Applications of Variational Inequalities in Stochastic Control
STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 12 ...
40 downloads
1015 Views
21MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Applications of Variational Inequalities in Stochastic Control
STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 12
Editors: J. L. LIONS, Paris G. PAPANICOLAOU, New York R. T. ROCKAFELLAR, Seattle
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NEW YORK
OXFORD
APPLICATIONS OF VARIATIONAL INEQUALITIES IN STOCHASTIC CONTROL
ALAIN BENSOUSSAN Universite Paris Dauphine and I N R I A JACQUES-LOUIS LIONS Collige de France, Paris and I N R I A
English version edited, prepared andproduced by TRANS-INTER-SCIENTIA P.O. Box 16, Tonbridge, T N l l a D Y , England
1982 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK
OXFORD
North-Holland Publishing Company, I982 All rights reserved. N o part of thispublication may be reproduced, stored in aretrieval system, or transmitted, in any f o r m or b y any means, electronic, mechanical, photocopying, recording or otherwise, without thepriorpermission of the copyright owner.
ISBN 0 444 86358 3
Translation of: Applications des Inequations Variationnelles en Contrble Stochastique Bordas (Dunod), Paris, 1978
Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors f o r the U.S.A.and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK. N.Y. 10017
Library of Congress Cataloging in Publication Data
Bensoussan, Alain. Applications of variational inequalities in stochastic control. (Studies in mathematics and its applications; v. 12) Translation of Applications des iniquations variationnelles en contr6le stochastique. Bibliography: p. 1. Control theory. 2. Stochastic processes. 3. Differential equations, Partial. 4. Calculus of variations. 5. Inequalities (Mathematics) I. Lions, Jacques Louis. 11. Title. 111. Series. 8 1-22404 QA402.3B4313 629.8’312 ISBN 0-444-86358-3 AACR2
PRINTED IN T H E NETHERLANDS
FOREWORD
This book treats second order partial differential equations and unilateral problems, as well as stochastic control and optimal stopping-time problems. It deals with branches of mathematics which may at first sight appear totally different and which have developed along quite independent lines, but which are in fact strongly inter-related and which are capable of cross-fertilising each other. The fundamental link lies in the interpretation of the solutions of certain partial differential equations. This interpretation is an extension of the method of characteristics which allows the solution of a linear first-order hyperbolic equation to be expressed explicitly as a functional defined along the characteristic trajectories. A similar phenomenon arises in the case of parabolic or elliptic equations, but the characteristic trajectories then become stochastic processes. In very general terms, it is absolutely necessary to resort to probabilistic models if we wish to be able to give explicit formulas for the s o l utions of partial differential equations (or of systems of.such equations). With regard to nonlinear equations, an important method (but not the only one) for expressing the solution of these equations consists of using the techniques of optimal control. Again, this forms an extension of the Hamilton-Jacobi method in the calculus of variations. The Hamilton-Jacobi equation is a nonlinear hyperbolic equation of first order. Stochastic control leads to quasilinear equations. The book by Fleming-Rishel gives an excellent discussion of the state of the art. Certain variational inequalities which likewise constitute nonlinear problems also possess a probabilistic interpretation. In this case we are dealing with control problems in which the decision variable is a stopping time. Chapter I, which is designed as an extended introduction, presents the problems in formal manner and gives a more detailed description of the contents of the book. We hasten to emphasise at this point that this book is by no means intended to be exhaustive in its treatment, either with respect to the probabilistic models used or to the control problems treated. The probabilistic models are limited to diffusions. The control can take effect via the drift or via the diffusion term, or it can even be a stopping time. We also investigate differential games problems, with or without stopping times. Other probabilistic models and other control problems will be considered in a second volume. In particular, we shall treat impulse control, which leads to quasi-variational inequalities. The book is designed so as to allow it to be read equally well by analysts and by probabilists, and we have followed a policy of using the formalism and the techniques from both disciplines. It is informative to be able to give, when possible, two proofs of a single result: an analytic proof and a probabilistic proof. We have endeavoured to do this in order to bring out the advantage of using the two types of approach in conjunction. The probabilistic methods undoubtedly are the more intuitive, in that in some circumstances they allow explicit formulas to be used for certain quantities. The analytic methods, on the other hand, are
vi
FOREWORD
undoubtedly t h e more powerful and more e l e g a n t when t h e v a r i a t i o n a l formulation I n t h i s c a s e , t h e y a r e c l e a r l y more econand energy techniques can be applied. The p r o b a b i l i s t i c methods a r e very omical a s f a r a s assumptions a r e concernEd. w e l l s u i t e d t o estimates i n t h e space L , and t h e a n a l y t i c methods t o e s t i m a t e s i n Sobolev spaces. However, our o b j e c t i v e i n t h e present book i s not t o i n v e s t i g a t e nonlinear problems of p a r t i a l d i f f e r e n t i a l equations; r a t h e r , it i s t o o b t a i n c o n s t r u c t i v e methods which w i l l allow us t o c a l c u l a t e , i f necessary by using t h e resources of Numerical Analysis, t h e s o l u t i o n of optimal c o n t r o l problems, i n p a r t i c u l a r those We have with stopping times ( a n d , i n t h e second volume, with impulse c o n t r o l s ) . not attempted t o t a k e t h e s u b j e c t m a t t e r as f a r a s it can be t a k e n , and we r e f e r t h e r e a d e r t o t h e bibliography f o r f u r t h e r developments ( u s i n g similar methods) ; numerous a p p l i c a t i o n s a r e described i n t h e r e f e r e n c e s c i t e d i n t h e bibliography; i n p a r t i c u l a r t h e reader may consult Goursat [l], Leguay [l], Maurin [l], F o r t h e nwnericaZ aspects we r e f e r t o Quadrat m, Quadrat [13 and Robin [I]. [2], Quadrat and Viot El], and Kushner [I] and, f o r t h e numerical s o l u t i o n O f v a r i a t i o n a l i n e q u a l i t i e s t o R. Glowinski, J . L . Lions and R. T r h o l i b r e s [11. The following t a b l e of c o n t e n t s shows t h e d e t a i l e d layout o f t h e volume.
T A B L E O F CONTENTS
Chapter 1 :
. 2. 3. 4. 1
. 6.
Synopsis
General i n t r o d u c t i o n t o optimal s t o p p i n g - t i m e p r o b l e m s ..................................
................................................................
Fomal description of stopping time problems
............................
........................ ..............................
1 1 1
Analytic characterisation by dynamic programing
4
Examples of optimal stopping-time problems 4.1 Bayesian formulation of a test problem 4.2 Modelling o f a break-down phenomenon 4.3 A warrant-pricing problem
6 6
............................ .............................. .........................................
8
10
............... 10 GeneraZisations ......................................................... 15 7 . Various characterisations of the optimal cost function .................. 18 5
Gptirnal stopping-time problems and free boundary probZems
Chapter 2 :
S t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s and l i n e a r p a r t i a l d i f f e r e n t i a l equations of second o r d e r
......................................
Introduction
.................................................................
Review of the calculation of probabilities and the theory of stochastic processes 1.1 Probability space Random variables 1.2 Expectation; conditional expectation 1.3 Distribution function : characteristic function 1.4 General discussion on stochastic processes 1.5 Concepts relating to martingales
....................................................
21 21 23
............................. 23 ............................. 24 .................. 27 ........................ 28 .................................. 30 Stochastic integrals .................................................... 32 2.1 The Wiener process ................................................ 32 2.2 Introduction o f stochastic integrals .............................. 33 2.3 Properties of the stochastic integrals as a function o f the upper bound ....................................................... 37 2.4 Ito's formula ..................................................... 38 2.5 Applications of Ito's formula ..................................... 44 Stochastic differential equations : strong formulation .................. 49 3.1 Definition of the problem ......................................... 49 3.2 Investigation of the Lipschitz-continuous case .................... 50 3.3 Investigation of the locally Lipschitz-continuous case ............ 52 .
vii
TABLE OF CONTENTS
viii
...................................... 59 ........................ 70 4. Stochastic differential equations : weak formulation ................. 78 Synopsis................................................................ 78 4.1 Fundamental lemma ................................................. 79 4.2 Girsanov's Theorem ................................................ 82 5. Linear elliptic partial differential equations of second order .......... 86 Synopsis ................................................................ 86 5.1 Preliminary results ............................................... 86 5.1.1 Sobolev spaces ............................................ 86 5.1.2 Trace theorems ............................................ 89 5.1.3 Green's formula ........................................... 91 5.2 Variational formulation ........................................... 93 5.3 H2 regularity and interpretation of the problem to be solved ...... 98 P regularity ................................................... 104 5.4 108 5.5 Elliptic P.D.E. ' s of second order in Rn ............................ 108 5.5.1 Unbounded coefficients of first order ..................... 5.5.2 Bounded coefficients ...................................... 117 3.4 3.5
Use of monotonicity methods Stochastic monotone multivalued equations
.
6.
..
Linear partial differential equations of second ordm, of parabolic type 123
................................................................ 123 Variational formulation ........................................... 123 126 Regularity ........................................................ 6.2.1 Regularity with respect to time ........................... 126 6.2.2 Regularity with respect to the space variables ............128 Parabolic P.D.E.'s of second order in Rn x 10. TC .................. 133 133 6.3.1 Unbounded coefficients .................................... 6.3.2 Bounded coefficients ...................................... 144 Positivity properties of the solution ............................. 147 Green's operator .................................................. 148
Synopsis
6.1 6.2
6.3
6.4 6.5
7
.
.
8
Probabilistic interpretation of the solution of boundary vaZue problems of second order 7.1 Dirichlet problem 7.2 Elliptic problems in Rn 7.3 Interpretation of parabolic problems in Q = 8 x 10.TC 7.4 Parabolic problems in Rn x 10. TC
................................................ 157 ................................................. 157 ............................................ 160 .............167 .................................. 169
Markov process associated with the solution of a stochastic differential equation 172 172 8.1 Interpretation of the function p(x.tl.S. t2) 8.2
8.3
................................................................ ....................... Some concepts relating to general Markov processes ................176 A generalisation of Ito's formula ................................. 183
Chapter 3 :
Optimal stopping-time problems and v a r i a t i o n a l i n e q u a l i t i e s .................................
................................................................. Stationary variational inequalities .....................................
Introduction 1.
1.1 1.2
187 187 189
............................... 189 . .................192
Various formulations of the problem Existence and uniqueness theorem Coercive case
TABLE OF CONTENTS
ix
...................................................... 193 ................................. 196 ............................ 197 1.6 ........................... 198 ............................................... 199 1.7 1.8 ........... 203 ........................................ 205 1.9 1.10 .................................................. 213 ....................... 216 1.11 1.12 ......................... 222 ..................... 224 1.13 1.14 ................................................ 229 .................................... ............. 232 1.15 2. Evolutionary variational inequalities ..................... ............. 235 2.1 The various formulations of the problems ............ ............. 235 2.2 Existence and uniqueness results for the strong s o l u t ons ......... 237 ............. 239 2.3 Penalisation ........................................ 2.4 Proofs of existence in Theorems 2.1 and 2.2 ......... ............. 243 2.5 Estimation of the "penalisation error" .............. ............. 247 ............. 250 2.6 Maximum weak solution ............................... 255 2.7 Some properties of the maximum solution ........................... 2.8 Elliptic regularisation ........................................... 257 258 2.9 Semi-discretisation ............................................... 261 2.10 Regularity of the solution ........................................ 2.11 A free-boundary problem and a one-phase Stefan problem ............ 265 2.12 Further discussion on regularity .................................. 270 2.13 Properties of the solution relative to the domain 0 ............... 273 2.14 Infinite horizon .................................................. 274 2.15 Unbounded open domain. bounded coefficients ....................... 278 2.16 Supports .......................................................... 278 280 2.17 Unbounded open domain. unbounded coefficients ..................... 2.18 Other inequalities ................................................ 287 2.19 Problems periodic with respect to t ............................... 290 iAu ......................................... 295 2.20 Estimate for at 2.21 Maximum weak solution as an upper envelope of sub-solutions ....... 298 2.22 Stability of the maximum weak solution ............................ 299 3. Optimal stopping-time prob'lems. Stationary case ....................... 303 3.1 Synopsis .......................................................... 303 304 3.2 Regular case .bounded open domain ................................ 308 3.3 Non-homogeneous problems .......................................... 3.4 Extension I. Weakening of assumptions concerning the coefficients ...................................................... 309 Extension I1. Weakening of the assumptions concerning J, and P and 3.5 interpretation of the 'penalised' problem ......................... 315 3.6 Extension I11. Additional weakening of the regularity assumptions on Ji and 0 ............................................ 336 3.7 Extension IV . Operators which are not in divergence form ......... 338 Synopsis .......................................................... 338 340 3.7.1 Assumptions - Notation .................................... 3.7.2 The penalised problem ..................................... 348 3.7.3 Investigation of the inequality and solution of the optimal stopping-time problem ..................................... 364 3.7.4 Application to diffusions. Additional results ............ 368 3.8 Probabilistic proof of certain properties of variational inequalities ...................................................... 378 383 3.9 Unbounded open domain ............................................. 1.3 1.4 1.5
Penalisation Proof of existence in Theorem 1.1 Estimation of the 'penalisation error' Monotonicity properties of the solution 'Non-coercive' case Properties of the solution with respect to the domain 0 Regularity of the solution The free surface Unbounded open domain. bounded coefficients Properties of the support of the solution Unbounded open domain. unbounded coefficients Other inequalities Estimates for Au
TABLE OF CONTENTS
X
........................................................... 383 383 3.9.1 Bounded coefficients ....................................... 398 3.9.2 Unbounded coefficients ..................................... 3.10 Investigation of a particular inequality ........................... 402 3.11 An extension of regularity ......................................... 408 4. Optimal stopping-time problems - evolutionary case ....................... 411 4.1 Synopsis ........................................................... 411 4.2 Regular case .bounded open domain ................................. 412 Extension I . Weakening of the assumptions concerning the 4.3 coefficients ....................................................... 415 4.4 Extension I1. Weakening of the assumptions concerning $I and Q and 416 interpretation of the penalised problem ............................ 4.5 Extension I11. Weakening of the assumptions concerning u .......... 418 4.6 Extension-IV . Further weakening of the assumptions concerning 9 , 0 and u ......................................................... 426 4.7 Extension V . Operators which are not in divergence form ........... 428 4.8 Infinite horizon ................................................... 435 4.9 Stopping-time problems in Rn .bounded coefficients ................ 442 4.10 Unbounded coefficients ............................................. 450 4.11 Problems which are periodic in t ................................... 452 4.12 The principle of separation for stopping-time problems ............. 453 4.12.1 Introduction .............................................. 453 4.12.2 Optimal stopping-time problem ............................. 456 5 . Stochastic d i f f e r e n t i a l games with stopping times ........................ 462 Synopsis ................................................................. 462 m e stationary case ................................................ 462 5.1 5.1.1 Assumptions - notation .statement of the problem .......... 462 5.1.2 Penalised problem .......................................... 463 5.1.3 Solution of the inequality ................................. 467 5.1.4 Estimation of the penalisation error ....................... 474 484 5.1.5 Weakening of the assumptions on .$,I $I2 ..................... 5.2 The nonstationaq case ............................................. 488 5.2.1 Operators which are not in divergence form .................488 5.2.2 Operators in divergence form ............................... 490 5.2.3 Principle of separation .................................... 493 Synopsis
Chapter 4 : Introduction
.
1
S t o p p i n g - t i m e and s t o c h a s t i c o p t i m a l c o n t r o l problems
.......................................... ..................................................................
Control by "continuous variable" and by stopping time
....................
495 495 495 495 498 498
........................................................... .................... ................................................ ....................................................... .................................. 501 ......................................... 502 ............................ 506 ..................508 ................................................... 508 ................ 509 .............................................. 509 ................................................ 511
Synopsis The case " 8 bounded" with bounded coefficients Proof of uniqueness 1.4 Penalisation 1.5 Proof of existence in Theorem 1.1 1.6 Regularity of the solution 1.7 Monotonicity properties of the solution 1.8 The case " (P unbounded" with bounded coefficients 1.9 Infinite horizon 1.10 The case " 0 unbounded" with unbounded coefficients 1.11 Maximum weak solution 1.12 Stationary problems 1.1 1.2 1.3
495
TABLE OF CONTENTS
2
.
Review material on the Hamilton-Jacobi equation
xi
..........................
512
................................................................. 512 512 2.1 Notation and aSS~.mptiOn6........................................... 2.2 Interpretation of the solution of the Hamilton-Jacob? equation ..... 516 2.3 Solution of the Hamilton-Jacobi equation ........................... 520 3. The Hamilton-Jacobi inequality . Operator not in divergence form ......... 524 3.1 Penalised scheme. Interpretation .................................. 524 3.2 The Hamilton-Jacob? inequality and the optimal stopping-time problem ............................................................ 526 Synopsis
.
................................. 531 ..................................................... 531 4.2 ................................................... 537 5 . Optimal control and stopping times with polynomial growth ................ 539 5.1 Assumptions - notation - the problem ............................... 540 5.2 Proof of Theorem 5.1 ............................................... 542 6 . The principle of separation .............................................. 547 6.1 Assumptions - notation - the problem ............................... 547 6.2 Preliminary results ................................................ 551 6.3 Variational inequality ............................................. 555 B I B L I O G R A P H Y ................................................................ 559
4
Hamilton-Jacobi variational inequalities
4.1
The games case The control case
This Page Intentionally Left Blank
CHAPTER 1 GENERAL INTRODUCTION
TO O P T I M A L S T O P P I N G T I M E P R O B L E M S
SYNOPSIS
1.
This f i r s t c h a p t e r i s i n t e n d e d t o g i v e a g e n e r a l i n t r o d u c t i o n t o t h e book a s a whole, and does n o t by any means a t t e m p t t o p r e s e n t a r i g o r o u s t h e o r y ; we i n t r o duce s t o p p i n g t i m e problems i n a s i n t u i t i v e a manner a s p o s s i b l e , and we g i v e a number of examples o f a p p l i c a t i o n a s w e l l a s some i d e a o f t h e t e c h n i q u e s which w i l l b e used and developed i n t h e l a t e r c h a p t e r s .
FORMAL DESCRIPTION OF STOPPING TIME PROBLEMS
2.
We s h a l l now g i v e a d e s c r i p t i o n of t h e b a s i c problems, i n i t i a l l y t a k e n t o be a s simple a s p o s s i b l e . We s h a l l d i s c u s s a number of e x t e n s i o n s and more c o m p l i c a t e d s i t u a t i o n s a l i t t l e l a t e r on. We c o n s i d e r a s t o c h a s t i c dynamic system, whose s t a t e y ( t ) ( 6 R n ) e v o l v e s i n accordance w i t h t h e f o l l o w i n g d i f f e r e n t i a l e q u a t i o n ( i n t h e s e n s e o f I t o ) :
(2.1
1
In
g(Rn;R
g.l),g ( x ) and a ( x ) a r e g i v e n f u n c t i o n s on R n , ).
1 . e . we have
(2.2)
i n ( r e s p e c t i v e l y ) Rn and F u r t h e r m o r e , w ( t ) i s a s t a n d a r d i s e d n-dimensional Wiener p r o c e s s ;
1
F,
w ( t ) i s a G a u s s i a n random v a r i a b l e w i t h v a l u e s i n R n , w i t h z e r o mean and w i t h v a r i a n c e
E wi(t) w j ( s ) = bijmin(t,s) ; i,j =
1
... n.
The f u n c t i o n g i s termed t h e d r i f t and t h e f u n c t i o n a i s t e r m e d t h e d i f f u s i o n . The i n i t i a l s t a t e i s x E R n , ( i n g e n e r a l non-random). Very f o r m a l l y , e q u a t i o n I f a = 0, ( 2 . 1 ) i s an o r d i n a r y d i f f e r e n t i a l e q u a t i o n . ( 2 . 1 ) s t a t e s t h a t i f a t t h e i n s t a n t t t h e system h a s t h e known s t a t e y ( t ) , t h e n over t h e i n t e r v a l ( t , t + A t ) ( A t sm a ll) t h e variation A y ( t ) of t h e s t a t e i s a Gaussian R.V. (random v a r i a b l e ) w i t h mean g ( y ( t ) ) A t and w i t h v a r i a n c e a a * ( y ( t ) J A t . N a t u r a l l y , i n o r d e r f o r ( 2 . 1 ) t o b e m e a n i n g f u l , it i s n e c e s s a r y t o make a number o f a s s u m p t i o n s w i t h r e g a r d t o t h e f u n c t i o n s g and u which e n s u r e t h e e x i s t e n c e and u n i q u e n e s s ( i n a s e n s e which w i l l need t o be d e f i n e d ) o f t h e s o l u t i o n of ( 2 . 1 ) . We assume t h a t we have a c c e s s t o a l l i n f o r m a t i o n on t h e p a s t and p r e s e n t s t a t e The i n f o r m a t i o n of t h e system ( * ) , ( b u t n o t , o f c o u r s e , on t h e f u t u r e s t a t e ) . such a t t h e i n s t a n t t i s t h e n (mathematically) d e f i n e d i n terms of a a-algebra
at,
(*)
T h i s w i l l be t h e c a s e i n t h e m a j o r i t y o f t h e s i t u a t i o n s c o n s i d e r e d i n t h i s book. Some c a s e s i n which o n l y p a r t i a l i n f o r m a t i o n i s a v a i l a b l e w i l l a l s o be treated.
