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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 HamiltonJacobiBeliman (HJB for short) equation,Leland equation and equations of pLaplacian 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 abote models:
1). Bernstein estimates ( especially
estimates ) and convexity of viscosity solutions
of the HJB equation: 2). monotonicity in time and in Leland constant of the viscositu solutions to the Leland equation and the relationship between Leland solutions and BlackSclioles 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 pLaplacian 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 siipersolutions 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 pLaplacian 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, BlackScholes equation,
free b o u n d q
Leland equation
pLaplacian
ACKNOWLEDGMENTS
Tliauks are due to my supervisor, ProfLuis 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 Convexitypreserving 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 CostViscosity 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 . . . . . . . . . . . . . . . . . . . 351 ,Monotonicity in time t . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Monotonicity in the Leland Constant . . . . . . . . . . . . . . .
3.2
Deltahedging 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 pLaplacian 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 AscoliArzela 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 HamiltonJacobiBehan 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 pLaplacian equation from nonNewtonian 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
ilkere 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 t7EP 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. HamiltonJacobiBellman 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
uhrre 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(B1ackScholes equation). We
are interested in studying its nonconvex 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 BlackScholes 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 nonconvex payoff functions. the equation
( Le) is mathematically illposed? 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) coridition f (S? = f (S) has no solution for generic, nonconvex 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 pLaplacian t y p e
(PLE)
of which F talces the form:
ut = div([DulP*Du).
p
>2
F ( x , t, r:q. X) = IqlP2tr{[I
q a + ( p  2)]X)q
lq12
nliere. q ~3(I is the tensor product of q; and more general form :
GW
of whicli F takes:
ut
= div(g([Dul)Du)
nliere g
C1 ((O, cm)) satisfying certain structure conditions, one of which is:
lVliat's more, w e consider the aaisotropic version:
(APL)
11hereQ = diag[leiqlp',
 .  le,qJp'1.
?
The above equations describe the motion of fluids with large velocity and nonNewtonian fluicls.(refer to [AsMa], [EsV], [An] and [PaPh].) W e will study the nonnegative 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.. socalled 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.Docs there exist a unique viscosity solution for Leland equation for nonconvex (not riecessarily piecewise 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 BaickScholes equation ?
4.Cnder what conditions do the solutions t o pLaplacian 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 HaniiltonJacobi 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, ne 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 socded 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 superdigerential of u at (z,). The subdifferential 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
iau 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 supersolutions, we can study their relationship . We
s a y conparison principle between viscosity subsolution u ( x , t ) and supersolution 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 diftnO 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 semicontinuous 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 superdifferentials of semicontinuous 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 ahich 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 . eg. 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 AscoliArzela 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
\iscositysolutions 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 cqiiations are required. which excludes many equations with nonsmooth 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 equationsHamilton.JacohiBellman equations, Chapter 2 wili be devoted to this topic.
2.HJB equations The Control of Ito Process and HamiltonJacobiBeUman Equation
The control of Ito process is the basis for the analysis of portfolio optimization problem.
Liben 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: HamiltonJacobiBehan 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 realvalued 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 optlmal 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 HamiltonJ acobiBeban equation
This is a nonlinear equation. if w e just require the nonnegativity 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 etal 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.
st.
< ( 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 continuoust irne finacial models
ivas
started hy Merton in a series of pioneer papers([Ml] and [M2] ). The application
of coritiniioustime 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 BlackScholes analysis of option pricing theory. In a complete financial market without
t rausact ion cost , the BlackScholes 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 Deltahedging argument he
derived au option price that is equa! to a BlackSciioles price with an augmentedvolatility
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 nonconvex payoff functions(e.g. for the
payoff of a portfdio of optionslike bull spread and butterîly spread), Leland equation can
uot bc reduced to BlackScholes equation and Leland equation is nodinear. and generally
nc cau riot
h d analytical solutions.
Hoggard et.al[HWW] gerieralized Leland's work to nonconvex(piecewise 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 illposed. 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 GConstantinides & Zariphopoulou [C.Z] etc):
herr
we
discuss Leland equation for
A 5 1 and establish
the viscosity solution theory for
rionconvex(not necessarily piecewise linear) payoff function.
