study 1

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

In (2.2): we set s = x

Proof of Lemma 2.5:

+h

z =y

+ 12 E Rn then

( ~ ( x h , t ) - u(xl t ) )- ( ~ ( y h , t ) - ~ ( y t,) )

+

+

< - &C(I x - y

replace h with hiei, where h; E

IPI

il

1 + 1 h Il+")

R ' ,

e; denotes the i-th unit vector o i Rn , thenefor

( ~ ( 2 hie,, t

+

) - U(Z, ) ) - ( u ( y t 51

+ hie,, t ) - u ( y 7t))

5

AC([- y 1" r

hi + h f f " )

i

Divide the above inequality on the both sides by hi and let hi

0: we get

By the symmetricity of xty, w e have

1 Diu(x) - Diu(y) 1 &C 1 x - y la 5

Proof of Lemma 2.6.

g(x. t ) E LOD([07 C ' @ ( R n )so Tlt )

S o t e 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

-4< = A<

defiuit ion of egenvalue:

~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

Proof:

ActuallyJet X i ( l

ma*,

then A2 is semi-definite positive

5i4

n ) be egenvalues of the matrix

A. then Xf (1 $ i 5

-4;).

n)

arc all egruvalues of .42.and -' is semi-definite positive. 4

Q.E.D.

3. Let = l i ( l 5 i $ n ) be real symetric man'ces,then

Proof:

We only prove for the case of rr = 2.

( x L l -4i)2$ 2"-'

Since --I1 is real symetric,so by the above Proposition 2.(-41 - -A2)' - -a2

2 O.and

Th

of

nest few propositions are about the computation of tensor product and derivatives

w e oxnit the proof because it c a a be checked directly.

2

IS - yl.

4.(s ::: s' = I ( x c ) x' 3

t r [ ( x :I: x ) A ] = xrrlx

g-&lr -

YI

-

I=,

T-Y

D,lr

- yl" = alz

- y l a - 2 ( x - y) = -DYIx - y ( a

2.5

Main Theorems

all sub indices ,B of

all

Before starting proving the main theorem.we make some simplifications: we only prove

oiir theorems for linear parabolic equations.namely,uve will &op

coefficients : the proof of H.JB case is completely similar.if we notice that for any small

~.tliere is

,o. that for Q < Bo, such

Ld.

wlirre F (x. t . u . p, X) = supass

Proof of Theorem 2.2: We assume that u is Lipschitz continuous in space variable.

il wliirh wl 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 ? xR", t~ [O,T]where ~

To prove (2.3), we fix any 6, 6 > O, and -kI > 0, and set 1

,-(.S.

y. z . .c.

t ) = Mik(s, y. 2. r )

+r (

I2 +*.

Where [ = (s? 2. x). y.

- y ) ( s , y:--, x.t) 5 O on U = R4" x [O. T].for r., ri > O norm of initial value uo siiiall eiiougli and M > O big enough.wlere LMdepends on

L\-e only need to prove that (w

aud 1.t' uornl of al1 coefficients of the equation.

'rote tliat

r

1 C 12.

play respectively the role of a barrier at infinity and t =

(u!

T.

aricl tllat

is bounded and p is nonnegative, so the function

- p)(s,y,z. x , t ) on ( ~ . y , ? . f ? fbe )

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

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

nest.

#

ij. It is obvious that

F # T.

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.

S o w by virtue of Lemma 1.9, here. b = 4.

U I = u ( s , t ) ,uz

<

= u ( ~ ) . ug = -.U(Z. t ) : = t. u4

- ~ l ( s . t ) . ( ( . F ) Ci. then V A > 0, 3 ( T ~ . X ) ~ ( F ~ ~ Y ) , ( T ~ , ZSn such ~ , S ) E E R x ) . ( ~ that

aud

< By the definition of viscosity solution,

E+XE?

72

-73

-T4

+ F(y.5. u(y, t),D , v ( i , t),Y ) 5 O + f ( 5 , f: ~ ( 5f ):. -l);v(f ,f ) : - 2 ) 2 0

U(X.

+ F(l? f.

t .- D = ~ ( f), -X) 3 O ) z~

Siibtracting the last two inequalities from the surn of the first two inequalities we have

.