1
INTRODUCTION TO STOPPING PROBLEMS
2
t
(CHAP. 1)
t .
t h a t y ( s ) i s 3 - measurable f o r a l l s 5 t . The f a m i l y 3 is an i n c r e a s i n g f a m i l y , which i n p r a c t i c e can be e i t h e r t h e family of o-algebras g e n e r a t e d by t h e process y ( t ) i t s e l f , t 3 I o-algebra g e n e r a t e d by y ( s ) , s 5 t , o r t h e family g e n e r a t e d by w ( t ) , o r even a f a m i l y with a wider d e f i n i t i o n . The d e c i s i o n v a r i a b l e ( t h e c o n t r o l ! ) i s t h e n a s t o p p i n g t i m e , i . e . a p o s i t i v e R.V. 8 such t h a t
(2.3)
event
{e5 t1
c
zt
.
The p r o p e r t y ( 2 . 3 ) means t h a t a t any i n s t a n t t , t a k i n g account o f t h e a v a i l a b l e ' ), we know whether o r n o t 8 5 t . information ( i . e . 3 Furthermore, l e t 6 denote a domain i n Rn and l e t T be t h e first e x i t time of t h e p r o c e s s y ( t ) from 0. , i . e . 5
= i d It 2
o I
y(t)
$ 01 (*)
.
We t h e n d e f i n e a c o s t f u n c t i o n
where t h e f u n c t i o n s f , jl, h , c , a r e g i v e n and where 1 i f e < z
xe 0
- Y;
'
othezle
Ply;+,
- Y;
< 01 =
0
,
E L2(Q,B,P)
.
< o
.
From (1.11)it t h e n f o l l o w s t h a t
( 1 . 1 2)
Y:
Y+
and
E Y + ~EX+
.
F u r t h e r m o r e , we deduce from (1.10) and from Lebesgue's Theorem t h a t
26
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
Considering X-, we define Y- s a t i s f y i n g EY-q = EX-q We then put Y = Y+
-
E Lm(Q,8,P)
.
Y- and hence Y s a t i s f i e s
EYq = EXq
( 1 .14)
vq
(CHAP. 2 )
Vq
.
E Lw(Q,B,P)
1 Since Y E L (Q,B,P) and s i n c e Lw(Q,B,P) i s t h e dual of d e f i n e s Y i n a unique manner. We put
Y=E@X.
s8X by i t s 1 .., .
1 I f now X E L (Q,O,P;Rn)we d e f i n e
Then #X
L1(Q,B,P)
, (1.14)
components
(#xli = #xi , i p 1 i s t h e unique element of L (Q,B,P;Rn) s a t i s f y i n g
( 1 .15)
~ ( 8 x . q )=
m.11 vtl
.
E L~(Q,B,P;R~)
I n p a r t i c u l a r , l e t 19 be t h e sub-a-algebra of U g e n e r a t e d by a s i n g l e element B , i.e. B = {B,CB,Q,$) A &-measurable R.V. i s n e c e s s a r i l y constant on B and on This i s t h u s expressed as follows CB.
.
We then s e e t h a t ( i f P ( B )
lO,l[)
E
j
J X dP B
= xB(")P(B)
+
B'
X dP
CB l-P(B)
/X dP The nwnber
B is realised.
B P(B)
i s termed t h e conditional expectation of X knowing that
It can r e a d i l y be shown from (1.15) t h a t (1 .I6)
E@ i s
1 a l i n e a r operator from L (p,o,P)
and c o n t r a c t i n g such t h a t
#I = 1 .
-
1 L (Q,B,P), and i s i n c r e a s i n g
We a l s o have t h e Hglder i n e q u a l i t y :
(1 .17)
Let p.q
E
Il,m[
1 1 , +-= P
9
1.
I f X E Lp
and
Y
E
Lq we have
We note t h e following p r o p e r t i e s
(1 .18)
(JenSen's Inequality):
a real-valued R . V .
E
l e t cp be a convex mapping from R
L1 and such t h a t
q(X) E L1 ; then
rp(#X)
-f
R and l e t X be
5 E@v(X)
a.s.
(SEC. 1)
Let X
PROBABILITY AND STOCHASTIC PROCESSES
E
XY
L 1 , and l e t Y be ,8 -measurable such t h a t
E[xY\B] = YE[XlB]
(1.20)
L1 ; t h e n
.
The p r o p e r t i e s (1.17)t o (1.20)extend t o R . V . ' s
R1, b2 of
We say t h a t two sub-a-algebras
E
aare
i n Rn.
independent i f
FXY = EXEY
(1.21)
V X b , m e a s u r a b l e , Y b measurable,
2
X,Y 2 0 .
Two R . V . ' s c , n with v a l u e s i n Rn a r e independent i f 3(5), 3(q) a r e independent. Bearing i n mind t h e f a c t t h a t any T(c)-measurable numerical R . V . X i s of t h e form
X =
'p(c) where
'p
: Rn -. R i s measurable
( s e e DOOB Ell, p. 6 0 3 ) , we s e e t h a t we must have
Erp(S)'Jdd
(1.22)
= Erp(S)W(d
Vrp,'? measurable 2 0 .
-
We s h a l l n b e making use of t h e following r e s u l t : suppose we have x C2 R measurable w i t h r e s p e c t t o t h e product o-algebra b ( R n ) x a 5 ( g e n e r a t e d by events of t h e form B x A where B E R(R ) a n d A E a ) . Let B be a sub-o-algebra o f 4; l e t 5 be a ,&-measurable R . V . w i t h v a l u e s i n Rn. We assume that
f(X;w) : R
and furthermore t h a t
( 1 .24)
vx f(x,w)
i s independent of 8 ;
then
1.3
Distribution function;
Let X be a R . V .
(1 .26)
with v a l u e s i n R n .
F(XI,...,x x
c h a r a c t e r i s t i c function
i
E R
We p u t
n = P{X, I x l ,
, X.
...Xn -< xn
t h e i t h component of X .
I t s fundamental i d e a The f u n c t i o n F i s c a l l e d t h e distribution function of X . i s e q u i v a l e n t t o t h e image l a w P of X ( a l s o c a l l e d t h e p r o b a b i l i t y l a w of X).
28
STOCHASTIC D.E.'s
P.D.E.'s
&
OF ORDER 2
(CUP. 2 )
We use the terminology characteristic function of X to denote the Fourier transform of dF, namely rp(ulp...u
(1.27)
=
n
L
...,x,) (u,. ..u,)) .
ex? i f u. xi dF(xl,
= E exp i
U.X
(u
The characteristic function defines dF uniquely (indeed dF is a temuered distribution on Rn and the Fourier transform is an isomorphism of the set of tempered distributions onto itself), We term normal law the distribution function F such that ( 1 .28)
dxl
..dxn
where A is a symmetric positive-definite matrix. The characteristic function of the normal law is
.
[p(ul,. .'un) = exp i u.m
(1.29)
(Abu) -2 .
We have m
E
A = E(X-m)(X-m)*
EX
,
and A is termed the covariance matrix ( * ) .
To $onclyde, we give the following useful result:
g : R
+.
R
, non-decreasing;
In particular if g(X) = A',
1.4
then for a t 0 we have
suppose we have
we obtain the BienaymQ-Tchebichev inequality.
General discussion on stochastic processes
Let ( Q , a , P ) be a probability space. A mapping t +. X(t) of R+ + the set of R.V.'s with values in Rn is termed a stochastic process with values in Rn. This is thus . actually a function X(t;w). We term the trajectory of the process, the family (dependent upon w ) of mappings t -.X(t;w) We interpret t as the time, which thus varies in a continuous fashion. We shall also consider the discrete-time case (the transpositions are straightforward).
.
and (n',C?,P',x'(t)) Suppose we have ( Q , O , P , X ( t ) ) processes. We say that these are equivalent if
(1.31)
(*)
P{X(tl)
E Bl,
...,X ( t n )
E B,]
= P'(X'(tl)
€ Bl,
which are two Rn-valued
...,X ' ( t n )
E Bn]
If A is not invertible, we can no longer take (1.28) as the definition of the normal law; we then u s e (1.29).
(SEC. 1)
If
PROBABILITY AND STOCHASTIC PROCESSES
(Pt,c7',P')
E
v
(1.32)
, we
(Q,Q,P)
29
say t h a t X ' ( t ) i s a modification of X ( t ) i f
t, xl(t) = X(t)
a.s.
A process i s s a i d t o be continuous ( r i g h t continuous, Left continuous) when a . s . i t s t r a j e c t o r i e s s a t i s f y t h i s property.
t
Let 3 be an i n c r e a s i n g family of sub-o-algebras
s s t - $ c
of
a, i . e .
3t.
a family such t h a t
t .
We say t h a t t h e process X ( t ) i s adapted t o t h e family 3 i f
t Vt, X(t) i s 3 -measurable.
(1.33) A R.V.
T 2 0 i s a stopping time with r e s p e c t t o t h e family
Vt, ( w / T ( w ) 5 t f c 3 t
(1.34)
(*)
at,
if
. 'p
We term o-algebra of the events previous t o T , t h e o-algebra 3 defined by
T t Ac3 o A n i T s t f c 3
(1.35)
Vt.
The following p r o p e r t i e s of stopping times w i l l be very u s e f u l l a t e r ( t h e family
3t i s f i x e d ) :
Let S , T be two stopping t i m e s ; ing times. Let T be a stopping time;
then t h e R . V . ' s
SAT, SVT, S+T a r e stopp-
T
then T i s 3 -measurable.
T
Let T be a stopping time and S a 3 -measurable R.V. then S i s a stopping time. Let S,T
be two stopping times,
T
~n
f 3
.
A E '3 ; we have
Let S,T be two stopping times such t h a t S
3s c 3*
such t h a t S 2 T ;
.
5
T;
we have
T
{S < T ] , (S E T), {S > T ] E 3' and 3 S I f < , n a r e two R"-valued R . V . ' s such t h a t 5 i s 3 -measurable and n i s
Let S,T be two stopping t i m e s ;
aT .
3T-measurable, then {< = ql n {ST] ~ 3 'and Let T be a sequence of stopping times; Tn 4 9 , then T i s a stopping time.
sup T i s a stopping time; n n
,
if
Let y ( t ) be an Rn-valued process which i s r i g h t We assume.that f o r a l l t , i s complete ( i . e . it contains a l l t h e events of p r o b a b i l i t y z e r o ) . Let 8 b e a Bore1 s e t i n R".
continuous and adapted t o
st.
~
(*)
More simply, we w r i t e
( T I t).
at
30
STOCHASTIC D.E.'s
&
P.D.E.'s
(CHAP.
OF ORDER 2
2)
We put
01
r ( ~=)i d {t : y ( t ) 6 = time of e n t r y of t h e process
(1.43)
y
i n t o 8.
=a$.
In Then r(0) i s a stopping time with r e s p e c t t o t h e family 3t+0 t t = 3 (we then say that+ t h e family 3 i s right continuous), then particular, if r(0) i s a stopping time with r e s p e c t t o 3
.
--
The process y ( t ) i s measurable i f t , o + y i s measurable f o r &R+) X ff i s s a i d t o be progressively measurable with r e s p e c t t o , i f f o r each t mapping 8,w y(s,w) from [O,t] x p * R" i s measurable with r e s p e c t t o t
d[o,tI)
at
.
the
It
x 3
I f y ( t ) i s a measurable process and adapted t o Zt , then t h e r e e x i s t s a modificIf y(t) is a t i o n of y ( t ) which i s p r o g r e s s i v e l y measurable with r e s p e c t t o 3;t r i g h t o r l e f t continuous and adapted, t h e n it i s p r o g r e s s i v e l y measurable.
.
t I f y ( t ) i s p r o g r e s s i v e l y measurable with r e s p e c t t o 3 and i f T i s a stopping time with r e s p e c t t o ( f i n i t e ) , t h e n y ( T ( w ) , o ) i s 3T-measurable.
at
l e t T be a stopping time with We should p o i n t out t h e following r e s u l t ; and l e t qt = 3T+t ; then for S t o be a stopping time with r e s p e c t it 1s necessary and s u f f i c i e n t t h a t T + S be a stopping time with r e s p e c t t o
at
to
R , C +
A process y ( t ) i s s a i d t o be separable, i f t h e r e e x i s t s a denumerable s e t i n such t h a t
a . s . , Vt E R+, t h e r e e x i s t s a sequence t j t h a t y ( t . ) -+ y ( t ) .
(1 .44)
E
C converging t o
t
and such
J
I f y ( t ) i s an a r b i t r a r y process, it i s p o s s i b l e t o f i n d an Rn-valued modificatI n g e n e r a l , we s h a l l be working with t h e separable modification of a process.
ion of y ( t ) which i s separable.
To conclude t h i s s e c t i o n , we g i v e t h e following c r i t e r i a f o r c o n t i n u i t y of a process, duq t o Kolmogorov; i f y ( t ) i s a s e p a r a b l e process defined on a compact subset of R such t h a t
(1.45) then y ( t ) i s a continuous process.
1.5
Concepts r e l a t i w t o martingales
#
Let be an i n c r e a s i n g sequence of sub-o-algebras ( s c a l a r ) R . v . ' s i s termed a martingale i f E I X < (1
E ( X m 1 8 ) (*) =
.46) The sequence X
(1.47) (*)
xn
2
.
I n
of d
m, xn
.
is
A sequence X Of -measurahe and
8
i s a submartingale i f (1.46) i s replaced by
E(X,I&
2
xn
vm
z n
This i s another way of w r i t i n g E!@ changeably.
.
Xm ; t h e two forms w i l l be used i n t e r -
(SEC. 1)
PROBABILITY AND STOCHASTIC PROCESSES
31
Finally, X is a supermartingale if -X is a submartingale. let Xn be a submartingale, then V h
The following results are much used:
PI:
( 1 .48)
ax X , B A ] S kka
>
0 :
EIX I y ,n g 0 .
Let Xn be a martingale, such that E I X
1' < -,
; then
a2 1
(which may be deduced from (1.48)by noting that IXnl" is a submartingale).
t
We now consider the continuous-time case. We take an increasing family 3 of sub-o-algebras of a which we assume complete,d itself being complete (i.e. it contains all the negligible partitions of i2, that is, those contained intan event of probability zero) ( * ) . Consider a scalar process X(t), adapted to 3 such that ElX(t) I < Vt We say that X(t) is a martingale if
-
.
E(X(t)
(1 .50)
I??)
n
X(S)
V t Z
,
8
or a submartingale if E(X(t)
(1.51)
I$)
Z X(s)
V t 2: 8
or a supermartingale if -X(t) is a submartingale. We shall assume that X(t) is separable. We then have the analogue of the estimates (1.48), (1.49). If X(t) is a submartingale then
PI: sup x ( s ) 2 hj 5 EIX(t)
( 1 .52)
h
&sst
.
1
If X(t) is a martingale, such that E I X ( t ) / ' PI sup
(1.53)
If
est
a > 1, we
Ix(s) 1
2 hl 5- E l X ( t ) ha
M) + PIlxZl >
Since, in view of the continuity of 5, PI sup IF;(t) CSST to show that, for any fixed M, we have
Ef
.
I > N]ri =0 , it is sufficient -L
p{\xzl >
(2.49) Now
In fact the double products vanish as a consequence of the property E [ ( j ( nnk +l)k b(t)dw(t))2
-[21)k gk] . b2(t)dtl
= 0
Now
Furthemore
The second summation clearly tends to zero. We thus have to show that (2.51)
’2’
IkO
E(
jkl’k
b(t)dw(t)l4
-
0
.
This is a consequence of the following lemma.
LEMMA 2.1
Let X(t) be a continuous, scaZar
3t martingale such that
43
STOCHASTIC INTEGFtALS
(SEC. 2)
We then have
Proof. We shall confine ourselves to proving this for t = T, t1 = 0, and for further simplification we shall assume that X ( 0 ) = 0. We tEus have to show that
(2.54)
E X ( T ) 4 5 C, T2
.
We discretise the interval (0,T) and we put
- X(nk)
Xk = X ( ( n + l ) k )
,n=
O..,
N-I
.
We shall henceforth suppress the index k to simplify the notation. We have
Now
(2.57) Since, from (2.52)
EgX;
,
ICT
the sequence X X2 remains bounded in probability. Now, from the continuity of the trajectories Ce Rave
mplxn16
-
o
,
a.s.
and consequently, from (2.57), we have
: ~
.
2 x2+6 0 n n Furthermore, for all a > 0 we have
Taking account of the fact that Z X3, 2 X4 n n n n least for a subsequence,
(2.58)
X(T)4 5 l i m (2a
p
2 Xm
+ 4a
+
0 , we obtain, from (2.55), at
nhXnXm x x x
m z .
We d e f i n e t h e following problem:
(3.1)
(3.3)
i s a n adapted process ( w i t h r e s p e c t t o
(1 - X , ( t ) ) y ( t )
(3.2)
y(t) y(t)
= 5
+
It
f i n d an Rn-valued process y ( t ) which s a t i s f i e s
t
3
)
i s a continuous process
t
(l-x,(s))$(y(s),s)ds
+ J 0 (l-x,(s))u(y(s),s)dw(a)
V t 6 [o,T]
I t i s c l e a r t h a t y ( t ) = 5 f o r 0 5 t 5 T ; what i s r e a l l y of i n t e r e s t i s t h e r e f o r e t h a t which t a k e s p l a c e a f t e r T (which i s i t s e l f random). For t h i s we w r i t e ( 3 . 3 ) i n t h e (more i n t u i t i v e , b u t l e s s p r e c i s e ) form
(( 33 .. 33 '' ))
dy(t)
=
Y(.c) =
5
g (y(t),t)dt
.
+
&(t),t)dw(t),t
>
'E
We say t h a t y ( t ) i s a s o l u t i o n of a s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n ( s t r o n g formulation
(*)I.
(*IThe
weak formulation w i l l be d e f i n e d i n t h e next s e c t i o n .
-
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
50
3.2
( C W . 2)
Investigation of the Lipschitz-continuous case
THEOREM 3.1 (3.4)
I n addition t o the assumptions of Section 3.1, suppose t h a t lg(xtt>
(3.5)
Idx,t)l
(3.6)
EISl2
- g(x'tt) I + 2
IU(x,t)
- U(xt,t) I -
1)
E / Y n ( t ) I 2 1 (1
+
EIEl2)(C
+
+ E/yn-'(s)l2)ds]
C 2 t +..+Cn-l
n
4 )5 n!
C(l
2 eCt
+ EiEl )
.
We can then proceed t o t h e l i m i t i n ( 3 , 1 3 ) , giving
and ( 3 . 7 ) .
3.3
1nvesti.qation of t h e l o c a l l y Lipschitz-continuous case
We s h a l l s t a r t by giving a uniqueness theorem which i s s t r o n g e r than t h e r e s u l t of Theorem 3.1. Let 8 b e an open domain i n Rn and l e t g l ( x , t ) , g 2 ( X , t ) , a , ( x , t ) , u,(X,t) functions such t h a t
be
We denote by y . ( t ) , i = 1, 2 , t h e process which i s a s o l u t i o n of t h e s t o c h a s t i c d i f f e r e n t i a l equation ( 3 . 3 ) , corresponding t o gi,ai,zi,Ci. We p u t
(3.23)
Q = 8 X]o,T[
53
STOCHASTIC D.E.'s : STRONG FORM
(SEC. 3 )
We assume h e r e t h a t
St
(3.25)
i s right-continuous
s o t h a t ( c f . ( 1 . 4 3 ) ) Bi from (1.41)we have
A =
i s a s t o p p i n g t i m e w i t h r e s p e c t t o St
h1 =
r21 n
{el = c21 c
,1
3
n3
2
.
Furthermore,
.