N î will study the follotving Leland equation :
wliere fo is the payoff function which may be nonconvex, 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 lineargrowth at infinity and for
1 5 1. We also study the properties of viscosity solutions of Leland equation and their
relationship tvith solutio~s BlackScholes 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 BlackScholes
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 BlackScholes analytic
solutions.
4.Some properties of Mscosity solutions for equations of pLaplace type
The pLaplacian equation(PLE for short) was first studied in comtriicted for p
[BI: where Barenblatt > 2 a class of selfsimilar 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 quasilinear 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
; 1diniensional. the estimate ( t IVr l~*),
2 f
plays the crucial role. In highdimensional
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 selfsimilar 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+IJXII, =
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 subsolution
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 quasilinear; 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 etal'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 pLaplacian
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 convexitypreserving property to uonliuear nonhomogeneous 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 nonconvex continuous viscosity solution (Theorem 3.4.1); relaxiation of the constraints of nonconvexity
and piecewiselinearity on payoff functions; finally, the properties of the viscosity so
liitions and tkeir relationship with solutions of BlacliScholes 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 pLaplacian 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 (JensenIshii CrandallLions) 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 pLaplace 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 convexitypreserving 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 BlackScholes 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 uppersemicontinuous (USC for short) (resp. lower semicootciirluous(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 eq. (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
(ll)
{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
= eCtu.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(rly,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 =
r1Y?
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 )  $(z. t ) ) = ( ~ ( f) . ll.(z'f)) 2
QT
Silice
if
is the viscosity solution of (0.1). we have at (2. f)
Son let r = r  ' y and by virtue of transformations (')
and (""), we get at ( t j . f ) :
To construct a viscosity solution. approsiarnatc approach is often used. For example.
Ive cari use a series of solutions to the uniformly parabolic equations ut EAU
+H(Du)= O
to obtain a solution to the equation ut
+ H ( D u ) = O by letting + O.
E
The follorving
proposition claims that this met hod works for viscosity solutions.
Proposition.l.S.(stability) Assume that
u, E
U S C ( Q , ) (resp.LSC) i a viscosity subs
solution (resp. super solution)^f the following
where Q , z tioninc~emingurith respect to s
E,
and u , > ~ Q , = Q, u, converges uniformly
to a furzction u on any compact subsets of Q . About F, , we assume that there eztsts a fr~rtctiorz F, such that for al1 sequences x,, t,, z,, p, to points x. t . z, p and X7 we have that
and
 Y that
converges respectively
zf
II
E LrSC(Q) r e s p .LSC(Q)):then, u i a viscosity subsolution (resp.supersolution ) of ( s
Proof of Proposition 1 8 .
For any
;j E
C2,1(Q)7such that sup(u  9) ( u  ii)(c . = f)
Q
and
(2. f ) E Q
(2. i f ive replace 9 with f)
11e GLU assume that
c7=$+
u
 9 attains its local strict maximum at
we assume that
I r  i IJ+lt1l2
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 enoughSincethat 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 ne 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 ne state the fundamental iemma of viscosity solution theory, wliich is given by
Y .G.Crandall and HISHII ([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 firstorder 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 WVsolution, 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
(AscoliArzela 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
nlirre (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
st.
(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 Wvsolutions.
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 1iscosity 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
corriparison 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
64
Q.for S. E < 1.  ~ ( r l   Iv(i.F)l 5 M. w e get ' Tt <
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(Za)cip2~(~oqa + L. then 1>y(Fia). and let ; 07weget ( c  pi)i\.12 5 p 2 M a 7so i l $ (A)&, + k cPI
< 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 te
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
St.
~
, > 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 12P2]
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 pLaplacian type with (&). p ( p ) = Cg(l)(p612
ne c w take
+ p6<').
,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 1.12.93 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 HISHII (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 pLaplacian 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 pLaplacian 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. we will apply the techniques and results obtained here to study Leland
ccpation.
Chapter 2
Regularity and Convexitypreserving 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.JacobiBellman(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 convexitypreserving property of ~ioiiliuearnonliomogeneous 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 (21), 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
convexitypreserving 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 ith 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 semidefinite positive
5i4
n ) be egenvalues of the matrix
A. then Xf (1 $ i 5
4;).
n)
arc all egruvalues of .42.and ' is semidefinite 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=,
TY
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
sr
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 12a2)
,<otite t hat each matrix above is semidefinite 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 = xh, z = z.