1 I

I I

-1 -1

-I

-1

s-r

I2

+2(1 + a )1 s - 1

12")

1 0 0 - I O O 0 O

O 0 - 1 0 0

O O

I

+

S (-

61

+ 4a(l + a))1 s - x 12a-2)

,<otite t hat each matrix above is semi-definite positive.only the coefficient of the second

term is negativejf we denote G =

D2$ subtracting the

/ second term.then D2 v S G

wr then choose

E = Y G + 2 r l . and it's not difficult to check that E2 < 2(gp 4r2I ) . + aud G" 5 CdG. where Cd is a constant dependent on a.&bi7 b2 and I - yl. Then.by r

clioosing X = r n i n { L J-):we have that 2hiCd 4r

1

2

3

aud in t lie following,for simplicity,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

above results. We get

~3Co taking the trace and using the

To show that the right Land side of (2.7) 5 Cf&for certain constant Cf depending on

14- tiioduli of the coefficients?wejust 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)

of

i

O as r*

+O

and C is a positive constant depending only on the

u

ILm(qi

W moduli

C . 6. c. f and 1 uo I C ~ . a ( R n )and

Thus Ive arrive

aiid 1wnce.noting Proposition 1.6, a e can always make Co >

C by suitable transformation.

theu -11 >

and

and

r are srnall enough, this inequality leads to a coutradiction.

Proof of Theorem 2.3.

t ha t

U(Z,t

To prove u ( x , t ) - u ( y ' t ) 5 L 1 x - y ' )- U ( Y . t ) 5 M ( S +

1 we need only to prove

I X - Sl 2 ) ~

Y 6>0

P(Z' y. t ) = ~ ( 6 r 1 z I2 the following argument is completely analogous to that of Theorem 2.2. M;e won't restate

Set

+

w)+

+*

here.

Proof of Theorem 2.4.

by (2.2). we have u(r

Set a = 1in Theorem 2.2. then let s = s+h. y = x-h, z = z.

+ h. t ) + u ( z - h o t )- 2u(r,t)5 2

~ &

lets=!/.

.r=y+h,z=y-hwehave

2u(y) - u(y

+h) - u(y -h)

2 ~ \ / 2h 1

thlis

1 D2u s C.By Theorem 2.3. 1 Du s C. Now by the equation 1 , 1 ,

2.6

Convexity Preserving Property

Corivesity is an important property of the value function of HJB equation. I this secn tiori.we will seek how a concave initial function evolves in time: w e 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) convex in X. However. th& method does not apply when F depends on time is

t or x. The main difficulty is that they have to get estimates for growing property of

\-iscosity 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 ~apply viscosity solution method,we u s u d y make a transformato

t ion u = cC'u

to guarantee the coefficient of u is positive or big enough,then unavoidably.

tiine variable t may appear in

eqiiat ion.

P. So it is

necessary to study a more general type of

In t his section. we consider the following Cauchy problem:

Brcause ~ v e study bounded viscosity solutions, rve can apply s i m ~ l e r test functions to

&rive the convexity preserving property under the following conditions:

( 1). F is degenerate parabolic

( 2 ) .F is continuous

(3).5i F ( t . q. -Y) convex on Sn for al1 t E (0:TI.qE Rn is

( 4 ).r

coustarit

i

G(r.) is concave on Rn for t Lc in Rrz

all t

E (O. Tl. G is also globally Lipschitz with

( 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

s cotrtit~uousviscosity solution of (2.1 1 ) and (2.12). If the initial fvnction rro i 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

l = r n a x { L . Lc). i

To prove Theorem 2.7 we need the following two lemmas,

Lemma 2.8 Suppose that function v (z) concave and glo bally Lipschitz with constant is

L in Rra.then

v(x)

f o r ail x,y. 2

+ .(y)

- 27-44 5 LIx

+y -2 4

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 i concave. s

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.

71

= 1 (')

is the assumption;

2. Assume that (') holds for n,now we prove tliat it liolds for n

+1

The second last inequality uses the induction assumption.

Step 2. Using similar method as above. we can prove that

2".

TL.

('1

holds for X = &, k 5

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 ~ (') h l d s for X = X ( n ? k ) ,and v is continuous.so by letting n 2.