We a r e i n t e r e s t e d i n what happens b e f o r e t h e i n s t a n t s el, i . e . b e f o r e e x i t from E A. For o E A , t h e i n i t i a l v a l u e s c o i n c i d e ; a s t h e c o e f f i c i e n t s of , it i s r e a s o n a b l e t o t h e s t o c h a s t i c d i f f e r e n t i a l equation c o i n c i d e on 8 x [o,T] c o n j e c t u r e t h a t t h e s o l u t i o n s y, ( t ) and y , ( t ) w i l l c o i n c i d e u n t i l e x i t from Q, and furthermore t h a t t h e e x i t times w i l l c o i n c i d e . This i s not obvious a p r i o r i , however, s i n c e t h e s t o c h a s t i c d i f f e r e n t i a l equation i s not s o l v e d a t f i x e d w, b u t g l o b a l l y with r e s p e c t t o t , W . We s h a l l now prove t h i s important p r o p e r t y , which forms t h e s u b j e c t of t h e following theorem.
Q, and for w
THEOREM 3 . 2
Proof.
Under t h e a s s m p t i o n s ( 3 . 2 0 ) , ( 3 . 2 1 ) , ( 3 . 2 2 ) , ( 3 . 2 5 ) we h o e
We denote by
xA
t h e c h a r a c t e r i s t i c function of A.
We n o t e t h a t
(3.27) and
Consequently, we deduce from ( 3 . 1 9 ) , w r i t t e n i n t u r n for y , , yz and t h e n d i f f erencing, t h a t
Likewise I3 = 0 , Since f o r t < 8 (y, (s),~)f Q and gl = g2 we see t h a t I1 = 0 . s i n c e t h e a r g d n t of t h e s t o c h a s t i c l n t e g r a l i s . as can r e a d i l y be s e e n , always equal t o z e r o . We t h e r e f o r e deduce t h a t
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
54
(CHAP. 2 )
We s h a l l now u s e t h e above theorem t o i n v e s t i g a t e t h e c a s e i n which t h e coeffi c i e n t s g , u a r e only l o c a l l y L i p s c h i t z continuous. We t h u s have t h e following theorem:
THEOREM 3.3 that
assume
I n addition t o t h e asswnptions of 3 . 1 , as well as ( 3 . 2 5 ) , we
then there e x i s t s one and only one process y ( t ) ( i n the sense of ( 3 . 8 ) ) , which i s
a solution of ( 3 . 1 ) , ( 3 . 2 ) , ( 3 . 3 ) . Proof.
(*II n
Uniqueness.
(*)
Let y l , y2 denote two p o s s i b l e s o l u t i o n s .
p a r t i c u l a r , we do not assume t h a t ( 3 . 6 ) holds.
We w r i t e
(SEC. 3 )
and
STOCHASTIC D.E.'s : STRONG FORM
eN
I
min
and from (3.34) It then follows that
we have
(eN, 1 e:)
.
55
56
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
Existence.
For
(CHAP. 2 )
i n t e g e r , we s e t
N
N
xN = ( x i ,
i = 1,
... , n )
and
= x . min (1,
xN
-0 N
).
We s e t
EN = 5
N N = g(x ,t)
gN(X,t)
N
uN ( x , t ) = u ( x , t ) It can e a s i l y be shown t h a t t h e Existence Theorem 3.1 can be a p p l i e d t o cN,T,gN,UN(*). Consequently, t h e r e e x i s t s y N ( t ) s a t i s f y i n g
(3.39) y N ( t ) = 5 , Let
eN =
+[:
(1
- X T ( s ) ) g N ( y N ( s ) , s ) d s +j:( 1 - X,(s)bN(y(s),s).dw(s).
i d {t2 T
We consider N' > N . Theorem 3.2 t h a t a . s .
1
lyN,i(t)l > N
, for
some i
,t
[o,eNl
E
.
jx.1 < N, it follows from
Since g N ' ( x , t ) = g N ( x , t ) f o r
y N ( t ) = y$t)
1
I
I n a general manner we t h u s have, t a k i n g N' = N + 1
(3.40) The events {ON < T} form a decreasing sequence.
P
(3.41
{e, < T J
-
w-
o
We s h a l l show l a t e r t h a t
.
It then follows t h a t
P o r i n o t h e r words
?7
N= 1
{e, < T J
o
3 No(w)
a.s,
However, from (3.40) we then have
a.s.
w
NZN~(w)
-
ouch t h a t
SUP
WST
N 2 N (w)
IYN(t)
0
0
eN=
- ~ , + , ( t )=\ o .
Consequently, a.s. t h e sequence y N ( t ) converges uniformly i n
(*I
T.
Here we a r e considering t h e L i p s c h i t z continuous c a s e and 6
t
towards y ( t ) .
2
N E L
(SEC. 3)
STOCHASTIC D.E.'s : STRONG FORM
Furthermore we have
( * ) This is the reason for introducing the function
+.
57
58
bY
STOCHASTIC D . E . ' s
THEOREM 3.4
(3.45)
&
P.D.E.'s
OF ORDER 2
(CHAP. 2 )
Under t h e s m e asswnptions a s for Theorem 3 . 3 and replacing (3.35) x.g(x,t)
+ 1? tr
u@
~
+ 6)s:
o
Proof
2 L ~ ( O , T ; R " ) such that one or other of t h e
la(t)I2dt
such t h a t
then we have (4.14)
E
E exp[jf a(t).dw(t)
,
1,
(4.24)
L
>
0
such that
where (4.25)
3 p, C
>
0 such that E exp p(\y(t)
i2
+
\P[t)l2)
01 = Q+ Oj n 6 - { y l y E Q, Y, < 01 = Q, os n r I Y ~ YE Q, ,Y, =.01 = Q~ 0,
n
6
-
. 6.n 8 J i
Also, t h e following c o m p a t i b i l i t y r e l a t i o n s hold: i f cpj(x)
=
‘pi(”)
, v
x
c v n ei 3
The s e t (OA,cpj)constitutes a system of with t h e 6. o u t s i d e C?’.)
J
J
a p a r t i t i o n of u n i t y ,
i . e . functions a
such t h a t
These c o i n c i d e w i t h WsYp(Rn)
.
local charts f o r
V
(*)
is invertible,
i f p = 2.
J
E
r
.
doj)
4 /d
then
We can a s s o c i a t e (extended by 0
(SEC. 5 )
Let
ELLIPTIC P.D.E.'s OF ORDER 2
u b e a f u n c t i o n on
Since a
j
.r
; we d e f i n e f o r ( y ' ( < 1
has compact support i n 8 t h e f u n c t i o n y ' +qj*(a.u)(yl) has compact j' J By extending t h i s o u t s i d e by 0 , we can d e f i n e it i n Rn-l Y' *
support i n Iy'(< 1. We then p u t , f o r
s
real,
Ha@) = (ulp.*(aju) E $(Rn;')
(5.15)
Y
J
V j = l,..v]
.
We equip H S ( i " ) with t h e norm (5.1
6)
f o r which H S ( r ) i s a H i l b e r t space ( n o t e t h a t t h e norm (5.16) depends on t h e system b.,cp , a . ) .
J
J
J
By i d e n t i f y i n g H o ( r ) with i t s d u a l , we have t h e following r e s u l t :
.
H-'(r) = ($(r))'
(5.17)
We s h a l l assume t h e following t r a c e theorem ( f o r t h e proof, s e e J . L . LIONSE. MAGENES [l]).
THEOREM 5.0. The mapping
(5.18)
Suppose t h a t the open subset 6 s a t i s f i e s (5.12) ( * ) and (5.13).
u
-.
{u,
bU
-
n.grad u)
(where n i s the u n i t normal pointing outwards from @ ) from do) extends by continuity i n t o a mapping, again written i n the form (5.18), which i s linear and continuous from
This mapping i s surjective.
8
We r e f e r t h e reader t o LIONS-MAGENES, l o c . c i t . , f o r t h e proof and f o r ext e n s i o n s t o ?(@), m > 2.
5.1.3
Green's f o m l a
Let @ be an open subset of R", of which t h e boundary
o.,
r = $0
is of Class C
2
( i . e . t h e diffeomorphisms TT1 defined i n t h e previous s e c t i o n a r e C2 i n s t e a d J J of being Cm ) , and l e t n be t h e u n i t normal pointing outwards from 8 . Suppose we now have functions a i j ( x ) , i , j = l.,n, such t h a t
(5.19) (*)
aij
In f a c t , @ o f class
E C'(0)
c2 .is
,
bounded, t o g e t h e r with i t s d e r i v a t i v e s
sufficient.
STOCHASTIC D . E . ' s
92
(5.20)
a . . = a.. 1.I Jl
We put ( f o r u,v
& P.D.E.'s
(CHAP. 2 )
OF ORDER 2
(*)
E &R"))
(5.21)
(5.22) We assume t h a t t h e following r e l a t i o n s n h o l d ( t h e s e may r e a d i l y be deduced from t h e formula f o r i n t e g r a t i o n by p a r t s i n R ) : r
i n which we have put
(5.25) We now suppose t h a t t h e open s e t 8 s a t i s f i e s ( 5 . 1 2 ) , ( 5 . 1 3 ) . The formulas They may t h e n b e extended by c o n t i n u i t y t o ( 5 . 2 3 ) , (5.24) hold for u,v E &a). u,v E H 2 ( 6 ) . In p r a c t i c e , we-make u s e o'f t h e f a c t t h a t a(b) i s dense i n H 2 ( 0 ) , and of t h e c o n t i n u i t y of t h e t r a c e operators defined i n t h e previous section.
The s u r f a c e i n t e g r a l s appearing i n ( 5 . 2 3 ) , (5.24) a r e i n t e g r a l s i n L
2
(r).
The r e l a t i o n s (5.231, (5.24) for u,v E H2(0) a r e c a l l e d ( g e n e r a l i s e d ) Green's fomtas. F i n a l l y , we conclude t h i s s e c t i o n by noting two fundamental p r o p e r t i e s of Hb(0) (we r e c a l l t h a t $ ( 0 ) i s t h e c l o s u r e of &0) i n H ' ( 0 ) ) ,
(5.26)
u E H;(O) o ulr
0
( i . e . t r a c e of
u
on
r
= 0)
I t then follows t h a t
(5.28) 1
1
i s a norm on Ho(6) equivalent t o t h e norm induced by H (0)
.
For t h e p r o o f , s e e f o r example J . DENY and J.L. LIONS 111 or J. NECAS 111. ( * ) This w i l l be s u f f i c i e n t for our needs, but is unnecessary i n t h e present s e c t ion.
(SEC. 5 )
5.2.
93
ELLIPTIC P.D.E.‘s OF ORDER 2
V a r i a t i o n a l formulation
Let 8 be a bounded open subset of Rn ( n o t n e c e s a a r i l y r e g u l a r ) . functions a i j ( x ) , a i ( x ) , a o ( x ) , i j = l . . n , i = l . . n , s a t i s f y i n g
We t a k e
(5.29)
We d e f i n e a continuous b i l i n e a r form on
(5.30)
a(u,v) =
Z
i j=l
Jo
a .-6U iJ axj
& +
axi
d ( e ) by ‘ 1 0 a i T v d ~+
i
It can e a s i l y be seen t h a t t h e r e exists A 2 0 such t h a t
s
*a0wcix
.
n
(5.32)
1 1 ~ 1 1= We also put
(5.33) We denote by
(5.34)
V
=
V
1 2 a H i l b e r t subspace of H (6)such t h a t i f H = L (6) we have
H , with continuous i n j e c t i o n and
V
i s dense i n
H.
We i d e n t i f y H with i t s dual and we denote by V’ t h e dual of V. from (5.34) t h a t we have
(5.35)
V c H c V ‘ , each space being dense i n t h e following space with continuous i n j e c t i o n .
The space
d(6) i s
ordered through t h e r e l a t i o n
u1 s u2 i f u,(x) 5 u,(x) I f ul, u2
(5.36)
It follows
E
1
H (6) then
a.e.
x E 0
.
94
STOCHASTIC D.E.'s
1
a l s o belong t o H ( 0 ) .
& P.D.E.'s
(CHAP. 2)
OF ORDER 2
I n p a r t i c u l a r , we have t h e formula
where
0
if
u
1
< u
2
, b (ul A
we have a similar formula f o r
u2).
I n p a r t i c u l a r , we put
u+=uvo
(5.38)
u- =
(4)
v 0
and we note t h a t
(5.39)
.
u,u+-u-
1
f
E
The ordering r e l a t i o n on H (6) induces an ordering r e l a t i o n on V. V', we say t h a t f 2 0 i f
(5.40)
f 2 0 r ( f , v ) 2 O v v EVyv2O
I n (5.40) ( , ) denotes t h e i n n e r product in H s h a l l assume t h a t
(5.41)
t h e mapping v
+
v- sends V i n t o V
We note t h e important r e l a t i o n
(5.42)
a(v+,v-)
=
ov
v
E H~ (0)
I f now
.
and t h e d u a l i t y V,V'.
We
(*)
.
THEOREM 5.1 Suppose t h a t ( 5 . 2 9 ) , (5.31) hold with A = 0 , and t h a t we haoe For a l l f E V' , there e x i s t s one and only one solution of the equation (5.41).
(5.43) I f f 2 0,
a(u,v) = (f,v)
then u
v
v E
.
v
2 0.
The e x i s t e n c e and uniqueness of t h e s o l u t i o n of (5.43) follow from Proof. Suppose t h a t t h e Lax-Milgram theorem ( s e e f o r example YOSHIDA Ell, p . 9 2 ) . f 2 0 ; then, t a k i n g v = u- i n ( 5 . 4 3 ) , which i s permissible from ( 5 . 4 1 ) , we o b t a i n , t a k i n g account of ( 5 . 4 2 ) :
a(u',u') so t h a t
+ (f,u-)
a
o
[Gl12 = 0
t h a t i s , u- = 0, and hence u 2 0.
m
1 This is t h e case f o r example i f V = {vjv E H (a), v = 0 on ro C r y rc of measure > 0 on r ( o r o f c a p a c i t y > 0 ) ) ( 0 r e g u l a r s o as t o allow t h e t r a c e t o be d e f i n e d ) . (*)
OF ORDER 2
ELLIPTIC P.D.E.'s
(SEC. 5)
95
We s h a l l now dispense with t h e assumption t h a t (5.31) holds with X = 0 . s h a l l assume i n s t e a d t h a t u,,u2EVru
(5.44)
if u
(5.45)
1
1
Vu
H (O),v
E
E V ,ul hu2E V
2
V a r e b 0 , then (u-v)-
E
We
E
V
.
f E Lye) 1 1 The assumptions ( 5 . 4 4 ) , (5.45) a r e s a t i s f i e d with V = H (0)or V = H (8).
(5.46)
1
Let us prove, for example, ( 5 . 4 5 ) i n t h e c a s e V = Ho(B).
n , u E C ( 0 ) ( a f t e r p o s s i b l e modification on a s e t of measure z e r o ) . E c:z(o) , i . e . b2" is locally We a c t u a l l y have r a t h e r more than t h i s :
-a
.
H8lder continuous of order a (from t h e i n c l u s i o n theorems f o r S$boi!ev spaces, c . f . ( 5 . 6 ) ) , and t h i s r e s u l t can even be obtained by assuming only t h a t f f 'C 1oc (a). We r e f e r t h e reader t o LADYZENSKAYA-URAL'TSEVACll o r MIRANDA [11. The assumption (5.77) which was used i n t h e proof can i n f a c t be weakened. It Co(@ (see loc. c i t . ) . 8 . . E Cl(&) ai, a
i s s u f f i c i e n t t o suppose t h a t
,
1J
,
We s h a l l now s t a t e , without proof, t h e g l o b a l r e g u l a r i t y r e s u l t s r e l a t i n g t o t h e D i r i c h l e t and Neumann problems.
Suppose t h a t 8 i s a bounded open set whose boundary r = bO is of THEOREM 5.6 class c2 Suppose that the coefficients a f C'(6), a , a E C0@) , a 2 0. there exist&one and on18 on& solution of'the Let p 2 2 ; then if f E Dirichlet problem ( 5 . 7 2 ) , ( 5 . 7 3 ) , or of the N e m a n n problem ( 5 . 7 2 ) , ( 5 . 7 5 ) which satisfies u E w ~ , P ( D ) ,and we have
.
@(a,
(5.91) Remark 5.4 The property (5.92) i s an immediate consequence of t h e e x i s t e n c e and uniqueness of u. I n f a c t t h e mapping f +. u from Lp(C9) +. W2yP(S) is c l e a r l y l i n e a r and of closed graph and hence it is continuous, t h u s g i v i n g ( 5 . 9 1 ) .
5.5
E l l i p t i c P.D.E.'s
o f second order i n Rn
it i s now necessary t o d e f i n e t h e behaviour We now consider t h e c a s e O = Rn(*); a t i n f i n i t y of t h e operator c o e f f i c i e n t s , of t h e right-hand s i d e and of t h e s o l u t i o n . We s h a l l d i s t i n g u i s h between two c a s e s , depending on whether t h e c o e f f i c i e n t s of f i r s t order a r e bounded o r unbounded ( t h e o t h e r c o e f f i c i e n t s , of o r d e r 2 and of order 0 , a r e always assumed bounded).
5.5.1
Unbounded coefficients of first order
We again consider
+
= A. where
A1 = Z a
-
b i bxi
A,
a.
,
*
The c o e f f i c i e n t s a i j , ai, a.
("I
+
satisfy
The following considerations adapt t o t h e c a s e i n which 8 i s an unbounded open s e t .
ELLIPTIC P.D.E.'s OF ORDER 2
a , . = aji E L"(R~)
(*)
IJ
(5.93)
(5.94) ao(x) 2 p > 0,
109
p being sufficiently large (see the estimates hereafter).
The functions a. can this time have "arbitrarily rapid growth" under the conditions which will %e defined below. We assume that
c c'(R~) v i ; i there exists a function x
a
such that
QZW
ing function of
(5.95)
da.
I&
I
J
(XI I Ic
[XI),
m(x)
v
-t
m(x), defined and continuous > 0 in Rn
c m(x)
sup m(tx)
(for example m may be an increas-
and with the ai satisfying i,j
,
for suitable co&.tants Bor c and c1
Example
ai(x) =
- xf
.
The conditions in (5.95) hold with m(x)=Zxi 2
,
~ o 3t ~ , ~ , 5 ~ , ~n = = 3 .
We shall now show that, under these conditions, we can solve the following equation uniquely:
(5.96)
AU
= f in R"
at least for sufficiently large (dependent on the space in which ( 5 . 9 6 ) is solved, the space itself being chosen as a function of the properties of f). We recall the following result, proved in LIONS 111, Chapter 3: Let F be a Hilbert space, 0 a subspace of F which may be closed or otherwise; denote a norm on 0 which makes it let 1 1 I [ denote the norm of F and let 111 a p r e - H i 6 e r t space; we assume that
111
(*)
The assumption of symmetry a.. = a.. is sufficient for our present purposes, J1 but not absolutely necessarytJ .
(CHAP. 2)
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
110
(* 1 Also l e t E ( u , cp)
i) u
1** 1
-t
be a b i l i n e a r form on F x 9 such t h a t E ( u , c p ) i s continuous on F
z aj/IvlI12
ii) ~ ( ' p , ' p )
and l e t
'p -. ~ ( ' p )
,
a
>
0
,v
E a
'p
be a continuous l i n e a r form on 9:
.
Then there e x i s t s u E F such t h a t E(u,'p) I L('p) v 'p E Q Furthermore, i f the soZution i s unique (which couZd possibZy be proved by an ad hoe argwnentl then
(5.97)
(5.98)
We put
2 1 which d e f i n e H i l b e r t n o m s on LT and HT r e s p e c t i v e l y . We introduce
F =
15.99)
{ V ~ VE
H,1
,
v
ID E Ljl]
2 11v11 = ~11v111 + F HZ
We t a k e Q we put
&an) , equipped
I
,
lW21 3
E
2
*
X
with t h e norm induced by F.
(5.100)
For f
equipped with t h e norm
LT, equation ( 5 . 9 6 ) i s equivalent t o
For u
E
F,
Q
E 0,
(SEC. 5 )
111
OF ORDER 2
ELLIPTIC P.D.E.'s
THEOREM 5.7 Under the asswnptions ( 5 . 9 3 ) , ( 5 . 9 4 ) , ( 5 . 9 5 ) , there exists u , the unique solution of (5.101) (and hence u E F, a solution of ( 5 . 9 6 ) ) .
Proof of existence. Let us now c a l c u l a t e E
('p,cp).
We have ( p u t t i n g dai diva = Z -)
axi
But
so t h a t
Then n
, l
IZ a. . i lJ
so t h a t , t a k i n g account of t h e f a c t t h a t
(5.1 02)
Ex('p,'p) 2
2
a,I1'pIlF
9
a
>
0
dx z i-1 I 5 c2 U
*
We t h e n deduce t h e e x i s t e n c e by applying t h e r e s u l t ( 5 . 9 7 ) .