+ h. t ) + u ( z  h o t ) 2u(r,t)5 2
~ &
lets=!/.
.r=y+h,z=yhwehave
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
xiscosity 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)
 2744 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
nith 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' +Tt
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  21.
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.
yoT2 < 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 CostViscosity Solution Theory of Leland Equation
3.1
Introduction
Lplancl rquation was first introduced by Leland in [Le] to incorporate the transaction
rost into BlackScholes analysis of option pricing theory. In a complete financial market
withoiit transaction cost, the BlackScholes 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 Deltahedging argument
lie derives an option price that is equal to a BlackScholes 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 nonconvex 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 BlackScholes 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 nonconvex(piecewise 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 illposed: 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 nonconvex(not necessarily piecemise linear)
payoff fiirictioris.
Iu this cliapter: Ive will study the following Leland equation
wlierr fo is the payoff fuiiction which may be nonconvex' 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 nonconvex payoff function f o ( S ) with lineargrowth 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 BlackScholes 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?
ne stucly the relationship between the Leland solutions and the BlackScholes 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 Europeaustyle 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 bidask 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 noninfinitesimal fixed timestep 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 riskless discount bond Bt
nith 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 riskless bonds w i t h value Bt. Subsequently the
portfolio is dynamicdy adjusted in a selffinancing manner. Its value at time t is
Theu the change in the value of the portfolio from t to t
+ St is
nherc. 6Bt = rBt6Bt and r is the riskless 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
wliere ive c m uot replace E with its e.xpectation E ( E : )= 1 because we can not let dt :
+ 0.
Son ive use deltahedging. following the same hedging strategy as BlackScholes' 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 BlackSchoIes 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
piccewise linear.
The Leland equation derived above is backward 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  $v'(x
 q ) . and
i2 = u 2 ( l
+ A.sgn(X  q ) )
Xow
WC
claim t hat F is continuous and satisfies ( F I ) ,( f i )required by Theorem 1.2.1
ancl Proposition 1.2.2.
F is coiitiuuous. if we note that
Scst w e
check the condition ( F , ) ,
F ( t , q , X + Y ) 5 F ( = , q , X ) . for Y 2 0
Son*w e discuss the sign of
X  q,(notice that Y
2 0,O 5 R 5
6
1)
1.If ,  q Y
> O.then
1,ccaiise each term in the brackets is nonnegative.
Sowwe can use Theorem 1.2.1 and Proposition 1.2.2, and get the follotving cornparison
principle for viscosity solutions of the equation (Le)'.
Theorem 3.3.1
Let u E c(Q).E C ( Q ) be respectzveLy a viscosity sub and superv
solution of the equation ( L e ) ' irr Q = [Ol +oo) x [O. T ) and satisfy (3.3). Then
sup(u(S:t )  u(S,t ) ) 5 e (1tr)T
Q
SUD
(u($ O )
 V(S. O))+
(34)
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 vniformlycontinuous module rn()? then
where na'() i s a unifonilycontinuozrs 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 piecewise 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 ) = erth(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
iie
Properties of The Pricing Functions
know that BlackScholes 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 BlackScholes equation is
couves. nondecreasing with respect to the time to maturity T  t and nondecreasing
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~ipersolution 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
f1,.
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 BlackScholes equation for Leland equation (Le) and
f ~ .5 s
f., for any A 2 O wliere fBS is BlackScholes 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 BlackScholes 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 BlackScholes
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
pointwisely 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 112! 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)
AO
5By 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 pLaplacian Type
In $5 of Chapter 1 ive have obtained the existence and uniqueness of the viscosity solution
for the folloning problem( see Theorem 152):
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 noncontractible.
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
nhere 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
st.
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
P2
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
QE.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 asmptotics of the l
4.2.1
Basic lemmas  monotonicity and symmetricity of viscosity soiutions
first prove the socded 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
ne 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.
LIe 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 hL.
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
11e 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 W2OD(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 nonincreasing
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
Lt
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 firstorder 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 WVsubsolution 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 maximurn 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 (0l)), 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
tbo
+ It  f[ 5 1 )
ILhich 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 (AscoliArzela 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 AscoliArzela 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 welldefined.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<
Ascoli4rzela 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 .
ndimensional 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 uppersernicontinuous functions in Q the space of lowersemicontinuous 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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