-t

m. w e get

Proof of Theorem 2.7:

We will prove that

for al1 ( = (x. z ) E R3".t E [O. Tl. Without loss of generality. we assume that r > 1 For y.

-!.6 F > O and I > 1 we set . i

with

b ( { ) = %lx

1

1 + y - 2~1'+ -

FE,

B(c.t) = s1(1'

6:y

7' +T-t

To prove (2.14) we only need to prove that for every

&(F.-,:

> O

:

there e'cists

=

A-) > O such that

if O < 6 < &. By virtue of Cauchy's inequality,

and the equality holds by letting

('2.15): w e got (2.14).

E

=l x

+ y - 2-1.

Taking this c and letting 7: d

-+

O in

Sow we prove (2.15) by contradiction, if it is false,there would exist

that

0:

> O such

(2.16)

sup @(f:t ) > O with r = 0:

O

= 70. IC = K0

Lolds for a subsequence 6 ,

+ O.By the boundedness

of u and (2.15): we have O < O for

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 ( O : T ) . E 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 ,

siifficiently large f , clearly @(J, ) = -oo a t = t t

<

Siuce <P attains its maximum over (i at ((: 5). so

with D ; Q ( ( ~ ~ ) A, Dt = (D,: $ D,:D=).

'o- by virtue of Lemma ;n

1.9,here,

k = 3, u , = U(I. t ) .U2 = u ( y . t ) , u 3 = - 2 4 ~ t): ( . ) E O. : (t

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

S o w ive compute the derivatives of \k,we denote r , ~= 2

+ i - 22. j

witli

c

=

and 6 = 4.

~ * \ = -S k

E

II'

+ 261

I

-21

-21

-21

. We take A = DZY. since S2= 6S, so llSJl= 6 and

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

Snsuch 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

assiiiiiptioii to get

3

Lc,then w e get a contradiction if we use the couvexity

ive prove (2.14) and complete the proof.

yoT-2 < O. Thus

Xow by

~.irtiie Lemma 2.9, u is concave. of

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 a t discrete times. B y a formal Delta-hedging argument

lie derives an option price that is equal to a Black-Scholes price with an augmentedvolatility

fi= 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 1 -is the strike price of the 1 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 b u t t e d y 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,e

> 1- the coefficient of the second derivative m a -

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 hedging problem. Here we study the c

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 b u t t e d y 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 -15 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 equat ioii(tj2). hen prove the comparison principle of the equation by transforming it into the t

forni to wliicli our results in Chapter 1 apply( $3); then in $4 w e 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 tion of Leland Equation

- Formulafo (S)

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

drpendiug only on the value of the underlying assets at the expiration time T. We will

&l (-oriibinr al1 techniques in [Le]: [WDH] and [ P to derive the model. We mmalie the

following assumptions:

1.Consider a market in which a security is traded with a bid-ask spread is fair to assume that:

-

=

k S t . where St is the average of the bid and ask prices and k is a constant percentage; it

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

ft

et is standard normal distribution: v is

the annualized volatility and p is the drift. 4.Ttie value

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. .

ft

= f(St.t)

.Assume that an investor sells an option with payoff f o ( S ) and t d e s a position consisting of At shares of the security and of risk-less bonds w i t h 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.

ft

= f (Sttt)

ive

expand f (St ) using Ito's lemma ,t

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 = 6f (S,.t ) 'we have

II depends only on S t ,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

wcA have that

-

&St = f ( S t . t ) - af (St' t )st

as

f (S.) satisfies the equation t

for S E ( O , m),t E ( O , T ) and

The Leland constant A =

fi--& an important role in this equation. plays

If -4 >> 1.

t h ~ the, transaction costs term dominates the basic tariance, this implies that transaction u

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 Black-SchoIes equation, Leland equation has one more term t h

where

I Iis w

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 will be studied in detail in constant ;\, the more vaulable the option is. This relationshi~

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 linearlygrowiug 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

as S 2

satisfies Leland equation,and we

constant from (3.3): so g is a

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 II ouiided continuous function.