Proof of uniqueness.
Let u
E
F be a s o l u t i o n of
Proceeding t o t h e l i m i t (by r e g u l a r i s a t i o n , f o r example) we s e e t h a t (5.103) i s t r u e f o r v 'p E H1 (Rn) with compact support.
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
112
8 = 1
Let 8 € &Rn) be defined by
eRb) =
(5.104)
We t a k e 'p = 8 u.
= 0
i f 1x1 2 2 , and l e t B R
.
e(x/R)
Then
R
.
E (u,e U) = o x R
(5.105)
8
i f 1x1 5 1 , 0 5 8 s 1,
(CHAP. 2 )
However
+ aox8,u2
We have : Ix Hence
x
+
1 I-. %t ( x ) I
n 8Ru2
- -12 z ai
and t h e r e f o r e ZR
c
3 dxi
-+
2
e U ] d x +
R
xR ,
0 from Lebesgue's Theorem.
dxl
and t h e r e f o r e (5.105) and (5.106) g i v e i n t h e l i m i t :
0
J
n
(5.107)
1
---(diva) 2
R n a ~i j x~
=, bU
J
i
+ aij
'
--a
2
du
bxi u
+ a0x
1
u2 --(diva) 2
x
u
2
j
!k
i dxi
U']dx*
0
and t h e c a l c u l a t i o n l e a d i n g t o (5.102) then g i v e s 2
a1 Ilull,
0
and hence t h e r e s u l t . I t i s p o s s i b l e t o prove t h e e x i s t Remark 5 . 5 - another proof of Theorem 5 . 7 . ence i n Theorem 5.7 by means of a ' r e g u l a r i s a t i o n ' method. Suppose we have y > 0 , we denote by Fo t h e space which w i l l u l t i m a t e l y tend t o 0 ;
F = { v ~ v E F , A Y E L2 ] 1
(5.108)
f o r u,v
E
Fo, we consider t h e q u a n t i t y
;
( S E C . 5)
ELLIPTIC P.D.E.’s OF ORDER 2
113
Y
Y ( A ~ ” , A ~ v+) En(~,v) ~ = d,(u,v)
(5.109)
We note t h a t , under t h e conditions of Theorem 5.7, we have, from (5.102)
(5.1 10)
2
+
2 yIA1vl,
&:(v,v)
2
alIbllF
,v
E Fo
,
and s i n c e t h e b i l i n e a r form g y ( u , v ) is continuous on Fo, we s e e t h a t t h e r e e x i s t s a u which i s unique i n F s8ch t h a t v
Y’
0’
(5.111)
*I
From (5.110)we have: when y -+ 0 , u remains i n a bounded subset of F and uy remains i n a bounded subset ofYL2. We can t h e r e f o r e e x t r a c t a sequence, a l s o denoted by u , ( i n f a c t , i n view o f t g e uniqueness, t h i s e x t r a c t i o n i s not needed) such t h a t Y
vy
-L ff weakly i n F Y and it can immediately be shown t h a t
u
En(ff,v)
I
(f,v),
v
v
.
E F
From t h e uniqueness, ii = u, t h e s o l u t i o n of t h e problem. t h e e x i s t e n c e and i n a d d i t i o n gives t h e following r e s u l t :
u -. u weakly i n F as
(5.112)
If
1
The spaces Lz and HT become p r o g r e s s i v e l y l a r g e r as f o r r0 = (1+ x‘)-*~
i s given i n L:
f
y -+ 0.
Y
Remark 5.6
This again demonstrates
on s ) we can t a k e
s t so;
s
increases.
,then as long as ( 5 . 9 4 ) holds ( 6 dependent
naturally, the solution
however it i s advantageous t o t a k e
s
u
i s independent of s ;
as small as p o s s i b l e .
I t follows from ( 5 . 9 6 ) t h a t l o c a l l y Au i s i n L2 - but not necessarily Remark 5.7 We can, however, o b t a i n a g l o b a l globally i n L 2 , because of t h e growth of t h e ai. r e s u l t r e l a t i g g t o Au i f we use supplementary assumptions; t h e following theorem: THEOREM 5.8
that
The assumptions are those of Theorem 5.7.
la
(5.113)
ij
( x ) / ( l + m ( x ) ) Ic
ba iJ(x)
”k
then the solution u
(*I
E
t h i s i s t h e s u b j e c t of
We assume in addition
(*)
I(1+IXIm(X)) Ic
;
F of t5.101) satisfies
The symmetry assumption, which was not involved i n t h e previous theorem, w i l l be u s e f u l h e r e , a t l e a s t i n t h e proof.
114
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
.
A u E L n2 (5.114)
(CHAP. 2 )
0
Proof.
g = f
We put
-a u ;g
A u + A u 0
1
=
g
E :L
and
,
s o t h a t (with t h e n o t a t i o n used i n Theorem 5 . 7 ) we have
BR(Aou
+ Alu) =
.
6#
We then deduce t h a t
We now c a l c u l a t e X = (fJ A L u , BRAou) and denote by O(1) q u a n t i t i e s which a r e -f m; we obaain r
bounded when R
We note t h a t
since
lak= ax 15 Cn(l+m(x))
and t h a t
and u
E
4
j
Similarly, since
25
C'm(x)
, we
j
and consequently i f
we have:
(5.1 16)
t o g e t h e r with ( 5 . 1 1 3 ) ,
x
= z Xk k
+
O(1)
.
have by v i r t u e o f (5.108)
115
STOCHASTIC P.D.E.'s OF ORDER 2
(SEC. 5 )
However, because of t h e symmetry a i j = a . .
we have
31'
f2
1
X k = -2 X =
R k
i, j
1
bu
2
( & bu I
b
r e a a
ij
xk dxi z.) dx 3
-b( n e bXk
2
R
a a k
ij
)CIX
.
But, a s above, we note t h a t
and hence
el ba
lak
and from (5.113),
with (5.116) shows t h a t X = O(1).
"lea
lek 'oUlf
s
hence Xk = 0(1), which i n conjunction
Then (5.112) gives
'oUln
and t h u s 18, Aouln
, and
Ic
+
.
C
Therefore, we can f i n d a sequence R -t m such t h a t 8 Au R but BR Au + A u i n p ( R ) f o r example, and t h e r e f o r e 2
L,
Au=YE
Remark 5.8
.
-+
2 Y weakly i n L n ;
B
We can a l s o o b t a i n estimates i n t h e spaces Lp, with weighting;
define
:L =
{VI
:1.1
0
5.
rvIp dx
0 such t h a t
2 y
> o , vu
E
v I.r
We t h e n have t h e following theorem:
THEOREM 5.9 We adopt the assumptions ( 2 . 1 2 2 ) . Then f o r a l l f e x i s t s one m d only one solution u E V n L such that i-1
(5.1 25)
a(u,v)
t
(f,v)
v
v E V
E
L"(Rn),
there
.
If, additionally, ( 5 . 1 2 4 ) holds with A = 0 , then (5.125) has one and only one s o h t i o n f o r a l l f E H or V' = dual o f V
v
1-I
1-I
.
Proof. The f i r s t p a r t o f t h e proof i s i d e n t i c a l t o t h a t f o r Theorem 5 . 2 . second p a r t i s w e l l known ( c o e r c i v e c a s e ) .
The
We n o t e however t h a t ( 5 . 1 2 4 ) may be t r u e ( w i t h A = 0 ) f o r a s u i t a b l y chosen v a l u e It can e a s i l y be seen t h a t i f of p.
(5.1 26) t h e n we can choose 1-1 such t h a t (5.124) i s s a t i s f i e d w i t h A = 0. We have t h u s found u
(5.1 27)
AU
E
8
such t h a t i n t h e d i s t r i b u t i o n a l s e n s e we have
V
1-I
= f a . e . i n R",
We s h a l l now suppose t h a t t h e c o e f f i c i e n t s o f A s a t i s f y t h e following supplementary r e g u l a r i t y p r o p e r t i e s :
15.1 28)
a
ij
,
E c ' ( R ~ ) ai E
c0(an) , a.
ba
E cO~R")
, 3 bounded. "k
(SEC. 5 )
ELLIPTIC P.D.E.'s
OF ORDER
119
2
We s h a l l now prove t h e following theorem:
Proof.
Let
=
I /x I
0
t h e constant C ' ( A ) being dependent only on R . We then apply (5.139) with v =
%.
Taking account of (5.134) and (5.136) and
by s u i t a b l y choosing A (independently of R ) we can show t h a t we have t h e following estimate
(5.140) However, t h e extension considered e a r l i e r i s such t h a t
We thus d e f i n e
GI an extension of
We then e x t r a c t a subsequence
for R
%
If
u in W2ypyp(Rn).
-+
n'
s u f f i c i e n t l y l a r g e (such t h a t Q R
r
t o a l l R ( w i t h compact support) such t h a t
3
n
support 'p
)
n
and t h u s Au - f = 0 a . e . This concludes t h e proof of Theorem 5.10.
m
'p
f S(Rn)
, we
have
(SEC.
6.
6)
OF ORDER 2
PARABOLIC P.D.E.'s
123
LINEAR PARTIAL, DIFFERENTIAL EQUATIONS OF SECOND ORDER, OF PARABOLIC TYPE Synopsis
I n a manner analogous t o t h a t used i n S e c t i o n 5 , t h e p r e s e n t s e c t i o n g i v e s t h e e s s e n t i a l r e s u l t s r e l a t i n g t o t h e t h e o r y of p a r a b o l i c P.D.E.'s, which w i l l be of use l a t e r . We s h a l l f o l l o w a g e n e r a l p l a n similar t o t h a t used i n t h e previous section.
6 . 1 Variational formdation Let 6 be a bounded open s e t in Rn and l e t Q = 6 x l0,TC. a . . ( x , t ) , a i ( x , t ) , a o ( x , t ) , ( x , t ) E Q which s a t i s f y 1J
For almost a l l t
E
lO,T[,
we d e f i n e a continuous b i l i n e a r form on
There e x i s t s a A 2 0 and a p > 0 such t h a t we have
(6.3)
a(t;u,v)
+
We t a k e f u n c t i o n s
2 h IuI2 2 ~.rllull
v u c H'(o), 1
a.e.t
$(a)
by
.
We denote by V a H i l b e r t subspace of H (@), c o n t a i n i n g $ ( B )
1 V = HI(@) o r H ( 6 ) ) . Let V ' denote t h e d u a l of V . i t s dual. We"thus have t h e sequence of i n c l u s i o n s
( i n practice 2 We i d e n t i f y H = L (6)w i t h
VcHcV' each space being dense i n t h e succeeding space, with continuous i n j e c t i o n . We s h a l l u s e a number of elementary concepts from t h e t h e o r y of vector-valued We w r i t e distributions. Let Z denote a Banach space.
&'(IO,TC;Z)
= space of l i n e a r continuous mappings from .@(lo,T[) + Z
.
The space t h e r e b y d e f i n e d i s t h e space of d i s t r i b u t i o n s on lo,T[ w i t h v a l u e s i n Z. I f p E & ( I O , T C ) and f E & ' ( I O , T C ; Z ) , t h e v a l u e f ( T ) , o f f a t t h e p o i n t cp, i s denoted by
(6.4) We d e f i n e t h e d e r i v a t i v e
df
if^
E
8 ' ( 1 0 , T [ ; Z ) by
124
STOCHASTIC D . E . ' s
&
P.D.E.'s OF ORDER 2
(CHAP. 2 )
1
We i d e n t i f y L (0,T;Z) = L1(Z) with a subspace of b ( l o . T r : Z ) , by t a k i n g f o r t h e right-hand s i d e of ( 6 . 4 ) , t h e u s u a l Lebesgue i n t e g r a l with values i n Z , i f 1 df Hence i f f E L ( Z ) , we can d e f i n e - as being an element of f E L1(2). dt b(l0,TC;Z). We then introduce t h e space
We equip W(0,T) with t h e Hilbert norm
It can be shown ( s e e LIONS-MAGENES [11 t h a t any f u n c t i o n z E W ( O , T ) , a f t e r H and p o s s i b l e modification on a s e t of measure zero, i s continuous from C0,Tl furthermore t h a t -+
t h e i n j e c t i o n from W(0,T) -+ C o ( C O , T l ; H ) (space of continuous f u n c t i o n s from [O,TI -+ H) i s continuous.
(6.7)
An e s s e n t i a l property of W(o,T) i s t h a t it permits i n t e g r a t i o n by p a r t s : more p r e c i s e l y , i f z 1, z2 E W(o,T); then we have
On t h e left-hand s i d e of ( 6 . 8 ) , ( , ) i s t h e i n n e r product i n H , and on t h e right-hand s i d e ( , ) r e p r e s e n t s t h e d u a l i t y of V,V'. We s h a l l now g i v e t h e following e x i s t e n c e and uniqueness theorem: THEOREM
elementu
(6.9)
E
2 Let f E L (V') and l e t u ( t ) O ~ W ( O , T ) such that
6.1.
- ( dum , v )
+ a(t;u(t),v) =
u
E
H; there exists
(f(t),v)
8.e.t
one and only one
C ]o,TI
yo f Q (6.10)
U(T)
E
u
We s t a r t by making a number of observations: t h e problem ( 6 . 9 ) , ( 6 . 1 0 ) i s backward i n time ( w i t h i n i t i a l v a l u e s a t T ) . We could equally w e l l consider du Since we a r e more i n i t i a l d a t a a t 0 , as long a s we t a k e d" i n s t e a d of dt dt l i k e l y i n p r a c t i c e t o encounter problems i n which t h e i n i t i a l d a t a i s f i x e d a t t h e i n s t a n t T , we have chosen t h e p r e s e n t a t i o n ( 6 . 9 ) , (6.16)even though t h i s may a t f i r s t s i g h t appear l e s s n a t u r a l .
-.
By p u t t i n g t = T
-
s , we immediately g e t back t o t h e u s u a l case.
y = exp - X(T - t ) u t h e problem ( 6 . 9 ) , (6.10) i s equivalent t o
I f we put
(SEC. 6)
PARABOLIC P.D.E.'s OF ORDER 2
and hence
125
i s replaced by a ( t ; y , v ) + A ( y , v ) .
a
We can t h u s r e s t o r e t h e s i t u a t i o n t o t h e c a s e i n which (6.3) holds with A = 0 , and we s h a l l assume t h i s i n t h e proof of Theorem 6.1. Since u E W ( O , T ) , we can consider i t s continuous v e r s i o n with values i n H , and t h i s j u s t i f i e s (6.10).
Proof of Theorem 6.1. Uniqueness. We consider t h e c a s e i n which f = 0 , = 0. We have t o prove t h a t u = 0 . Now t a k i n g v = u and using t h e integration-by-parts formula, we o b t a i n
and hence from ( 6 . 3 ) we have u = 0.
Existence.
...
Let wl...w be a b a s i s of V (we assume V s e p a r a b l e , which i s not s t r i c t l y necessary and fs, i n any c a s e , s a t i s f i e d i n a l l t h e a p p l i c a t i o n s which we have i n We seek an approximate s o l u t i o n mind).
t h e gim(t) being s o l u t i o n s o f t h e following system of l i n e a r d i f f e r e n t i a l equations:
(6.1 1 )
+ a, 1 and Bm + T i i n H as m Multiplying equations (6.11)by g . and adding, we o b t a i n
where E
m
[w l...w
E
-f
m
im
-- - 1 ~ m ( t ) I I 2
(6.12)
d dt
2
+
a ( t ; u m ( t ) , u m ( t ) ) = (f,u,)
so t h a t by i n t e g r a t i n g between 0 and T and using (6.3) we o b t a i n
From t h i s we can r e a d i l y deduce t h e e s t i m a t e
Let 9
E
I
C (CO,Tl) b e such t h a t q ( o ) = 0 and l e t q . ( t ) = q ) w J j'
Let u
)u
be a
STOCHASTIC D . E . ' s
126
m' f o r m = p and we m u l t i p l y by c p ( t ) .
- (u
(up,cpj)dt
P.D.E.'s OF ORDER 2
5:
On i n t e g r a t i n g , we o b t a i n
(~),cp~(T)+ )
P
+
We can t h e n proceed t o t h e l i m i t a s I.! +
5:
(6.14)
- (c,cpj(T))
(u,cp;)dt
(CHAP. 2 )
2 weakly convergent i n L ( V ) , t o u .
subsequence e x t r a c t e d from u
5;
&
m.
So
T (f,cpj)dt
This g i v e s
jo
+
=
a(t;up,cpj)dt
We w r i t e (6.11)
=
a(t;u,vj)dt
As j i s a r b i t r a r y , we r e a d i l y deduce from ( 6 . 1 4 ) t h a t we have
5:
(6.15)
-
(u,v)cp'dt
(C,v)cp(T) I
j:
+
5
T
a(t;u,v)cp(t) d t
v v
(f,v)cpdt
In p a r t i c u l a r , we can t a k e and f o r a l l cp as above. t h a t i n t h e s e n s e of d i s t r i b u t i o n s we have
- F d( u ( t ) , v ) ,
+ a(t;u(t),v)
= (f(t),v)
E
v
p E B ( I O , T C ) , which i m p l i e s
.
vv E v
2 E L2(0,T), which i m p l i e s d" E L ( V ' ) and dt dt By i n t e g r a t i n g by p a r t s ( i n W(0,T)) and t a k i n g account o f (6.151, we can (6.9). r e a d i l y v e r i f y (6.10).
V v E V
Hence
COROLLARY
6.1.
d('(t)'v)
The soZution
u
of (6.9, ( 6 . 1 0 ) satisfies the estimate
Proof. This i s an immediate consequence of (6.13) (which remains t r u e on proceeding t o t h e weak l i m i t ) and of t h e p r o p e r t i e s of t h e b i l i n e a r form. 6.2
RepdaritL
6.2.1
Regularity with respect to time
We now suppose t h a t we have
~
(6.1 7)
ba.. bt
J
E C'(Q), i , j = l...n
.
We t h e n have t h e following r e g u l a r i t y theorem: THEOREM 6.2
then
$f
L2(H)
Under the assumptions ( 6 . 1 ) end ( 6 . 1 7 ) , if f and u
E
Lm(V).
E
L 2 ( H ) and
u
E
V,
(SEC. 6)
PARABOLIC P.D.E.'s OF ORDER 2
127
0
We shall now obtain a supplementary a priori estimate on (6.11); assume, as is permissible, that
U
(6.20)
m
-.
ii
in
v.
We write (6.11)in the form:
(6.21)
- (rlwj)
+
ao\t;um,wj) I (f
- A 1 ~ m l w j ) lj = l1...,m .
Multiplying (6.21) by (-g! (t)) and summing over j, gives: Jm du m 2 + ao(t;uml u;) = (f AI um1 u') m (6.22)
-
ld~I
-
-
.
If we put
( 6.23)
a;(t;u,v)
,Z
Bu dv
Consequently, by virtue of (6.20) and (6.13) we then deduce that
1
this time we
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
128
(CHAP.
2)
and
The theorem then follows from t h i s .
rn
I n a d d i t i o n , we have a l s o obtained (6.27)
The above r e s u l t s do not a s s m e t h a t t h e o p e r a t o r i s parabolic of Remark 6 . 1 They a l s o hold f o r p a r a b o l i c systems. second order ( s e e LIONS [ll).
Remark 6.2 The proof of Theorem 6.2 i s based e s s e n t i a l l y on t h e symmetry of a ( t ; u , v ) ; however t h e r e s u l t s t i l l holds without t h i s assumption, on t h e basis 09 a d i f f e r e n t proof f o r which we r e f e r t h e r e a d e r t o C. BARDOS C11. 6.2.2
Regularity with respect t o t h e space variables
Theorem 6.2 allows u s t o i n v e s t i g a t e t h e r e g u l a r i t y p r o p e r t i e s with respect t o t h e space v a r i a b l e s , a s i n t h e e l l i p t i c case.
THEOREM 6.3 f
E L2(Q), ii
satisfies
Proof. towards +
m.
€ V
suppose t h a t ( 5 . 6 1 ) , ( 6 . 1 ) hold a d t h a t aij, ai, a0 E cl(q) ( V + H1(6) or H35)); then the solution u of ( 6 . 9 ) , (6.10)
,
..
We d i s c r e t i s e (0,T) i n t o o , k , , , Nk = T , where N w i l l l a t e r tend du 2 which belongs t o L (Q) i n view of Theorem 6.2. We put g = f
-
We define a sequence un, n = 0 , ,
..