5h :

So,C the is

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 lt ) - C e r ) . the Leland equation (Le)' becomes:

1 ht - ;.h, - - ü Z ( h , - h,) = - r h

2

V' = v2(l

a2s + A . s g n (as2 -))

= u2(1

+ 1 2 s g n ( s 2a29 -)) dS2

= v2(1+ Asgn(h,, - h,))

i1i.i te the above equation into the general form: (Lch)

ILl + F(?z,. h,,)

=O

h ( x ,O ) = fo(ez) - C e z >

in Q' = sER

R

x (O'T)

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

6

1)

1.If , - q Y

> O.then

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).E C ( Q ) be respectzveLy a viscosity sub- and superv

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

Q

SUD

(u($ O )

- V(S. O))+

(3-4)

SE[O.+=)

If

tr zs

a viscosity solution satisfying above conditions. then

Iii(S

+ AS. t ) - u(S, t)l 5 e(lfrlT

sup

s'[O,+-)

(u(S

+ AS, O ) - u(S. O ) ) + + 2C'IASI

(3.5)

for A S E R such that A S

and

+ S E (0, +m), C'

i a constant depending on C ' T and r . s

for d l S E (O. =x;).t,t T (O:T),r )_ O . In particular, i j

+

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:

ri(cr. t

.Ifter malüng the transformation h(x. t ) = f ( e x .t ) -Cez, we have that u h ( z .t ) =

v(er. t

) - Cer. u t l ( x .t ) =

)

- Cer

are respectively the viscosity sub- and super-

solution of the equation ( L e h ) . and uh E c(Q'), c(Q') bounded in Q' = R x ( O . Tl. vh are

Bj- virtue of Theorem 1.2.1 and Proposition 1.2.2, we have

s u p ( u ( e Z . ) - u ( e z . t ) ) 5 e('+'IT s u p ( u ( e Z 0 ) - v(eX?O ) ) + t Q' theu let S = er.we get

ZER

To prove ( 3 . 5 ) . notice that

uh(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

siip(il(S Q

+ AS. f ) - u(S.t) - C A S ) 5 e(l+r)Tsup

SE[O.oo)

(u(S

+ h S , O ) - u(S.0)- C A S ) +

so

I T

get ( 3 . 5 ) by simple calculation of the above inequality.

As to (3.6) notice that u t , ( 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 i a s

vrrique cor~tinuoz~s~ c o s i t y u sohtion f (S.t ) satisfying l i n e a ~ l ~ - ~ ~ o condition: wing

1f (S.t ) - CS]5 K'

.where

in Q

I' i a constant depending on the parameters C.' of payofl function fo(S) and r. i s h

Proof:

If ive write h (z. t ) = crzg(er.t ) = ert(f (el' t ) - Ce'). the Leland equation (Le)' beromes:

1 hl - rh, - -i2(h,, - h,) = -rh 2

Likite the above equation into the general form:

wharr F ( z . q7X ) = -rq - ?ü2(X- q). and

fi2

= v2(1

+ Asgn(X - g ) ) + CS is the unique

Q.E.D.

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 ) solution to (Le)'. and f (S.t ) is linearly grotving.

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 - X I ) +- (S - &Y2)+. obvioudy it sat isfies above conditions,t hus relevant results hold. Sin?_ilarly,abovetheorems 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 I - XI; Strip consists of a long S

and a call with the same

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

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

ii-e

Properties of The Pricing Functions

know that Black-Scholes equation has an analytic solution for c d payoff function

c = S N ( d l ) - ~ e - ' ( * - N(d2) ~)

( S - -Y)+.

-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 e g a of V

Leland equation have

siiuilar properties: and they hold for more general payoff functions.

3.5.1

Monotonicity in time t

I tliic section rve study hovv the d u e of the option evolves witli respect to tinie. Uë n

claini that if the p - o f f function f o ( S ) is a viscosity subsolution of the Leland equation

and linearly growing a t infinity, then the pricing function has monotonicity property.

Theorem 3.5.1 Assuming that the payoff function f o ( S ) i a viscosity asbsolution of s

( 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

f o ( S ) - f ( S .t ) 5 e('+ 'IT

Sest. by

S ~ Pf o ( S ) ( SE(O,P)

f ( S .O ) ) +

=O

( 3 . 6 ) .we have

f ( s . t ) - f ( S - t + ~ ) S ~ ( l + ' ) ~f ( S . 0 ) - f ( s . ~ ) ) + = O SUP ( for r > O

s'[O,c~)

Q.E.D.