,N
- 1 by solving
(6.30) From t h e e l l i p t i c r e g u l a r i t y , t h i s being a p p l i c a b l e i n view o f t h e assumptions, we have
(SEC. 6 )
PARABOLIC P.D.E.'s
OF ORDER 2
129
(6.31) t h e constant C being independent of n , k. We next put g k ( t ) = gn u,(t)
t E r(n-i)k,nk[
for
= un
.
II
It follows from (6.31) t h a t uk l i e s i n a bounded subset of hence, up t o an e x t r a c t e d subsequence we have
(6.32)
Uk -.
Let v
where n
E
2 L ( 0 , T ; V);
2
L (0,T; H
2
(s))and
w weakly i n L2(0,T;H2(0)) we deduce from ( 6 . 2 9 ) t h a t
t = i n t e g e r p a r t of t / k .
*
Now, when k = 0 ,
Proceeding t o t h e l i m i t i n (6.33) we o b t a i n
s:
a(t;w(t),v(t))dt =
1:
-
(g(t),v(t))dt
Hence
(6.34)
a.e.t.
Now, f o r f i x e d
t
a ( t ; w ( t ) , v ) = (f
--,TIdu dt
v v E
v
.
( o u t s i d e a s e t of measure 0 ) , we a l s o have du
and t h e r e f o r e a.e.t.
u(t) = w(t),
i.e.
(6.28).
We can t h e r e f o r e i n t e r p r e t t h e problems t o be solved. Using arguments analogous t o t h o s e used f o r t h e e l l i p t i c c a s e , it can be shown t h a t u s a t i s f i e s
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
130
-
(CHAP. 2 )
a.e. in 6
+ A(t)u = f
(6.35)
ii
=
u(T)
and t h e boundary conditions
(6.36)
o
uIr
a.e.t.
or
(6.37) . .
depending on whether V = H
1
1
or V = H
.
Care must be exercised if V = ( v l v f H ' b ) , v = 0 on r c r}, as t h e regulari t y r e s u l t no longer holds i n g e n e r a l , i f r i s a boundary m k f o l d ; t h e boundary value problem i s i n t e r p r e t e d formally as foylows:
If
hood of
0, i s an a r b i t r a r y open s e t
ro
n
r,,
then
c
6' contained i n the complement o f a neighbour-
.
u f L2( 0,T;H2(Dl) )
, we
F o r Sobolev spaces constructed on Lp, p & 2, 1 < p < g l o b a l r e g u l a r i t y r e s u l t s analogous t o Theorems 5 . 5 and 5.6. an a r b i t r a r y open subset of Rn and t h a t Q = 0 x ]0,T[
have l o c a l and We assume t h a t 6 i s
.
We denote by
t h e space of t h e f u n c t i o n s
"(Q)
u
such t h a t
equipped with t h e n a t u r a l Banach- o r Hilbert-space norm i f p = 2 ;
i n t h e notation
, t h e "1" r e f e r s t o t h e number of d e r i v a t i v e s with r e s p e c t t o t which b2" "(0) a r e i n Lp and t h e "2" r e f e r s t o t h e number of d e r i v a t i v e s with r e s p e c t t o x ; i f p = 2 we w r i t e
V
such t h a t
'p
b2'l
(a)
E s(Q)we
We t a k e functions a . .
.
1J '
Let v
E
Lyoc(Q).
We denote by
have a;,
b2" "(Q)
1oc 'pu f b2""(Q)
a.
We denote by
on Rn
Lv
x
.
t h e space of t h e f u n c t i o n s
l0,TC which s a t i s f y
t h e following d i s t r i b u t i o n on Q :
u
131
PARABOLIC P.D.E.'s OF ORDER 2
(SEC. 6 )
where Y E s ( Q )
THEOREM 6.4 that
. Suppose t h a t the asswnptions ( 6 . 3 8 ) hold. Lu=--
dU
+A(th =f E
If G i s a bounded open s e t have
c
qOc(Q) , then
Let u
E
L y o c ( Q ) be such
.
u E lt~:Ah,~(Q)
Q and G' i s an open s e t such t h a t G I
c
G , then we
the constant C being dependent on t h e bounds on t h e c o e f f i c i e n t s of L on G , as well as on G and G ' . Remark 6.3
Under t h e assumptions ( 6 . 3 8 ) , and i f we a l s o have
and Lu = f , then w e have
t h e n o t a t i o n being s e l f - e x p l a n a t o r y .
I n p a r t i c u l a r i f p > n , then u
E
C2''(Q).
We s h a l l now g i v e a r e g u l a r i t y r e s u l t f o r t h e D i r i c h l e t o r Neumann problem.
Suppose t h a t B i s a bounded open s e t of which the boundary r =A6 We assume i n addition that the c o e f f i c i e n t s a ( x , t ) , a i ( x , t ) , ij I f f E L ~ ( Q )and = 0, then there e x i s t s one and only one a ( x , t ) f C'(Q) f h c t i o n u such that THEOREM 6.5
.is of c l a s s
c2.
.
u
(6.40)
and u i s a solution of ( 6 . 3 5 ) , ( 6 . 3 6 ) . function u such t h a t
u E LP(0,T;W2'P(O))
(6.41)
and (6.42)
u
,
i s a solution of ( 6 , 3 5 ) , ( 6 . 3 7 ) .
Similarly there e x i s t s one and only one
E LP(Q) I n both cases we have
132
STOCHASTIC D . E . ' s
--bW at (6.43)
=
+ AW = h
wlc
g
& P.D.E.'s
( C W . 2)
OF ORDER 2
9
9
w(x,T) =
w
we can reduce t h i s t o t h e homogeneous case when t h e d a t a g , h , are sufficiently r e g u l a r and s a t i s f y c o m p a t i b i l i t y conditions. I n f a c t , if we can f i n d 1
E LP(~,T;W29P(0)) with d t
E Lp(Q)
,
such t h a t
Y = h
(6.44)
Y(X,T)
on I
2
2
on
,
c+
U = W - Y
then
i s a s o l u t i o n of t h e 'homogeneous' problem a s i n Theorem 6.5 with f
g
hY - (-E + A(t)Y)
f Lp(Q)
,
and consequently, from ( 6 . 4 0 ) :
and we have
(6.45)
The necessary and s u f f i c i e n t conditions t h a t t h e r e e x i s t s Y f bZr1*"(Q) s a t i s f y i n g (6.44) a r e r e l a t i v e l y complicated; we r e f e r t h e reader t o , f o r example, DA PRATOGRISVARD [ll, LIONS-MAGENES [11 and-to t h e works of P. GRISVARD c11. ( I n p a r t i c t h i s i s what we term a 'compatu l a r , it i s necessary t h a t h(x,T) = w(x) i f x E r ; i b i l i t y c o n d i t i o n ' between t h e d a t a ) .
Remark 6.5 Let u be a s o l u t i o n of ( 6 . 3 5 ) , 16.36J (under t h e assumptions of Theorem 6.3) and l e t us suppose i n a d d i t i o n t h a t u E L (@) and f E L p ( Q ) , p 2 2 ; then it can e a s i l y be seen t h a t u E L p ( Q ) . I n f a c t i f we m u l t i p l y , formally t o (uIP-'u and i n t e g r a t e with r e s p e c t s t a r t w i t h , t h e f i r s t equation i n ( 6 . 3 5 ) , by t o t h e v a r i a b l e x over t h e domain 8,we o b t a i n ( u s i n g Green's formula)
(SEC. 6)
PARABOLIC P.D.E.'s OF ORDER 2
133
Since, possibly after multiplication by an exponential, we can always assume that . a 2 5 (sufficiently large), then it follows from (6.1)that
and from Gronwall's inequality we have
The above calculation can be justified by approximating the coefficients and f by regular functions for which the above integrations have a meaning, and then proceeding to the limit, taking account o f the a priori estimates. 6.3. Pa~abolicP.D.E.'s of second order in Rn
x
10,TC.
In a manner analogous to that in Section 5.4, we distinguish between two cases, depending on whether the coefficients of the operator are bounded or unbounded.
Unbounded coefficients
6.3.1. We write
A(t)
5
Ao(t)
+ A , ( t ) + aoI
where
BOr c, c1 suitable constants, m
satisfying the properties of
(5.95), E
Lm, and (though this is not a restriction in the parabolic case),
ao(x,t) L 5 > 0.
134
STOCHASTIC D . E . ' s
&
P.D.E.'s OF ORDER 2
(CHAP. 2 )
2 1 The n o t a t i o n s LT, H and F, 0 a r e defined i n Section 5 . 5 . 1 THEOREM f
6.6.
.
Suppose t h a t the asswnptions (6.46), (6.47), (6.48)hold. Let 2 u Fzen there e x i s t s one and only function ( * ) and Li E LT.
2 E L2(0,T;ItT)
such t h a t 2
(6.49)
u
( 6.50)
_ -au at
(6.51)
u ( T ) = B.
L (o,T;F),
E
+ A(t) u = f,
Remark 6.6. I t f o l l o w s , for example (what follows i s d e f i n i t e l y not t h e optimum r e s u l t i n t n i s s e u s e ) from (6.49), (6.50) t h a t , f o r a l l G c R n , t h e r e s t r i c t i o n u, of u t o G x 10,TC s a t i s f i e s
and hence
uG ( T ) i s meaningful, and
E
2 L (G);
so u ( T ) i s defined by t h e (compatible) s e t of t h e uG(T) and (6.51)i s meaningful..
Proof of Existence. We u s e n o t a t i o n analogous t o t h a t used i n Section 5.5.1: 3 = ~ v i vE L2 (o,T;H) , v vm E L'(o,T;L:)I = L ~ ( o , T ; F )
i
= space of t h e functions
E
C"(6)
( * * ) , with support i n a compact s e t , and
which a r e zero i n a neighbourhood of t = 0 ; t h e norm
1I
II
o f 'p i n
0 is
where
For u
E
i and
11 1, cp
E
and
i,we
1 ,1
denote t h e norms i n H
1
and L
2 71.
introduce
It can then e a s i l y be shown t h a t t h e problem reduces t o f i n d i n g u (6.53) where
(*)
(**)
I t would be s u f f i c i e n t t h a t f
Q = Rn x C0,Tl.
E
2 L (0,T;F').
E
$
such t h a t
(SEC. 6 )
PARABOLIC P.D.E.'s OF ORDER 2
135
=
$(P,'p)
A s we have been a b l e t o r e p l a c e a
+
by a.
k , with
k
a r b i t r a r i l y l a r g e , we
can s t i l l assume t h a t ( c f . ( 5.102) )
and i n t h i s c a s e
(6.56) Then, using t h e v a r i a n t of t h e p r o j e c t i o n theorem mentioned e a r l i e r i n Section
5 . 5 . 1 , with F , @, E i n s t e a d of F , solution of the problem. proof of Uniqueness, (6.57)
ixbl,'p)
Let u
= 0 v
0, E , we thereby deduce
the existence
Of
u, a
E B , satisfying 'p
E
.
6
By extending u by 0 for t > T and q by 9(T) f o r t > T (extensions which we a l s o denote by u and 9 ) , we can w r i t e ( 6 . 5 7 ) i n t h e form:
(6.58)
1; l(u,g)
x
We introduce t h e functions
x is xn(t) p ,
+ Ex(v,'p)]dt
xn
= 0
.
and pm as follows:
continuous on R , and piecewise l i n e a r ,
= 1 i f t t 2/n ,
E F(R)
We introduce BR = 6
R
(x)
, p,(t)
xn(t)
= 0 if t
= p,(-t),
5
l/n;
\ p m ( t ) d t = 1, p,
c
0
if It
I2
1
as i n Section 5.5.1.
We p u t :
( 6.59)
vn,m
= xn8R((x,8Ru)*pm)
,
m
> n
;
I n ( 6 . 5 9 ) , (xnBRu)is extended by 0 f o r t < 0 and t h e convolution i s taken w i t h respect t o t. By supplementary r e g u l a r i s a t i o n with r e s p e c t t o l i m i t , we see t h a t it i s permissible t o t a k e 'p = V
( 6.60)
xn,m
+ Y
n,m
+ z
n,m
= o ,
t n .m
and by proceeding t o t h e i n ( 6 . 5 8 ) ; we o b t a i n
~
.
STOCHASTIC D.E.'s
136
P.D.E.'s OF ORDER 2
&
(CHAP. 2 )
where
When m
+
m
,
Hence (6.60) gives Xn + Zn = 0 and t h e r e f o r e
But when n
( 6.62)
-
zn s o .
(6.61) -+
m,
Zn
{:
Z
t
[:a
2 E (u,e u) d t
.
Ea(u,ORU) 2 dt I 0
However, e x a c t l y as i n Section 5.5.1, c o e f f i c i e n t s a. being used h e r e ) t h a t
1:
Ex(u,8$)dt
-L
1:
and t h e r e f o r e we a c t u a l l y have
R
it can b e shown ( t h e assumptions on t h e
M(u)dt
,
where
M(u) =
i j
'p
+
,
~ - ~ ) x2 u ] d x ' p = b R U .
+ ( a O - 1 d1 i v a - - - 2 a1 2
dxi ax
i
axi
Hence
and we then deduce, s i n c e M(u) 2 allul12
, that
u = 0.
8
Remark 6.7. We can g i v e another e x i s t e n c e proof for T h e o r y 6 . 6 , based on a semi-discretisation with r e s p e c t t o t . We introduce A t = k = N a n d we put
we t a k e analogous n o t a t i o n f o r a:,
an.
We t h u s d e f i n e
(SEC.
6)
137
OF ORDER 2
PARABOLIC P.D.E.'s
We f u r t h e r put
S t a r t i n g from
( 6.63)
{$')'
,
fn(x)
%
=
i, we
--
u
-u n
n+l
f(x,t)dt
define u
+ Anun
At
by t h e implicit scheme from u n+l
= fn
;
Equation (6.63) corresponds t o t h e s o l u t i o n of t h e s t a t i o n a r y e l l i p t i c problem
(An
+ T1 ~un)
f o r which we apply Theorem Taking t h e i n n e r product of
= fn
1 + T~ u
~ + ~
5.4 ( t h i s i s p e r m i s s i b l e f o r A t s u f f i c i e n t l y s m a l l ) . 2 (6.63) w i t h u i n LT, we o b t a i n
Iu~-u~+,
+
If] + k(Anuntun)n
= k(fn,un)%
A s we have s a i d p r e v i o u s l y , t h e g e n e r a l i t y w i l l n o t be r e s t r i c t e d i f we assume that 2
(An(p,cp)n 2 aI/(pljF We t h e n deduce from
,
a
>
0
independent of n.
(6.64) that
( 6.65) I f we t h e n d e f i n e
q(t)and
u ( t )= u k
f k ( t ) by
i n [nk, ( n + l ) k [ ,
f k ( t ) = f n i n [nk,(n+l)k[,
2 2 t h e n f k + f i n L (O,T;L,)
(6.66)
l%(t)l:
Hence, when k
(6.67)
-+
+
and we deduce from
C
st' Iluk(s)ll,2 a s s
(6.65) t h a t C[
{ f Ifb)l: ds + lclf]
0:
2 2 uk l i e s i n a bounded s u b s e t o f L (0,T;F) n Lm(O,T;LT).
We can t h e n e x t r a c t a sequence, a g a i n denoted by u k , such t h a t uk
-t
2
2
u weakly i n L ( o , T ; F ) and weakly i n L ~ ( o , T , L ~ ) .
.
STOCHASTIC D.E.'s
138
I f now 9 (6.63) with
&
OF ORDER 2
P.D.E.'s
(CHAP. 2)
5, we define 9 = a)(., n k ) ; t a k i n g t h e i n n e r product i n ( p n , and summing Ever n , we o b t a i n :
E
Lf of
) t h e piecewise-constant f u n c t i o n ( r e s p . t h e I f we denote by 'p, ( r e s p . continuous and piecewise l i n e a r $unction) such t h a t
,
cpk(t) = 'p, i n [nk,(n+l)k[ (resp. q k ( n k ) =
9,).
then we can w r i t e (6.68) a s f o l l o w s ;
and we can proceed t o t h e l i m i t . We then deduce t h a t uk where u is t h e s o l u t i o n of problem ( 6 . 5 3 ) , ( 6 . 5 4 ) .
+
u weakly i n L 2 (0,T; F)
We have thus proved once again the existence o f a solution. I n fact:
i ) s i n c e we have uniqueness, it i s not necessary t o e x t r a c t a subsequence: t h e e n t i r e sequence uk converges t o u; i i ) we o b t a i n by means of t h i s procedure a supplementary piece of information:
u E L"(o,T;L,~)
.
rn
Remark 6.8. We s h a l l here g i v e a t h i r d e x i s t e n c e proof f o r Theorem 6.6, based For y > 0, we consider on t h e 'regularisation' method introduced i n Remark 5.5. t h e equation
-
+yA*A u 1 1 y -
(6.69) uy(T)
f
,
=
which admits a unique solution such t h a t (with t h e n o t a t i o n of Remark 5.5)
uY E L ~ ( o , T ; F ~ ),
dU
E L~(o,T;F;)
.
I n f a c t we apply t h e method of Theorem 6 . 1 , with V replaced by F and a ( u , v ) I n a d d i t i o n , we have t h e following estPmates, t h e (cf. (5.109)). by &'(u,v) proof'of which i s immediate:
I
J.
2
l i e s i n a bounded subset of L (0,T; F ) and 2 2 bounded subset of L (0,T;L,).
when y
+
0, u
Y
vy u
Y
lies
in
(SEC. 6 )
PARABOLIC P.D.E.'s
OF ORDER 2
We can e x t r a c t a subsequence, a l s o denoted by u
Y
2
i f we t a k e (6.69) t h a t
'p
E
139
, such
that
u + 6 weakly i n L (0,T; F); Y Q (defined i n t h e f i r s t proof of Theorem 6 . 6 ) , we deduce from
so T
+Y
GZ(Uy,?)
(A1~y9Al'p)n dt
and, i n t h e l i m i t , we o b t a i n
in(Gl'p)
= L('p) tl
'p
E
= L(cp)
.
6
Hence fi i s a s o l u t i o n , and i n view of t h e uniqueness, 5 = u and we have t h u s again proved e x i s t e n c e , with, i n a d d i t i o n , t h e following r e s u l t : 2
u --u
(6.70)
weakly i n L (0,T; F) as y
Y
Remark 6.9.
-t
0.
B
A f o u r t h proof could be based on e l l i p t i c regularisation which b2 b~a by -dT-b 4 0 ( t h i s can a l s o be used f o r
amounts t o r e p l a c i n g proving Theorem 6.1)
.
-
--
bt2
,
8
1s
We s h a l l now g i v e a regularity r e s u l t :
Suppose that t h e conditions for application of Theorem 6.6 h o l d THEOREM 6.7. and that i n addition we have
For
f
E L2(0,T;L:)
and ii E: H
the solution u given i n Theorem 6.6 s a t i s f i e s
(6.73)
Proof * (6.74) We put:
We introduce e R ( x ) as b e f o r e , and we w r i t e (6.50) i n t h e form:
-
bU
+
0
A (t)u R o
+ 6R A1 (t)u =
BR(f-aoU)
.
140
STOCHASTIC D.E.'s
&
P.D.E.'s OF ORDER 2
(CHAP. 2 )
(6.74) with X(eRu); t h i s g i v e s
and we t a k e t h e inner product i n L2 of
We have:
XU = x A U
-x
Blu
where bU
Blu = Z n-'
" Ti 'ijF j
and
Ao(BRu) = e A u
R o
+ rl ,
With t h i s n o t a t i o n , we can express ( 6 . 7 5 ) i n t h e form (6.76)
-3
a:(eRu,
+ = (eR(f'aou)
eRu)
(e,Aou
,
+
~
e
2
+~( e ,~~ , uu,eRAou)x l ~
~
+ BRAIu;r,
eRAou
+ rl -
- BIuIx
Bidn
.
We now i n t e g r a t e (6.76) from t t o T. We n o t e (as i n Section 5.5.1) t h a t
.
(eRAlu,e A U) d t = O ( 1 )
R o x
We have
and s i n c e
-
we have, i n view of t h e expression f o r rl,
(eRA1u,rl)n a t
0(1)
.
(SEC. 6 )
141
PARABOLIC P.D.E.‘s OF ORDER 2
st
lai(.)
Similarly, since f o r B u , we have
1
n-l-g, J
I
C(l+m(x))
.
= O(1)
T (BRAlu,B1u)X d t
and i n view of t h e expression
We t h e n deduce from ( 6 . 7 6 ) t h a t , by v i r t u e of t h e f a c t t h a t f have
We deduce from t h i s t h a t a s R
eRAou l i e s
+
-
E
2
2
L (0,T; L n ) we
we have 2
2
i n a bounded subset of L (0,T; L n ) ,
u l i e s i n a bounded subset of Lm(O,T; F),
8
R
from which we deduce t h e theorem.
w
Remark 6.10.
We can a l s o o b t a i n t h e above r e s u l t by semi-discretisation, a s
Remark 6.11.
We deduce from (6.73) t h a t
i n Remark 6.7.
-
( 6.78)
dU
+
A, (t)u E L ~ ( o , T ; L $
.
We do n o t know whether, s e p a r a t e l y , we a l s o have (under t h e conditions of 2 and A l ( t ) u E L (0,T;Lf) ( i n t h i s r e s p e c t s e e Remarks 6.14 and
Theorem 6 . 7 )
$
7.7). Remark 6.12. for example
N a t u r a l l y , with supplementary r e g u l a r i t y assumptions on f and 2 , we can o b t a i n supplementary L 2 ( ~ , ~ ; ~ n, )A ( T ) u 6
f E bt
L,‘
r e g u l a r i t y r e s u l t s on u , f o r example: 1
bt
Remark 6.13.
E L ~ ~ o , T ; L ~ )(see If
a l s o Section 7 . 4 ) .
f E @(O,T;L;(R”))
then
u E Lp(O,T;LP(Rn))
Remark 6.5, so t h a t , a l s o using Theorem 6.4, we have
u
analogously t o
E b2”’*(Rn x ]O,T[). l0C
S i m i l a r l y i f t h e c o e f f i c i e n t s s a t i s f y t h e assumptions of Remark 6.3 and i f
,
g,E qoc(Q)
bt 1 implies, if p
Remark 6.14.
then u E I$:’”(Rn
>
n
,
u,
x ]O,T[)
u E C2” (R” x ]O,T[)
.
and
E
.,’iL’p(Q)
, which
Further notes on the regularity with respect to t .
I f we d i f f e r e n t i a t e equation ( 6 . 5 0 ) with r e s p e c t t o t , and i f we put
142
STOCHASTIC D.E.'s
&
P.D.E.'s
(CHAP. 2)
OF ORDER 2
t h e n we o b t a i n
( 6.791 where
w i t h t h e " i n i t i a l " c o n d i t i o n (deduced from
(6.80)
w(T) = A(T)ii
(6.50), (6.51))
- f(T) .
assumptions
We t h e n adopt t h e
a t E L'(o,T;F*) (6.81)
A(T)C
- f(T)
I
E L,'
(6.82) together with
( 6.83) S i n c e t h e right-hand s i d e o f
(6.79) i s
following theorem:
t h e n i n L 2 ( 0 , T;
F), we have t h e
Under the assumptions of Theorem 6.6 and w i t THEOREM 6.8. (6.83), the solution u obtained i n Theorm 6.6 s a t i s f i e s
(6.84)
(6.81), (6.821,
au E L 2 (o,T;F) . at
COROLLARY 6.1.
Under Che conditions o f Theorems 6.7 and 6.8, we have
(SEC. 6 )
u E L~(o,T;F), au E L 2 ( o , T ; F ) , (6.85)
143
P M O L I C P.D.E.’s OF ORDER 2
.
~~u E L ~ ( o , T ; 2L ~ )
A ~ UE
L 2 (o,T;L:)
,
m
We n o t e t h a t t h i s completes Remark 6.11, but only under a d d i t i o n a l assumptions With d i f f e r e n t assumptions, a r e s u l t of t h e type (6.85) i s given i n Remark 7.7 through t h e use of p r o b a b i l i s t i c methods. N a t u r a l l y , it i s p o s s i b l e t o i t e r a t e t h e above procedure.
Remark 6.15. Further notes on the reguZarity with respect t o x . We now d i f f e r e n t i a t e ( 6 . 5 0 ) with r e s p e c t t o xk, and t h i s time we put
We o b t a i n bW -+ (Ao+ dt
(6.86)
A l + aoI)w
bf bXk
bAo
(-
bXk
bA1
ha
bXk
bXk
+ - + 2)u
where
with t h e “ i n i t i a l “ condition
(6.87)
w(T) =
dii . bxk
We adopt t h e assumptions
,
(6.88)
m(x) = 1
(6.89)’
lJ , 0 E bXk dXk
ba..
ba
Lm(Rn x ( O , T ) ) , V k
We t h e n have t h e following theorem: THEOREM 6.9. (6.901, we have:
Under the assumptions o f Theorem 6 . 6 and with ( 6 . 8 8 ) , ( 6 . 8 9 ) ,
144
STOCHASTIC D . E . ' s
where u
& P.D.E.'s
OF ORDER 2
(CHAP. 2 )
i s the solution obtained i n Theorem 6.6.
Proof. It i s s u f f i c i e n t t o apply Theorem 6 . 6 t o equations (6.86), ( 6 . 8 7 ) , provided we can show t h a t t h e right-hand s i d e of (6.86) i s i n L2(0, T ; (H')').
af E
But s i n c e f E L 2 (0,T;L2) we have
L2(O,T;(H:)l).
bAO u
and hence by v i r t u e of (6.89),
dxk
E L2(0,T;(H;)')
S i m i l a r l y by v i r t u e of (6.47) with m(x) = 1, we have and hence t h e r e s u l t . 8
6.3.2
H i we
Similarly i f Y E '
bxk
have:
. 2
E L2(0,~;~:)
"k
Bounded c o e f f i c i e n t s
We s h a l l e s s e n t i a l l y u s e t h e same n o t a t i o n as i n Section 5.5.2. We consider t h e family of d i f f e r e n t i a l o p e r a t o r s introduced i n t h e preceding section. The c o e f f i c i e n t s now s a t i s f y t h e following assumptions
z
ij
a . . ( x , t k i 5 z- a 1J
j -
a o( x,t) 2 fi
z ti2
i
( s u f f i c i e n t l y l a r g e ) > O,(which i s not a c o n s t r a i n t ) .
We d e f i n e on V a f&ly(indexedby t E C0,Tl) of continuous b i l i n e a r forms a ( t ; u,v) by meank of formula (5.122) i n which t h e a i j , ai, a. now depend upon t . On t h e b a s i s of a proof i d e n t i c a l t o t h a t of Theorem 6.1 ( * ) , we have t h e following theorem: THEOREM 6.10.
Suppose that (6.92) holds.
then there e x i s t s one and only one element u u E L~(VJ
(6.93) (*)
,
E L~(v;)
+ u(T)
I
u
,
E
L2(V;)
and u
E
Hi ;
such that
satisfying
a(t;u(t),v)
.
Let f
P
( f ( t ) , v ) a . e . t E ]O,T[
v
v E
8
I n f a c t , we h e r e have two examples with t h e same a b s t r a c t s e t t i n g .
v'
P
(SEC. 6)
We a l s o have t h e equivalent of Theorem THEOREM f
L
E
2 dt E
2
6.11.
(H ) and u J !
L‘(H,)
Under t h e asswnptions of Theorem 6.10 and E
.
6.2:
V
!J
E L‘(O,T;D(A(t)))
da
\-$.I
$
C
,
for
t h e solution u ( t ) of (6.93) belongs t o L m ( V ) and !J
The proof i s i d e n t i c a l t o t h a t f o r Theorem IL
145
PARABOLIC P.D.E.‘s OF ORDER 2
6.2.
It t h e n follows t h a t
and t h a t
F i n a l l y , we can e s t a b l i s h a theorem analogous t o Theorem 5.10: THEOREM
bounded.
6.12.
Suppose t h a t (6.36), (6.92) hold and t h a t
n
Let f E LP(O,T;Q’P1p)
L2(0,T;Hp) and u = 0 ;
da
ba
bX.
I(
,2 dt
are
then t h e solution of
(6.94) s a t i s f i e s t h e regularity property u E L ~ ( O , T ; W ~ , ~and ,~)
* d t
E L~(O,T;#’~’~)
.
A d d i t i m a l l y , we have t h e estimate
Proof. We proceed e s s e n t i a l l y as i n t h e e l l i p t i c case (Theorem 5 . 1 0 ) . denote by uR t h e s o l u t i o n of t h e D i r i c h l e t problem
--+ dt
A(t)uR = f
in
We
’fi
(6.96)
Using t h e p a r t i c u l a r system have
(6.97)
Oil ‘pi
defined i n (5.131) it can be shown t h a t we
146
STOCHASTIC D . E . ' s
& P.D.E.'s
(CHAP. 2)
OF ORDER 2
where C i s independent of R .
IU!~-'
Multiplying t h e above r e l a t i o n (6.97) by u exp - pplxl and i n t e g r a t i n g o v e r 8 , we deduce by means of a c a l c u l a t i o n similar t o t h a t performed i n Remark 6.5 ( w h c h i s completely j u s t i f i e d here s i n c e i s regular t h a t
5
Furthermore, from (5.137) it can be seen t h a t if w
E
then
Lp(o,T;w2'p(Rn))
and a l s o t h a t i f v E L p ( 0 , T ; W 2 ' p ' ~ ( ~ R ) )
%,
By applying t h i s l a s t r e l a t i o n with v =
we deduce from (6.97) (as i n t h e
e l l i p t i c case) that
(6.98)
We extend
%
over t h e whole of
Rn i n t o
bounded subset of Lp(0,T;W2'pa'(Rn)) and
%
"ik at
i n such a way t h a t f$,
lies in a
l i e s i n a bounded subset of
By e x t r a c t i n g a subsequence and proceeding t o t h e l i m i t , we LP(O,T;WoYPr'(Rn)). obtain t h e stated result. rn
Remark 6.16.
We p u t f
on
]o,T[
0
on
]-T,O[
.
and ] T , P T [
We extend t h e o p e r a t o r A C t ) i n such a way t h a t it i s defined over l-T,2T[ We then consider t h e solw h i l s t r e t a i n i n g t h e p r o p e r t i e s of t h e c o e f f i c i e n t s . u t i o n B of
(6.99)
It i s c l e a r t h a t
i=
theorem 6.4, a p p l i e d t o
0 on [T,2TI and
i, we
have
u E Co(Rn x [O,T])
Z
u' E (if
p
u on C0,TI
>
n+l)
.
From t h e l o c a l r e g u l a r i t y
x ]T,2T[).
W2""(Rn
.
a
Hence
(SEC. 6)
6.4.
147
PARABOLIC P.D.E.'s OF ORDER 2
P o s i t i v i t y p r o p e r t i e s of t h e s o l u t i o n 2
For f given i n L ( O , T ; V ' ) ,
(6.100)
f 2 0 o
we say t h a t
f
is 2 0 i f
2 (f,v)dt Z 0 V v E L (0,T;V)
5 0
with
v 2 0
.
We t h e n have t h e following theorem:
THEOREM 6.13.
The conditions are those of Theorem 6.1.
We assume t h a t
(6.101) t h e mapping v + v- = s u p ( - v , o ) i s L i p s c h i t z continuous from V +. V and t h a t f and ii are given with f L 0 and Then ii 2 0 . z 0, E H.
Proof.
(*)
0
148
STOCHASTIC D.E.'s
&
(CHAP. 2 )
P.D.E.'s OF ORDER 2
so t h a t
Similarly
and hence t h e r e s u l t then follows. Then l e t u. be a sequence of f u n c t i o n s 2) J W(0,T) (such a sequence e x i s t s ) .
E C'([O,T];V)
such t h a t u. J
+
u in
We have
and we can proceed t o t h e l i m i t i n t h i s e q u a l i t y , by v i r t u e of 1).
I
6.5. Green's o p e r a t o r We consider t h e s i t u a t i o n of Section 6.3.2, and i n p a r t i c u l a r we adopt t h e assumption (6.92).
We t a k e
li
du -=+ A(t)u (6.1 06) u(t,) Since
at
E
c
P
L2(Vt), u
E
P
0,
.
< t2
t
. f o r a l l tl 5 t 2 and hence
W(t,,t,),
.
u E Co([tl,t2];H) Furthermore, t h e mapping
, VP = V t H 1 ( F ) .P E H, u E L 2 ( V ) , a s o l u t i o n of
= 0 and hence H = H = L2(R")
By applying Theorem 6.8; we d e f i n e f o r
+ u ( t ) i s continuous from H + H.
1
If we put
dt,) =
G(tl,t2)c
we t h u s d e f i n e a family ( w i t h two parameters tl,t2 with tl 5 t 2 ) of o p e r a t o r s
G(tl,t2)
E 2 (H;H)
.The f u n c t i o n tl,t2 + G ( t ,,t2) i s termed t h e Green's operator
a s s o c i a t e d with A ( t ) . From t h e theorem of k e r n e l s due t o L. SCHWARTZ Ell, we can r e p r e s e n t G ( t t ) by 1' 2
G(tl,t2)(P =
iRn
P(x,tl;S,t2)'p(C)dS
V 'p f S(Rn)
where p ( x , t , c , t 2 ) i s ( f o r t ,t f i x e d with tl 5 t 2 ) ,a d i s t r i b u t i o n on :R 1 1 2 defined uniquely by G ( t l , t 2 ) .
x
(SEC. 6)
Taking
149
PARABOLIC P.D.E.’s OF ORDER 2
u = 9 , we deduce from (6.106)t h a t t h e d i s t r i b u t i o n
p
satisfies
(6.107)
where A (t,) r e p r e s e n t s t h e d i f f e r e n t i a l o p e r a t o r A ( t l ) a p p l i e d with r e s p e c t t o t h e Let A * ( t ) be t h e a d j o i n t of A ( t ) ( * ) i . e . v a r i a b l g X.
A*(t) = A, ( t )
(6.108)
+ A;(t)
+ aoI
where (6.109) (note that A o ( t ) is self-adjoint). We consider t h e equation
+
A*(t)v
P
(6.1 10)
v(t,) = ? E H
t,
0
1 ;
- dV
+ A(t)v
= f
(6.112)
v(T)
=
0
belongs to L~(O,T;E?+'). Proof.
We know from t h e v a r i a t i o n a l theory (Theorem 6.1), t h a t v
D i f f e r e n t i a t i n g formally with r e s p e c t t o xl, that
w
and p u t t i n g
satisfies
- -dw dt w(T)
+ A ( t ) w + A:
1
(t)V
=
w
E
2 1 L (0,T;H ) .
av
= x, it can be
seen
df -
= 0
( t ) i s an o p e r a t o r defined l i k e A ( t ) but f o r which a l l t h e c o e f f i c i e n t s =1 aaij bai bao a i j , ai, a. a r e replaced by , Hence A,: ( t ) v E L2(0,T;H-') -, ax1 dXl dXl 1 so t h a t by applying t h e v a r i a t i o n a l t h e o r y once a g a i n , we o b t a i n
where A.!
-,
w E L'(o,T;H') 2 1 Similarly * f L (0,T;H ) ax.
-
.
v
i
and hence f i n a l l y we have v
E
2
2
L (0,T;H ) .
,
TO
1
j u s t i f y t h i s c a l c u l a t i o n , we u s e t h e method of d i f f e r e n t i a l q u o t i e n t s ( s e e Theorem 5.3). I n view of t h e assumptions adopted for t h e c o e f f i c i e n t s and f o r
f
,
it i s permissible t o d i f f e r e n t i a t e m times, so t h a t v E L2(0,T;?+l). By applying t h e closed graph theorem, or d i r e c t l y from t h e above c a l c u l a t i o n s , we have:
(6.113)
(SEC. 6 )
PARABOLIC P.D.E.'s
151
OF ORDER 2
We now
The asswnptions are t h e same as those i n Theorem 6.14.
THEOREM 6.15.
take f, E ~ ~ ( 0 , ~ ; H m and - l ) g, beZonging t o L ~ ( 0, T ; gm+' ),
E
(duaZ of
H-m
at
p); then thcre e x i s t s a unique )
E L ~ (0, 'T; "'B
u
such t h a t
(6.114)
Proof. We n o t e t h a t ;AT) has a meaning as an element of H-m, s i n c e u i s We u s e t h e t r a n s p o s i t i o n procedure ( s e e LIONS-MAGENES continuous from [(),TI + HEll). Let f+f L ( 0 , T ; e I). I n view of t h e above theorem, t h e r e e x i s t s v E L~(o,T;? ), a solution o f
.
The mapping
f
- 1;
< v,f+ >
dt
+ < v(T),g,
>
2 -1 i s , i n view of (6.113) c l e a r l y l i n e a r an$ continugys on L ( 0 , T ; p ) . ) such t h a t t h e r e e x i s t s a unique u belonging t o L (O,T;H-m
dt =
v Let write (6.1 1 7 )
'p
E b(Q)
j i < v,f,
Consequently,
A(t)u,cp
>
dt =
j
dt
.
The right-hand s i d e of (6.117) i s t h e i n n e r product i n t h e sense of t h e d u a l i t y
We can t a k e i n (6.116)
f
P
2+
A*(t)'p
, and
naturally v =
'p
.
We o b t a i n , s i n c e ' p ( T ) = 0 ,
(*)
To apply Theorem 6.14, it i s s u f f i c i e n t t o r e p l a c e A*(T-t) i n s t e a d of A ( t ) .
t
by T - t and t o consider
STOCHASTIC D.E.'s
152
&
P.D.E.'s OF ORDER
(CHAP.
2
2)
which, compared with (6.117)shows that
- BU
+
A(t)u = f,
in the distributional sense. continuous from fo,T1
"
bt
hence u
is
Integrating by parts, we have dt
E
S O
=
lo'
-
d t at
which compared with (6.116)shows that
< u(T),Y(T) >
< ~,,P(T) >
JT < > + I
+
< u(T),v(T)
- < u(~),v(~) ( 6.1 18)
, and
€ L2(0,T;H-m-1)
u9 u,f
8V
>
+ A*(t)v >
dt
dt
.
But, from the trace theorem (see LIONS-MAGENES [ll, vol. 1, p.25) the mapping v
+
(v(T!,v(o))
from w,(o,T) I {v E L ~ ( O , T ; $ P + ~ ) , E L ' ( ~ , T ; P ' ) ~ We can therefore, in (6.117),take v(T) arbitrary in
into
9,
so
(el2 is
surjective.
that u ( T ) = g,.
For fixed 6, the distribution 6(x-S) belongs to H-m, for m =
[$ + 11.
We have
We can
thus confirm the existence, for (6,t2) fixed, of a distribution (with respect to x,t ) , denoted by p(x,t ;E,t ) , which is a solution of (6.107)and which 1 1 2 belongs to L2(0,t2;H-m+')
n
Co(0,t2;H-m)
.
8
We put ( * )
then v1 is a solution of
( * ) This change of function is natural in the context of a n a l y t i c semi-groups: see Remark 6.17 below.
(SEC. 6)
Thus v1
PARABOLIC P.D.E.'s OF ORDER 2
E
L2(0,t2;H-m+3) n Co(0,t2;H-m+2).
sequence yk E L2 (O,tp,.H-m+2k+l ) by the recurrence formula
--byk + A(t)vk dt =
Vk(t2)
n
153
We can then iterate and define a
Co(0,t2;H-m+2k)
,% E L2(0,t2;H-m+2k-1)
- yk-l
E
0
and Vk(t) = (t-t2)vk-l Hence, finally
We can, in particular, take k = m.
vm
E
The function
L2(o,t,;P+l) n Co(o,t2;Hm).
Since the elements of
9 are
(to within an equivalence) continuous functions,
and since the injection from fl Into the set of continuous functions on Rn (equipped with the Frechet space structure) is continuous, we can see that x,t1 +. vm (x,t1 't2 ) is continuous on R" x CO,~,I. We shall now investigate the continuity with respect to the four variables x,tl,E,t2. We assume that t2 E [O,T] and we put t1= tt2, 0 d t d 1. We successively define x(x,t,c,t2)
= P(X,tl , S , t 2 )
w k ( x , t ; S , t 2 ) = (t-1)
Hence in particular if 0 I t
The distribution
TI
k
n(x,t,E,t*:
< t2 we have
is defined on Rn
x
[0,11 x
( 6.1 21 )
-
It can easily be shown that the mapping
S,t2
n(x,t;S,t2)
.
Rn
x
C0,Tl and satisfies
154
i s continuous from Rn
STOCHASTIC D . E . ' s
& P.D.E.'s
[O,TI + W - m ( O , l )
(*) using i n particular t h e f a c t t h a t
x
5 + 6 ( x ) i s continuous from R~ 5
I
dw.
I -t h e mapping S , t 2
-+
dt'
-+
H-".
OF ORDER 2
Since w1 i s a s o l u t i o n of
+ t2A(tt2)wl =
-
I
w l ( x , t ; & , t 2 ) i s continuous from Rn
can be shown by recurrence t h a t S , t ,
m
x R" x [O,T]
[0,1]
x
C0,TI
+
W_,+~(O,~).
x
[O,Tl
+
Co(O,l;?);
we t h e r e f o r e
.