Reniark The ~ a y o f 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:

.21+j&q

=

{

{O) x {O) x [O1cc) S c K

4

{O} x {1} x

[o. 00)

s = rc s>r-

D2,-fo(.S) =

{

S<K {O) x ( 0 ) x ( - ~ . O I { O ) x (Ol1) x R u { O ) x {O, 1) x [O' os) S = Ic s > ri{O) x (1) x (-00.01

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 ( L e ) ' , and let

f-1,.

i = 1.2 respectively be a viscosity solution of the Leland equation v i t h Leland constant

111

-1;. = 1,2. then i

5 il2

* fA1

5f~*

Proof: Let LeA4,. the Leiand operator with Leland constant ili,i = 1.2, then we only be

rieed to prove that

fAIl

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

Remark:

1.In particulas. for A = 0,we get Black-Scholes equation for Leland equation (Le) and

f ~ .5 s

f., for any A 2 O wliere fBS is Black-Scholes solution. a 2. By virtue of the fact that -1 fssl 5 fss 5 1fs l and simil r argument

f ( s . t . (1 - A ) ) I f.\(S.t) 6 f(S,t? (1

as above,

~ t . e

lia\*cthat

+ 11))

where

G =

f (S.t. (1 + A ) ) is the Black-Scholes solution for the BS ecpation with volatility v and f(S,t;(i Li)) is the BlacBScholes solution for the BS equation -

wi t h volatility I/ =

coustarit

ud=,

f ,(S.) is the solution to the Leland equation with Leland . t

and the Black-Scholes

-1.

Sest ive derive the relationship between the Leland solution

soliition as following:

Theorem 3.5.3 fBs(S, ) = lim fA(S.t)locally unifol.mly in t

A 4 0

Q

78

Proof:

1.By Theorem 3 - 5 2 , for any (S, t ) E QI fA (S,t ) is nondecreasing with respect to A:so

there is a function

f(S?t) that such

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

in A and Rudin's theorem 1-12! we have that

4.By the nionotonicity of

fa,

f (S? = lim fA(S, locally unif o r m l y irt Q t) t)

A-O

5-By the stabilit- Theorem 1.8. w e 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: T l .

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:

41 .

Properties of the support

we

Iii tliis section

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 is

F ( q , X ) E C(*fo), degenerate ellipticoi.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. Y' E Sn Ar,

and

where

& and sl are constants. Next, we study the condition (G), it 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. B = B ( n l , p ) : p > 2 . p).

Remark. Here we define IlXII = t 7 f X ) for X 3 O or - t r ( X ) for X 5 O , which is actually

to cq~~ivaletrt 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)t . s-

L '

iS a classical supersolution of problem (4.1) and there ezists a

trnrnber C = C(A. B , p , n o1, 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)'

Soting that the l s inequaiity above employs the fact p > at

41:

I

=2

+

for large

4 . ~ ( x . 2 O on O)

For

Let

Rn:hence v ( x t O) 1 1 5 R,alx12 - t 5 a R 2 x

= a R 2 + 1. then

> g(x) as 1x1 2 R.

7'

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 )

nl.r12

> O

s-t.

u(z.t) =

O as

- t >_ aR"

1.

Xest ive study the positivity. W first construct a class of subsolutions as following: e

Lemma 4.1.4 F satisfies

of problem (4.1) and sati.$es

(Fi) (Fs)', then 3 and

u E C2.'(Q)i a classical subsolution s

Proof:

Let

mhre s = n p 2 P 2, >

+ A,large enough and a > &:r is a constant to be chosen. k

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 < Non-

- 1) and g(0) =1.

Sou-

whvre we take s =

2. we If

P-2

substitute g , g and g" with their representations,then ive '

QP

L(v)5 - ( 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 L S C ( Q ) be a viscosity s u b s o k t i o n of (4.1) and let F satisfy ( F i )

and (F6)'.u ( z o .t o ) > O

:

u ( z . t o ) ts continuous at x = xo' then l l ( x o lt ) > O for t > t o .

Proof:

u(to? o) t

> O and the continuity of u(x,to)at xo imply that there exists a positive

Q.

nuniber po s.t. u(z,t o ) > co > O for x E B ( x O ,pO) for some constants

Now define

.S.

O

and g are chosen as above in Lemma 41.4, Blit

T

> ro

is to be chosen. Then v is a

viscosiry subsolution of problem (4.1).

Sow ive clmose

7

so large that

$5

Q.

Then by g' 4 O we obtain that c ( z . to)

5

r*(.ro.o ) 5 t

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

O

<-oiiilxuisonprinciple . ive have that o(z. t) 5 u ( x , t ) in Rn x [to.T ]then

< 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

{.L.E

Let u E U S C ( Q ) Le a viscosity solution of problem (4.1) and R ( t ) =

Rn l u ( s . t ) > 0: 0 < t < T } . then

4.2

Lipschitz continuity of the interface

o ~ d ~

Iu last sertion? w e prove that the viscosity solution of problem (4.1) has a fiee b

if the initial function has. In this section, we fist study the monotonicity and syametricity of the viscosity solution by the moving plane method( [GNNi] and [Li].) Then the iiioliotonicity resdt

interface.

d be applied to study the regularity and the as-mptotics of the l

4.2.1

Basic lemmas - monotonicity and symmetricity of viscosity 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

. r

(Fi) (FT).E C ( Q ) be and u

E R then

a uiscosity solution of problem

(4.1). deirote D = suppu(x, O ) , a compact subset in Rn. and D C S = {x E Rn(

- : O . n > < O ) . for certain

20

<

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 - zoon parallel to n.

>= O) is

a plane in

Rn and

E

x

- y is

Thus y = x and v ( x , t ) = u(x?t) on

u ( . l - . O).

as.

6 D,

therefore v ( x ~ 0 = O )

Wliile D c S,then y E R I S if x E S. so y $ D as x

5

S o w we consider prob.(4.1) in

S. - simple calcdation shows that 4

D:V

D,u = rrDYu

then by virtue of

= I'rDy~I?,

(FI),z 7t ) is a viscosity subsolution of prob.(4.1) in S. v(

u(x,t) - U(Z. t ) 5 m ( l x - z()

By Proposition 1.2.2

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;):

1 1x2 .

Rn.wzth 1 1 1.1221 > &,where & = inf{R > O : suppu(x, O) x satisjij (CUo): and

E

C B R ( 0 ) ) , l e tu ( x , O)

Proof:

Let

-1.~1x 2 .

H be a plane passing the rnidpoint of x i , and being orthogonal to the segment x2

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 z l 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.

w c cari get the

gIobal monotonicity of the viscosity solutions

:

Proposition 4.2.3. Let uo E BUC(Rn)?andsatiqfy ( M U o ) 7 u a u ~ c o s z t y is solution of

problem (4.1).F satisfij

( f i) and (F;),then u

is nondecreasing

i xl for x 1 n

< 0 and :

[TL

particî~lar.if o ( - z l : y ) = uO(x19) . then u ( - X I ,y, t ) = u(xl, . t ) . u y y

The proof is similar to that of Lemma 4.2.1.

4.2.2

Let

Lipschitz continuity in spat i d variables and asymptotic symmetricity of the interface

S o = { t > ~ I ~ h c Q ) t ) )f. l ( t ) = { x ~ R " l u ( ~ . t ) ~ O ) ( ~ (

aiid assi.iriie tliat

Theorem 4.2.4

So is not empty,let To= inf So, then we have: 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 APcis not empty for

E

small enough.

LI-e concentrate on proving that x E Rn/(& UIi:)for 2 E

it clearly implies that the free boundaq

ï(q.

.

l -il x

< aR,because

r

can be represented as r = f(0, ) . mhere t

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

cos(2 - X o , z o )

Ro >10 x

1

wliicli deduces that

3

Xo E

[o. 1)

s.t

cos(X0l - . , xo) q.

11x0l

R

Howver.by the definition of To and Proposition 4 - 2 2

theri

rl(ro.

t) 2 i r ( X o I .

f ) > 0.