However (6.120) then implies t h a t on t h e s e t 0 i tl < t 2 i T , x,S function p ( x , t l ; S , t
2
It
wm(x,t;S,t2) i s continuous from
+
Rn x C0,TI -+ W m ( O , l ) and hence a l s o from Rn have t h e r e s u l t t h a t w i s continuous on
R" x
(CHAP. 2 )
E
Rn t h e
) i s continuous.
We have t h u s proved t h e following theorem: THEOREM 6.16. Suppose t h a t t h e c o e f f i c i e n t s a i j , a i , a. are continuous with respect t o a l l t h e variables, bounded, m-times continuously d i f f e r e n t i a b l e with respect t o x w i t h m = 1" + 11, a l l t h e d e r i v a t i v e s being bounded; then t h e Green's operator associgted with A ( t ) can be represented bx means of a f u n c t i o n t 2 5 T , x , s E R , which i s continuous p ( x , t , , g y t 2 ) y defined on t h e s e t 0 < tl with respect t o the f o u r variables.
Remark 6.17.
I f x,S
E
K , a compact subset of Rn, we have n
(6.122) This r e s u l t i s not t h e b e s t p o s s i b l e , which i s not s u r p r i s i n g s i n c e we have n o t , i n t h e above, made use of t h e f a c t t h a t t h e p a r a b o l i c o p e r a t o r i n question i s of aecond order. We s h a l l make use of t h i s f a c t i n Theorem 6.17 below. We s h a l l give here, without t h e f u l l p r o o f , another method which i s v a l i d f o r t h e case i n which the c o e f f i c i e n t s are independent of t ( * * ) and which extends t o p a r a b o l i c operators of arbitrary order. We t a k e t 2 = T and we put G ( t ) = G ( t , T ) ; t h e o p e r a t o r G ( t ) s a t i s f i e s
--dd t
(6.1 23)
G(t)
+
AG(t)
I
0
,t
5;" then D(A-Y) c
with-y >
t and
y
D(AY+~) c co
+
.
s >
Co and if we choose
t,
so
s >
that
2
we can see that
The operator G(t)'p = j p ( x , t ; y , T ) ' p ( y ) d y then maps the space of measures with lies compact support into the space of continuous functions and (T-t)n'2+E@(t) in a bounded domain (fixing the supports) so that
The above is justified if the coefficients a. are in Cm, m being as in ij Theorem 6.16. THEOREM
6.17.
-
The assumptions are those of Theorem 6.16.
-
The function p i s p o s i t i v e and i f . a = 0 it s a t i s f i e s JP(x,t,,c,t,)dE
=
JkX,tl,Fit2)dr
1
2 n Proof. In fact we consider in (6.106) that E L (R ) , u t 0. corresponding solution u(t) is then t 0 (Theorem 6.13). Hence
which implies p
Z
The
0.
We next consider the function eR defined in (5.104), and we suppose that We take = OR in (6.106) and if % denotes the corresponding solution,
a = 0. we have
u
156
STOCHASTIC D . E . ' s
& P.D.E.'s
(CHAP. 2)
OF ORDER 2
= 0 , A ( t ) e R involves only t h e d e r i v a t i v e s of and hence A ( t ) e R = 0 f o r 1x15 R Znd A ( t ) B R + 0 i n LP(0,t2;LP(Rn)). By a
Now, by v i r t u e of t h e f a c t t h a t a OR,
c a l c u l a t i o n analogous t o t h a t performed i n connection with Remark 6.5, we can
From theorem 6.4, it then deduce t h a t w + 0 i n , f o r example, LP(0,t2;LP(Rn)). R 2' yp(Rn x ]0,t2[) and t h e r e f o r e then follows t h a t w + 0 in R oc wR(x,tl) 0 v x , t, < t 2 Furthermore p ( x , t l i S , t 2 ) B R f S ) 2 0 P(x,tl,S,t2)
-
.
5
r
.
From t h e Monotone Convergence Theorem it then follows t h a t
.s,
= lim
p(xtt,;C,t2)dE
A n analogous proof shows t h a t
jRn
%(x,tl)
= 1
P(X,tl;5,t2)&
.
= 1
.
8
We have assumed t h e c o e f f i c i e n t s of t h e operator A t o be Remark 6.18. m-times d i f f e r e n t i a b l e with r e s p e c t t o x . The assumptions can i n f a c t be weakened considerably. It i s s u f f i c i e n t t o assume t h e c o e f f i c i e n t s t o be once continuously d i f f e r e n t i a b l e with r e s p e c t t o x , with t h e i r d e r i v a t i v e s s a t i s f y ing a H8lder c o n d i t i o n , and t o be H8lder continuous with r e s p e c t t o t , and bounded, t o g e t h e r with t h e i r d e r i v a t i v e s . It can be shown t h a t p ( x , t l ; 6 . t 2 ) i s continuous over 0 5 t
1
< t 2 2 T , x.6
E
Rn,
(For t h e d e t a i l s , and twice continuously d i f f e r e n t i a b l e with r e s p e c t t o x , c . We can a l s o prove o r A. FRIEDMAN DI). s e e I L ~ I N , KALASHNIKOV, OLEINIK t h e following estimates
t11
(6.132)
where M ,
3,M2, a ,
al, a 2 , X a r e c o n s t a n t s which a r e > 0.
(SEC. 7 )
157
PROBABILISTIC INTERPRETATION
Remark 6.19.
1
I f a . . ( x , t ) = s6ij, ai = 0 , a. 1J
= 0 , it can e a s i l y be shown by
direction calculation that P(X,tl,S,t2) =
1 -
1
exp
1x5l2 -.
The estimates ( 6 . 1 2 4 ) , (6.125) a r e obtained by comparing t h e equation for p with t h e h e a t equation and applying a s e r i e s of changes t o v a r i a b l e s .
7
PROBABILISTIC INTERPRETATIOL' OF THE SOLUTION OF BOUNDARY VALUE PROBLEMS OF SECOND ORDER
7.1
D i r i c h l e t problem
r
Let 6 be a bounded open subset of R n , with consider t h e second-order e l l i p t i c o p e r a t o r
b
A = - '
ij
b
~
~
j
+ ' Fi i
ai
By a p p l i c a t i o n o f Theorem 5.4 and from on f u n c t i o n u such t h a t
(7.3)
u E w2'P(~) n ~ ~ (n 0~ )' ( Au = f a t any p o i n t of 8 u/$
=
0
b i
+
= Bb
'0
Remark 5.2,
of c l a s s C
j
2
.
We
~
t h e r e e x i s t s one and only
5)
.
Since t h e matrix a = ( a ) i s symmetric p o s i t i v e d e f i n i t e , it possesses a ij U2 which i s i t s e l f symmetric (so that a = 2 )
square r o o t , denoted by positive definite. example KATO C11)
72
Furthermore, it has t h e r e g u l a r i t y of a, s i n c e ( s e e for
t h e i n t e g r a l being uniformly convergent s i n c e
(7.5)
1\(2a(x)+ k1-Y
(2a+'')A
.
158
STOCHASTIC D.E.'s
& P.D.E.'s
OF ORDER 2
(CHAP.
2)
We now adopt a r e g u l a r i t y assumption which i s s t r o n g e r than ( 7 . 1 ) r e l a t i n g t o t h e c o e f f i c i e n t s a i j and ai ( * ) :
(7.6)
a. E ij
.
c2iii) , ai E c 1 (5)
We d e f i n e
(7.7) and hence, by v i r t u e of ( 7 . 6 ) we have g
j
E
C1(&).
The f u n c t i o n s ~ ( x and ) g ( x ) belong, by c o n s t r u c t i o n , t o Cl(8). By s u i t a b l e extension t o Rn we can suppose t h a t they a r e i n C1(Rn) and, together with t h e i r
f i r s t derivatives, are bounded.
We now seek a p r o b a b i l i s t i c i n t e r p r e t a t i o n of t h e f u n c t i o n u by constructing a s t o c h a s t i c d i f f e r e n t i a l equation f o r which t h e t r a j e c t o r i e s y ( t ) a r e t h e " c h a r a c t e r i s t i c s " of A. We s h a l l show t h a t t h e r e i s an i n f i n i t g number of ways of doing t h i s , with a strong formulation, i f t h e c o e f f i c i e n t s a r e r e g u l a r ( i n t h e sense ( 7 . 6 ) ) ( i f t h e c o e f f i c i e n t s a r e l e s s r e g u l a r a weak formulation i s p o s s i b l e , s e e Remark 7 . 1 below). We t a k e a p r o b a b i l i t y space (il,a, P ) , an i n c r e a s i n g family of sub-a-algebras of 0 and an Rn-valued standardised Wiener process w ( t ) , which is a martingale. We can then consider, on an a r b i t r a r y f i n i t e i n t e r v a l , t h e s t o c h a s t i c d i f f e r e n t i a l equation
at
(7.9)
at
y(0) = x
E R"
where x i s f i x e d , non-random. The s o l u t i o n i s denoted by y x ( t ) ( * * ) . by ' I t~h e e x i t time from 8 ,i . e .
We note t h a t i f x E 8 vention T = + m ( * * * ) .
, then
'cX
= 0 and i f
yx(t)
E 8 , V t
, then
We denote
by con-
(7.11)
This is not s t r i c t l y necessary ( s e e Remark 7.1), but allows t h e presentation t o be s i m p l i f i e d . (**I Or, f o r b r e v i t y , by y ( t ) i f t h e r e i s no p o s s i b i l i t y of confusion. ( s e e Theorem 7 . 1 ) . (***)We s h a l l see i n f a c t t h a t a . s . 'cX < +
(*)
-
(SEC. 7 )
159
PROBABILISTIC INTERPRETATION
Proof. If x 6 r , T = 0 and u(x) = 0 and t h e r e f o r e (7.13) i s c l e a r l y s a t i s f i e d . We t h u s assume t h a t x ~ ~ Let 0 VE. be a closed neighbourhood of r such t h a t i f 5 E 8 and 5 1 V, then d ( 5 , r ) > E . Let v be a f u n c t i o n E C2(Rn) which coincides with u on 6VE,2. We put
-
-
5:
= ao(y(t))z
,
z(t)
(7.1 2)
exp
[
;
ao(y(s))ds]
thus z ( t ) i s a s o l u t i o n of dz
(7.13) A s t h e point
is fixed i n
x
be t h e e x i t time from
.
~ ( 0 =) 1
8, we
can always assume t h a t x € 0 -V
8-VE of t h e process y ( t ) .
.
Let T~
We apply I t o ' s formula t o t h e
process ( y ( t ) , z ( t ) ) and t o t h e f u n c t i o n a l v ( x ) z which belongs t o C2(Rn+1). have i f T i s a r b i t r a r y but > 0:
We t h u s
The expectation of t h e s t o c h a s t i c i n t e g r a l i s zero, and we t h e r e f o r e have
E V ( Y ~ ( T T~ A ) ) Z ( Z ~T)
(7.14)
But f o r 0 s t
so t h a t we have
We make
E +.
6
162
STOCHASTIC D . E . ' s
& P.D.E.'s
where C1 depends only on t h e c o n s t a n t s KO, K1.
OF ORDER 2
(CHAP. 2 )
Hence
and hence i f f3 i s s u f f i c i e n t l y l a r g e t h e right-hand s i d e of (7.25) i s w e l l defined. We put fN =
(7.26)
.
f A ' N
The f u n c t i o n f N i s not continuously d i f f e r e n t i a b l e , but
bx bfN E
qOc .
t h e l e s s t h i s i s s u f f i c i e n t t o ensure t h e e x i s t e n c e of a f u n c t i o n % ( x )
UN f
2
(R
n
,
f
UN
such t h a t
1
H,
(7.27)
A%
= f N a t any point of R".
We s h a l l now show t h a t
F i r s t , we show t h a t we have (7.29)
IUNb)\
5 YN
We note i n f a c t t h a t i f w
, E
YN
T
R -
for a c e r t a i n R o ( w ) and R
b
Ro(w).
% of
t h e process y X t ( s ) ;
172
STOCHASTIC D.E.'s
(7.72)
z
a.s.
R
A T = T
& P.D.E.'s
R > R~(w)
for
Application of I t o ' s formula t o %M
OF ORDER 2
(CHAP. 2 )
.
leads t o
(7.73)
But from
(7.72) = 0
t h u s a.s. t h i s converges t o 0 as R +. + l e a d s immediately t o (7.71)
-.
for
Rz
Ro(w)
,
Application of Lebesgue's theorem
We next prove t h e estimates lu,,(x,t)
I5
C(1+/xlrn)
f
By proceeding t o t h e l i m i t s u c c e s s i v e l y i n M and N , we o b t a i n (7.69).
Remark 7.9.
The s o l u t i o n
of problem ( 7 . 6 5 ) s a t i s f i e s t h e e s t i m a t e
u
I=dU b , t )I s C(l+lxlrn) and hence, as f o r Remark 7.5, A . u
€
2 (O,T;Hn), 1
E
2 2 L (O,T;L,)
-=+ du
Ao(t)u
from which it follows t h a t
€ L2(0,T;Lf)
.
We then deduce, i n a manner analogous t o t h e proof of Theorem 6.2, by using t h e symmetry of t h e form a T ( u , v ) (and by multiplying by %) t h a t
au
E
2 2 L (O.T;L,)
at
and hence a l s o Aou
E
2 2 L (O,T;L,).
This supplements t h e r e s u l t of Theorem 6 . 6 , without having t o b r i n g i n assumpt i o n (6.721, b u t n a t u r a l l y t h i s i s a t t h e expense of having t o u s e t h e assumptions of Theorem 7.4.
8.
MARKOV PROCESS
ASSOCIATED WITH THE SOLUTION OF A STOCHASTIC DIFFERENTIAL
EQUATION
8.1
I n t e r p r e t a t i o n of t h e f u n c t i o n
p(x,tl,S,t2).
We now propose t o g i v e a p r o b a b i l i s t i c i n t e r p r e t a t i o n t o t h e f u n c t i o n p ( x , t l , S , t 2 ) introduced i n Section 6.5, t h e k e r n e l of t h e Green's o p e r a t o r associ a t e d with A ( t ) .
We assume t h a t t h e conditions f o r a p p l i c a t i o n of Theorems 7.4
(SEC. 8 )
MARKOV PROCESS
173
and 6.17 a r e s a t i s f i e d .
= 0.
I n p a r t i c u l a r , we assume a (Cauchy) problem
- - dU + d t
We consider t h e s o l u t i o n
of t h e
= 0
A(t)u
.
u(x,T) = u(x)
From formula (7.67) we have (8.2)
u
.
u ( x , t ) = E E(Yxt(T))
But, from t h e d e f i n i t i o n of t h e Green‘s o p e r a t o r we have u ( t ) = G(t,T); and t h e r e f o r e , s i n c e
p
i s t h e k e r n e l o f G , we have
(8.3)
u satisfying
Relation ( 8 . 3 ) holds f o r a l l
u cb(Rn).
By d e n s i t y , ( 8 . 3 ) i s ;rue f o r
(7.64) and hence i n p a r t i c u l a r f o r E
C o ( R n ) bounded.
6.17, p ( x , t ; S , T ) i s a p r o b a b i l i t y d e n s i t y on Rn.
Now from Theorem
The e q u a l i t y (8.3) t h u s s t a t e s
t h a t t h e random v a r i a b l e yxt(T) ( w i t h v a l u e s i n R n ) possesses a p r o b a b i l i t y d e n s i t y given by p .
(8.4)
Hence i f
P{yXt(T)
E
B
=
s
i s a Bore1 subset of R n , we have p(x,t;C,T)dS = P(x,t;B,T)
.
From t h e p r o p e r t i e s of t h e Green’s o p e r a t o r , we have, i f t < s < T: G(t,T) = G(t,s)G(s,T)
and hence i f
‘p E
&(Rn)
it then follows t h a t
Now t h e f u n c t i o n p ( x , t ; n , s ) p ( n , s ; S , T ) ~ ( S ) i s i n t e g r a b l e over Rn
n
x
Rn (by
5
v i r t u e , f o r example, of t h e upper bound ( 6 . 1 2 4 ) ) . By a p p l i c a t i o n of F u b i n i ’ s theorem, we can change t h e order of i n t e g r a t i o n on t h e right-hand s i d e of ( 8 . 5 ) . It then follows, s i n c e 9 i s a r b i t r a r y , t h a t
(8.6)
p(x,t;S,T) =
5
Rn P (x ,t ;11, SIP (n, 8 ;C T) drl
V t < s < T , x , S f R ”
.
Equation ( 8 . 6 ) i s c a l l e d t h e Chapman-Kolomogorov equation. The f u n c t i o n P(x,t;B,T) d e f i n e d i n ( 8 . 4 ) i s , using t h e terminology introduced i n DYNKIN Ell, Chapter 4 , p.96, a probability transition function ( t h i s terminology w i l l be j u s t i f i e d i n t h e following s e c t i o n ) . It can e a s i l y be shown t h a t t h e
174
STOCHASTIC D . E . ' s
& P.D.E.'s
OF ORDER 2
(CHAP. 2 )
following p r o p e r t i e s a r e s a t i s f i e d : (8.7)
P ( x , t ; B , T ) , a s a f u n c t i o n of B , i s a p r o b a b i l i t y measure on R n ,
(8.8)
x + P(x,t;B,T) i s a measurable f u n c t i o n on Rn,
(8.9)
P(x,t;B-{x},t)
(8.1 0 )
P(x,t;B,T) =
= 0,
P(x,t;dq,s)P(q,s;B,T)
.
( n o t e t h a t P(x,t;dq,s) = p(x,t;q,s)dq)
if t 5 s S T
We s h a l l now e s t a b l i s h a very important supplementary property of P. by y ( t ) t h e process which i s t h e s o l u t i o n of
The asswnptions are those of Theorems 6.17 and 7.4. THEOREM 8.1. be two stopping times such that 05 T < T c T 1 2-
a s s . , r 2 is
3
Let 'p E S(Rn) and l e t u ( x , t l , t 2 ) ,
tl
(8.1 2)
We denote
Let
T~
'
T2
" measurable;
then we have for any Bore1 set B
Proof.
S
t 2 , be a s o l u t i o n of
(8.1 4)
We have
(8.1 5) The f u n c t i o n
u
i s continuous on Rn
x
I0 i t
i
t2 5 T I and C 2 ' l on
Rn x l 0 , t [ (for f i x e d t 2 ) , From I t o ' s formula, we have 2 (8.1 6)
I t i s t h e r e f o r e permissible tl i t 2 S T. Since r 2 i s $1 Furthermore, i s bounded. ax
The e q u a l i t y (8.16) holds a.s. t o t a k e tl = r1, t 2 = r2.
V 0
5
au
measurable, t h e process
(*)
y(s) d i f f e r s from y x t ( s ) through t h e i n i t i a l c o n d i t i o n s .
(SEC. 8)
1 75
MARKOV PROCESS
i s adapted. The c o n d i t i o n a l expectation of t h e s t o c h a s t i c i n t e g r a l with r e s p e c t t o 3 " zero, so t h a t (8.1 7)
E[(P(Y(T~))
(df] = ~ ( Y ( T l ) s ~ l ; T 2 )
Let r- be a 8"measurable i n t e r p r e t e d i n t h e form
(8.1 8)
Eq (p(Y(7,))
and bounded R.V.
=E
k
qp(Y(rl 1
By r e g u l a r i s a t i o n , (8.18) i s v a l i d f o r Let 9
E
Cp
Co(Rn),
is
The e q u a l i t y (8.17) can a l s o be
s
; E s~ T . J ~c P ( Z ) d t
9 = xB(E;), hence (8.13).
bounded; consider t h e q u a n t i t y
We n o t e t h e following property (which i s an immediate consequence of t h e c o n t i n u i t y p r o p e r t i e s of t h e f u n c t i o n p , s e e Theorem 6.16)
Remark 8 . 1 .
Processes stopped on e x i t from an open domain.
I n s t e a d of s t a r t i n g from t h e Cauchy problem (8.1),we can consider t h e D i r i c h l e t problem i n an open domain ( c f . ( 7 . 4 7 ) )
(8.20)
where
'p
E
B(b)
.