Below w e prove (")

inzplies

3. Let r o E

Ki .then Proposition 4.2.2 is used

to derive that u(xo: ) = O f

for

x0 E h-L.

Q.E.D

N e s t we study the asyrnptotics of the free boundaq. Deuote

R , ~ f ( t ) sup(IxI: x E R ( f ) ) =

R,(t) = inf (1x1; E 8fi(t)) x

Theu we d a i m that

Proposition 4.2.5

Proof:

L e only need to prove that V

Dcuote .rt - x

n=

lxt - z [

z 0 =

xf + x - S = { ~ RnI : E 2

< y - z ~ ~ n > c O ) = { y ~ R " II x - y l <

Ix-~'~)

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 (

lh,,

R:o ( t). limt3m &( t ) exist( may be infinity). 2. If lirn,,, Rltl(t)= ca then from Proposition 4.2.5, w e have that limt,,

If l i ~ n R,&)~ = RI < w, hmt+, &(t) = r < oo ,then RI ~ +

2=1

n(t)

:

tviiicli nieans that O ( t ) becomes more and more like a sphere as they expand to infinity:

< + 2&

and

tcud to lie in the area BR,/Br t goes to infinity. as

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 transformation. 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) ( Q ) .u i a u i s c o s i t ~ ( fl C s

.wL~~tZ'on problem (4.2). then the fiee boundary of u can be represented in the f o r m of

r = f (O. t)

2'71

for

9E

Sn-'t > To.To i defined as before. f i s locally Lipschitz continuous . s

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 + ~ )+ to),tthen ~

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 ~ ( Ltc , 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. 1 1 > &. x

u,(J:.

O ) - U ~ = ~ ( O ), = -Z

'5

l+&

u((1

+ E)X, t o )+ u ( ( 1 + c ) ~t o,) - u ( q t o ) 5 O

froni Proposition 4.2.2.

u , ( z . t ) - U(X' t

+ fol

+ ) (1 + ~ ) +~to)]f ~ , + [ u ( ( l + E ) I . (1 + ~ ) ' t+ f o ) - (1 + ~ )+ to)] t ~ + [u(x'(1+ ~ )+ to) - U(X. t + to)] ~ t

= [-E

1+,2

~ ( ( 1i

U(X.

A

~ ( ( 1 t x (1 ).

so

E

-

II + 1 2 + 13

i ) Proposition 4.2.2 implies that

+ < O.

+~

) ~tO)5 U(X, l + (1

+ ~ )+ to) t ~

fur

1 1 > &,thus x

for

1 1 = &: x

ii) Let to 2 t l > To: then there is a c = ci(ti, i &:T) , such that u

2 ci > O on

for 6 smaU enough, thus u((1

+ E)X. (1 + a jZt + t o )

< t < h:

3 i f u 2 ci > 0. n

Dd

iii).Tlieorem 1 . 5 2 implies that u E W I J * m ( ~thus I3 = e(2 ).

3 Il 1~~ 11- j -

+ e)till 5 E

~

C(Q ~

=

Then for c < 1 on 1x1 = & , O

for.

Ir

5

-CO

From t h we conchde that

3. From Theorem 1.5.2. u E W ~ ~ ' . ~and ) , (Q

for

< 1. C depends on

IIuoll w 2 . =

.

4. The cornparison principle is now used to foUow that

for t a > t i

> To7h5

E.

sud f r o ~ 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,

+

012

+ 6,.

2.Lemma 4.2.5 implies that

for a.e.1~1 &.To 2

< t i 5 t o , t o< t < t o + h.h 5

then

we can choose to = t -

c u h u t ( ~ . t ) + x - D u - u < O for

tl+-Ct 2 -

ah

a.e.

1z1>&

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-? , t ). 8

Sxnoothen this function:

Fis r > R,. t 2 f. choose 6 < min{&

(ii - & ) 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.

u ( ~ e h ( ' - O . t) = O 8.

4. For u ( ? .B. f) = O, i.e.

1:8E I'(f): then

thiis r = f(0, ) 5 Fe= t theri

L-t

for To < ? < t

< T.where

s

E (& t ) ; nom*by virtue of Corollary 4.1.6. ( B oi) - f (8' 9 1 0 f . Q.E.D

so f ( B . t ) is Lipschitz in t .

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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, ) E Q t

suppose u R ( x t ) < .

r;

for (x, ) E Q.then u is a WV-subsolution of (0.1) t

Proof: By the definition of weak viscosity solution. we need to prove that for al1

fiiiiction 6 E C 2 * ' ( Q ) . if

m - ( *- 4 ) = (u' - 4)(5? a-. t)

Q

1.IfTithoutthe loss of generality. assume that ( u * - d)(5, = O.we can replace 6 with t)

o(.r.t )

+ ( u * - 9)(5,) to achieve this; f

2.St.t C l ( r ? t ) o(z.t ) =

at

+ Ir - 1 " + It - il2,then u* - ?J

attains its strict maximum in Q

(2.0so

3.By the definition of u - , there is a sequence of

k400

( x k ,t k )

E

Q?

( x ~ , -+( 2 , t) such that tk)

iim at = ( u oe o)(Z: -

t)

with an: = ( u - O ) ( x k ? ~ tk)

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 a t

(fi. k ) E s

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:

S, = { for all

V ~ U

Let g : Q

+R

be a supersolution of (0.1),

v ( x , t ) 3 w ( x ; t ) ?(x.t ) E

5 g. u is subs~lution of (0-l)), i f v E S, and

S,then u is a supe~solution (0.1) of

Q

u1E

Proof:

1.If

(.E.

1.

is not supersolution of (0.1), then there is a function 6 E C2**(Q). and a point

m..u(v- - 6 ) = (z', - +)(Z, ) = O f

Q

f ) E Q s.t.

since the function 6 can be modified as d 2.Clearly c.

+ lx - 212+ It - f12 if necessary

would contradict the

Ive

5 g.

in Q. so v. (f

t) = Q(Z: f ) < g - ( 5 . 0. otherwise it

fact tliat g is a supersolution of (0.11.

3.F is coutinuous and O E CZvl(Q).for 6 > O small enough

have

d(x.t )

for y E

+ J2/2 5 g (z. ) . t

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 l - 3 = x 1

t-bo

+ It - f[ 5 1 )

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 (0.1). If f 5 g in Q ,then there of

f 5u

e z i s t s a soiution u of (0.1) satisfying

<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 #

t) = sup{v(x. t)lv E

a.

Dcfine

U(X,

Sg)

5 g. Then by

Leuima 2 u

By

Proposition 1, u is a subsolution of (O.l),so u E S since u

is a supersolution of (0.1) and w e have

Appendix B

Ascoli- Arzela Theorem on Unbounded Domain

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

space.f,, E

Rn is separable

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,,) i a sequence of functions on the countable set E,. and for any s

.r E

Ec.{f,z(x)} i s

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.

Soir

f (x)?foral1 z E Ec-

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

XOWfor

z,' z,,

E

Ec7 is (2,)

a Cauchy sequence, we have

so { f ( z , ~ ) } is

also a Cauchy sequence?so lim =nEEc f ( z , ) is well defined. =.-=

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 a 1 Ive get l

Lm f&)=

&-+Cu

fz. ()

ZEE

in 1. by

2 . For any comapct subset I<

-Ascoli--4rzela t heorem

c E, considering the subsequence (f,,)got

K ; now by 1. g ( z ) = f(z) for

z E

[RI, have that we

K.

there exists a subsequence { f, } uniformly ,,

couverges to certain function g on

K. so {f,,,)

Q.E.D.

iiuiforrnly converges to f on

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 ? ) be a function on Q t

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

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[+ If - 4 ) )

the space of bounded functions with bounded first and second order derivatives

r,1.-?,l " ( Q ) {u E ~"+'(li)(u~. Du.D2uE L n f l ( V ) , V

{U

V

CC

Q}

C2.' ) (Q

E C(Q)Iut? Du,D2u C(QH E

the transpose of the matrix A the trace of the matrix A the norm of the matrix -4 and defined as:

sup ,ER.

Isl= 1

1 < - 4 ~ . > 1 = max{lXl : X

is an eigenvalue of -4)

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 a l i there exist (s) deuote ... as or is denoted by

respect ively such that

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