I n view of formula (7.57) ( n o t e t h a t a. (8.21
u ( x , t ) = E 'p(Yx,(T))
XTgT
= E 'p(yXt(T
A 7))
= 0 ) we have for x
+ E 'p(Yxt(T))
I t i s convenient t o introduce t h e process yx,(s)
on e x i t from 6") defined by (8.22) which g i v e s
fxt(S)
= yXt(s A
Txt)
s
s 2t
s
€5, t
E
[O,Tl:
XT< T
( c a l l e d t h e "process stopped
176
STOCHASTIC D . E . ' s
& P.D.E.'s
OF ORDER 2
(CHAP. 2)
It can be shown t h a t P(x,t;B,T) i s a l s o a p r o b a b i l i t y t r a n s i t i o n f u n c t i o n . s h a l l now simply prove t h e arialogue of Theorem 8.1. Let
y
We
be defined by (8.11);l e t T be t h e corresponding e x i t time and l e t
f(s) =
Y ( s A T)
BY considering u ( x , t l , t 2 ) ,
u(x,t,;t2)
=
a s o l u t i o n of
/a
cp(S)d%,tl;dS,t2)
and by reasoning as i n t h e proof of Theorem 8.1, we s e e t h a t we have, i f 0 5 TI
5 T~ 5
(8.26) 8.2
T,
%2 $l-measurable: P(~(T~ E )B 1 6 1 ) = P ( ~ ( T ~ ) , T ~ ; B , T ~ )
m
Some concepts r e l a t i n g t o general Markov processes
Let E be a t o p o l o g i c a l space, and l e t 8, be t h e Borel u -algebra on E ( E w i l l be e i t h e r Rn or b i n t h e examples). We u s e t h e terminology probability transition f u n c t i o n t o denote a f u n c t i o n P(x,t;B,T),
x E E, B E
5,
0
5
t
5
T
such t h a t IP(x,t;B,T) as a function of B , i s a p r o b a b i l i t y measure on E ,
iE
if t 5 s 5 T. P(x,t;dg,s)P(g,s;B,T) t An E-valued adapted ;F C Q Let (O,Q,P) be a p r o b a b i l i t y space, 3it f process y ( t ) i s a Markov process, i f t h e r e e x i s t s a p r o b a b i l i t y t r a n s i t i o n f u n c t i o n P(x,t;B,T) such t h a t IP(x,t;B,T) =
J-
,
.
(SEC. 8 )
MAFiKOV PROCESS
177
t
Since y(t) is 3 -measurable, it follows from (8.28) that we also have
P(y(T) C B l y ( t ) ) = P ( Y ( t ) , t ; B , T )
(8.29)
which can be rewritten in an intuitive manner ( * ) in the form
The manner in which ( 8 . 3 0 ) is written justifies the terminology "probability t r a n s i t i o n function" (probability that y(T) E B, knowing that y(t) = x). Formula (8.30)shows that there can be only a single probability transition function associated with a Markov process. On the other hand, several Markov processes can have the same transition function.
A Markov process is called a strong Markov process on COXTI, if, given two ~ that 0 5 T~ 5 r2 5 T and T~ is 1'3 measurable, we have stopping times T ~ , Tsuch (8.31)
P(Y(T,) f B13'')
= P(Y(~~),~~;B,T~)
which is clearly a generalisatioq of ( 8 . 2 9 ) .
A Markov process is said to be a Feller process if its transition function satisfies the property (8.32)
I
if x -. y
,t
s and
'p
f Co(E),
'p
bounded.
The arguments developed in the previous section lead to the following theorem: THEOREM 8.2. Under the asswnptions of Theorem 8.1, the process' y , the solution o f (Bell),i s a strong Markov process and a Feller process.
Remark 8 . 2 . It is clear (from (8.26))thatf defined in Remark 8 . 1 is a strong Markov process. It is slightly more difficult to see that it is also a Fellgrprocess. Let cp f s(0) ; then since the solution u(x,t) of ( 8 . 2 0 ) is in C (Q), it is clear that we have
If now
(*)
'p f
C"(B), 3
'pn C
s(6) such that
This is rigorous if {y(t) = x) is not of measure zero.
178
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
(CHAP. 2)
and therefore
.
uniformly with respect to x,t, from which it readily follows that (8.33) is still true for 'p E @(a)
A transition function is said to be stationary if
(8.34)
P(x,t;B,T)
I
P(x,T-t;B)
.
Let B(E) be the (Banach) space of functions which are measurable, scalar and bounded on E. The function P(x,t,B) defines a semi-group on B(E) by means of the formula
It is clear that T(t) is a contraction semi-group.
of T(t) is defined by
(8.36)
ff
3x1
=lim hi 0
The infinitesimal generator
u
T (h) (x ) -E( x ) h
The domain D(A) of A is the set of functions h exists unifonnZy with respect to X .
E
B(E) for which the limit (8.36)
Let us take as an example the transition function P defined by (8.4). More precisely, in order to have a stationary transition function, we assume that A(t) = A, independent of time (cf. (8.1)). Let uh be the solution of
We then have
- f:
A %(x,t)dt
( * ) yx(t) is the solution of a stochastic differential equation o f which the
coefficients are independent of time.
(SEC. 8 )
MARKOV PROCESS
from which we can e a s i l y deduce t h a t
(8.39)
a
ffii=-Aiivc€J(Rn).
The property (8.39) can be g e n e r a l i s e d t o t h e nonstationary case. the l i m i t (8:40)
Q (t)ii(x) = l i m
--I:
c(c)p(x,t;ac,t+h)
J2
ht 0
We consider
- ii(x)] .
It can be shown, a s above, t h a t we have
ff ( t ) G
(8.41
=
- A(t)c v c
E &Rn)
We now introduce t h e following concept: i f we have (8.42)
3 6 > 0
'I
such t h a t lim ?;
h s
There e x i s t functions hi(x,t)and
(8.43)
lim h h s
[
. we say t h a t P(x,t;B,T) i s a diffusion
/ X - Y / ~ + ~P(x,t,dy,t+h)
! . i i j ( x , t ) , i , j = 1,
(Yi-Ti)P(X,t,dy,t+h)
= hi(x,t),
...
= 0
,n
i = 1 .a
.
such t h a t
,
We have t h e following theorem: THEOREM 8.3.
with
If P(x,t;B,T) is a diffusion, then P satisfies equation (8.411,
(8.45)
Proof. (8.46)
and
Let xo, to be f i x e d .
T a y l o r ' s formula a p p l i e d t o
< gives
180
STOCHASTIC D.E.'s
& P.D.E.'s
OF ORDER 2
(CHAP. 2)
We t h u s have
(8.47)
L[ 2
so t h a t , proceeding t o t h e l i m i t , t h e r e s u l t then follows. We know t h a t t h e t r a n s i t i o n f u n c t i o n P defined by (8.4) s a t i s f i e s equation We s h a l l now show t h a t , i n a d d i t i o n , it i s a d i f f u s i o n .
(8.41).
THEOREM 8.4. Under t h e assumptions of Theorems 6.13 and 7.4, t h e t r a n s i t i o n function defined by (8.4)i s a d i f f u s i o n with
Proof. We content ourselves with proving ( 8 . 4 4 ) ( a c c e p t i n g ( 8 . 4 3 ) , which i s proved i n similar f a s h i o n ) . We put g(x,t)
1
. I
-g- SEl, E l
.
p(x,t;C,t+h)dE
We t h e r e f o r e have t o prove t h a t
(8.49)
g(x,t)
x x -7 k l 2akl(X,t)
Furthermore, f o r s
5
t+h, \(x,s)
+
Xk
gl(xst)
'+ X l
Bk(x,t)
d X , t )
-
i s a s o l u t i o n of
(8.50)
Let 0 s 0s 1 and s = t + Bh. Wh(X, 0)
=
We put
x x
UJX,")
k.8 -h ;
x x
(*I+
!) H1(Rn) but belongs t o Hi(En); (8.50) follows ( f o r example) from 2 a p p l i c a t i o n of Theorem 7.4, o r can be shown d i r e c t l y by working with LT i n s t e a d of
L2, and by applying t h e technique described i n Section 6.5.
(SEC. 8 )
181
MARKOV PROCESS
then (8.48) i s equivalent t o
(8.51) But wh(x,9) is a s o l u t i o n of t h e Cauchy problem dW
--d: + M(t+8h)wh = v(x,t+8h) (8.52)
Since t h e c o e f f i c i e n t s o f t h e o p e r a t o r A a r e bounded, we can u s e t h e techniques We multiply (8.52) by m: wh and i n t e g r a t e with r e s p e c t t o X. o f Section 6.3.2. This g i v e s d 2 - z l w h ( e ) I + h a(t+8h,wh,wh) = (q(t+8h),wh) and hence
(8.53)
We then deduce ( a t l e a s t f o r a subsequence) t h a t W
h
"
-
hwhwh(0)
weakly star i n
Lm( 0 , l ;Hp)
0
weakly i n
L2( 0,1 ;vp)
w(o)
weakly i n
W
Hll
Proceeding t o t h e l i m i t , ( i n t h e d i s t r i b u t i o n a l s e n s e ) i n ( 8 . 5 2 ) , we o b t a i n
and hence
w(x,o) = + , t )
s o t h a t we have proved (8.52) i n t h e sense o f weak convergence i n H !J have
(8.54)
wh(x,o) P w h -
Indeed, we have
-
cp(x,t)
strongly i n H
u
2
s t r o n g l y i n L (0,l;V ) . 1!
.
I n f a c t we
(CHAP. 2 )
STOCHASTIC D.E.’s & P.D.E.‘s OF ORDER 2
182
which t o g e t h e r with t h e weak convergence f i n a l l y gives ( 8 . 5 4 ) . I n order t o have a p o i n t convergence r e s u l t , we r e q u i r e supplementary estimates. P u t t i n g zh = wh - w, we see t h a t zh s a t i s f i e s
b - b ~ + ~ hh A(t+bh)wh = cp(x,t+@h)
(8.55) Zh(X’t)
t
We m u l t i p l y (8.55) by m; account of t h e f a c t t h a t
0
- cp(x,t)
P
cph(x,8)
.
IzhlP-2zh and i n t e g r a t e with r e s p e c t t o x .
Taking
lzhlmP s by v i r t u e of t h e l i n e a r growth with r e s p e c t t o x of t h e f u n c t i o n cp
, we
have
and t h e r e f o r e from (8.54)
h
Since
jRnA(t+Bh)w hmpIz P h
I
2 0 i n L (O,1;WoypYu)
‘h
IPdx]dB
4
0
.
we can r e a - i l y deduce from (8.55) t h a t
wh(x,o) + w(x,o) s t r o n g l y i n w ~ ~ P ~ ! ’ .
(8.56)
We next d i f f e r e n t i a t e (8.51) with r e s p e c t t o x . , j = 1, permissible i n view of t h e r e g u l a r i t y of t h e c o e f g i c i e n t s .
Xh
=
--.
this is We o b t a i n f o r
h bW
bX
j
a problem of t h e same type as f o r wh.
xh
... n;
2 (x,t)
By t h e same type of reasoning we o b t a i n
strongly i n
P,P,P
.
j
Hence f i n a l l y we have
We t h e r e f o r e c l e a r l y have ( 8 . 5 1 ) a t any p o i n t x , which t h u s provides t h e r e s u l t . 8
(SEC. 8)
MARKOV PROCESS
183
Remark 8 . 3 . There exist transition functions which are not diffusions or which do not satisfy (8.41). This is the case with discontinuous Markov processes such as the Poisson process. We shall have occasion to use such processes in Volume 2. The general structure of the infinitesimal generators of Markov semi-groups has been investigated in particular by BONY-COURREGE-PRIOURET C11 and VENTZEL Ell. Remark 8.4. The existence and uniqueness of a diffusion corresponding to 'general' coefficients A.(x,t) and p..(x,t) has been investigated by STROOCK1J
VARADHAN El1 in the form of the martingale problem.
8.3
A generalisation of Ito's formula
Ito's formula is applicable to functions belonging to C2'l (cf. Theorem 2.4). We shall now extend this result (or more precisely the version obtained after proceeding to the mathematical expectation) to continuous functions belonging to
We shall first prove the following lemma:
LEMMA 8.1. Let y(x,t) E L2 ( o , T ; < ~ ~ ( R ~ ) ) . Let ylt) be u solution of (8.11) and l e t T~ be the e x i t time from 6,.Let o I @ 6 8' s T be two stopping t h e 8 and f o r h > 0 , elh = Min(e*,O+h) We then have
.
(8.57)
the constants being independent of R. The meaning of the expectations in (8.57) and (8.58)is as follows: Proof. since the function Y is in fact an equivalence class we can take a representative of the class, Hence Y(x,s) is defined at any point x, s . We can consider for any s Y(y(s),s) which is measurable as a function of s and w (by composition); then (8.57) and (8.58)hold, which implies in particular that the values on the lefthand sides do not depend on the choice of the representative of Y. We put
$
=
0,x [O,T]. Y E
It will suffice to prove (8.57)and (8.58) f o r
b ( $ ) and
We define Y = 0 outside
$.
Y
2 0.
We have
184
(CHAP. 2 )
STOCKASTIC D.E.'s & P.D.E.'s OF ORDER 2
Also
But t h e n
or
(8.59)
E
Xs E j E
ds
jbR
Y(E,s)p(y~o),o;E,s)dC
.
From t h e Cauchy-Schwartz i n e q u a l i t y and t h e e s t i m a t e ( 6 . 1 3 1 ) i t t h e n follows t h a t
-1 /2
To prove ( 8 . 5 8 ) we use Htjlder's formula with parameter p .
We o b t a i n
(SEC. 8)
185
MARKOV PROCESS
a
1 and therefore But for p >(n+l) we have 2
converges when h When h + 0, 6;
+ +
0
.
7 1 and the integral (8.60)
0 ) of t h e operator Ao:
(1.1 in
0)
6 ,f o r
t h e D i r i c h l e t boundary condition on
r.
We a l s o n o t e t h a t ( i n g e n e r a l ) (1.6) does not imply ( 1 . 5 ) ; r e f e r t o case (1.6) a s t h e "non-coercive case". Notat i o n : We p u t :
(1.1
1)
(1.12)
V
t
K=
1
Ho(0) {v]v
E
v ,v
s
I
a.e. in
We s h a l l always assume t h a t
For example if JI
E
H1(S), (1.13) holds i f
The v a r i a t i o n a l i n e q u a l i t y problem We seek a function u
where
E
K such t h a t
93
f o r t h i s reason we
(SEC. 1)
STATIONARY V . I . ' s
lo
(f,v) =
(1 .I 6)
f v dx, w i t h f given i n L 2 (6)( * ) .
Problem ( 1 . 1 5 ) i s what i s c a l l e d a stationary variational i n e q u a l i t y ( s t a t i o n a r y v.I.).
Remark 1 . 3 . I f i n t h e above we r e p l a c e s t u d i e d i n Chapter 2 .
K by V , we o b t a i n t h e Dirichlet problem
Remark 1.4. I n t h e very s p e c i a l c a s e (which i s not of g r e a t i n t e r e s t i n t h e p r e s e n t c o n t e x t ) i n which e(u,v)
I
v
a(v,u)
u,v €
v
problem (1.15) i s e q u i v a l e n t t o seeking
-
1
inf - a ( v , v ) 2
(1 .17)
(f,v), v E K
.
*
Synopsis I n Chapter 2 we obtained a p r o b a b i l i s t i c r e p r e s e n t a t i o n of t h e s o l u t i o n of t h e D i r i c h l e t problem. One of t h e o b j e c t i v e s of t h e p r e s e n t c h a p t e r i s t o o b t a i n a p r o b a bilistic representation - through t h e intermediary of a st o c h a st i c control problem of "the" s o l u t i o n of ( 1 . 1 5 ) ( * * ) . I n t h i s s e c t i o n , we s h a l l s t a r t by i n v e s t i g a t i n g ( 1 . 1 5 ) independen'tly of any concepts r e l a t i n g t o c o n t r o l .
-
Other v a r i a t i o n a l i n e q u a l i t y f o r m u l a t i o n s we s t a r t We can g i v e two o t h e r f o r m u l a t i o n s of problem (1.15), w i t h u E K ; with t h e 'strong' formulation. We observe t h a t (1.15) i s e q u i v a l e n t t o
(Au-f, v-u) 2 0 V v E K
(1.18)
,
where t h e b r a c k e t s denote t h e i n n e r product between V ' and V;
,
v=u-cp
cp,o
,
cpCa(0)
taking
#
we t h e n deduce t h a t
(Au-f,cp) S 0 V cp2 0 and hence t h a t
Au-f
(1.19)
0 i n 9.
The s t r o n g formulation i s t h e n
(1.20)
(*)
Au-f
0
,
u E Hi(@)
.
u-JI
0
,
(--L-f)(u-+)
More g e n e r a l l y , we could t a k e f i n Ii-l(S) = V'
., 1J
( * * ) A t l e a s t when t h e a .
a i , a.
.
in 6
.
are sufficiently regular.
OPTIMAL STOPPING PROBLEMS & V . I . ' s
192
(CHAP. 3)
This assumes t h a t a regularity r e s u l t of t h e t y p e (1.21
1
holds.
u
H2(S)
E
To prove ( 1 . 2 0 ) we t a k e v = $ i n (1.18) which i s permissible i f JI
i f (1.21) holds, from which we deduce t h a t (Au-f, $-u) P 0 .
E
L
2
(8),
Since Au-f S 0 , JI-u
2 0,
we a l s o nave (Au-f, $-u) S 0 and hence (Au-f, $-u) = 0 which gives (Au-f) ($-u) = 0 a.e. and t h u s ( 1 . 2 0 ) .
The 'weak' f o m Z a t i o n i s not u s e f u l h e r e , but t h i s t y p e of remark w i l l play an important r o l e i n t h e following s e c t i o n f o r evolutionary problems. I f we assume that a(v,v) 2
(1.22) then i f u
E
,
v E K
K and s a t i s f i e s (1.15),we have: a(v,v-u) 2 (f,v-u)
(1.23) CIn f a c t
ov
a(v,v-u)
- (f,v-u)
v
v E K
= a(u,v-u)
-
(f,v-u) + a(v-u,v-u)]
.
Conversely i f u E K and s a t i s f i e s ( 1 . 2 3 ) t h e n , i f (1.22) holds, we have (1.15); w i s taken a r b i t r a r i l y i n K , then i n ( 1 . 2 3 ) we can t a k e v = ( i - e ) w + BU , e E [ o , i ] ;
in fact
r e l a t i o n (1.23) gives:(1-8)a((I-e)w + BU,w-u) 5 (i-e)(f,w-u); then deduce a ( ( l - e ) w + eu,w u) 2 (f,w-u)
assuming
e
< 1, we
-
and making
1.2
-f
1 we then deduce (1.15).
Existence and uniqueness theorem.
Coercive case
We s h a l l now prove t h e following theorem: THEOREM 1.1. Suppose t h a t ( l . l + ) , (1.5) hold i n addition t o ( 1 . 1 2 ) , There then e x i s t s a unique u E K, t h e solution of t h e V . I . (1.15).
(1.13).
Proof of uniqueness. Let ul, u2 be two p o s s i b l e s o l u t i o n s . We t a k e v = u2 ( r e s p . v = u,) by adding we o b t a i n
( 1 . 1 5 ) r e l a t i n g t o u1 ( r e s p . u 2 ) ;
- a(ul-up~u1-u2) 2
o
i n the V.I.
(SEC. 1)
193
STATIOXARY V . I . ' s
which g i v e s t h e r e s u l t t h a t u1 = u2.
a
We s h a l l g i v e i n S e c t i o n s 1.3 and 1 . 4 one e x i s t e n c e proof based on p e n a l i s a t i o n . For o t h e r p r o o f s , s e e LIONS-STAMPACCHIA [l] and t h e book by LIONS [ 2 1 where f u r t h e r b i b l i o g r a p h i c r e f e r e n c e s a r e given.
1.3
Penalisation
We c o n s i d e r , f o r
E
> 0 , t h e f o l l o w i n g problem:
1
a(uC,v) +-$(u,-
(1.24)
+)+,TI)
= (f,v) v
find u
E
e
c V , t h e s o l u t i o n of
v
This i s t h e p e n a l i s e d problem a s s o c i a t e d w i t h t h e V . I .
(1.15).
Remark 1.5. We s h a l l g i v e i n S e c t i o n 3 a p r o b a b i l i s t i c r e p r e s e n t a t i o n , u s i n g optimal c o n t r o l t h e o r y , of t h e s o l u t i o n uc of ( 1 . 2 4 ) . a Under the assumptions of Theorem 1.1, problem ( 1 . 2 4 ) admits a
THEOREM 1 . 2 .
unique solution. Proof. (1.25)
Uniqueness.
We p u t :
p(v) = -(TI-+)+ 1
(penalisation operator)
C
Let u and u' be s o l u t i o n s of ( 1 . 2 4 ) .
We t h e n deduce t h a t
But t h e o p e r a t o r p i s monotone i n t h e sense t h a t
We t h e n deduce from ( 1 . 2 6 ) , ( 1 . 2 7 ) t h a t a ( u - u ' , u-u') 5 0 , so t h a t u = u ' . We u s e t h e Gazerkin method or, more g e n e r a l l y , an i n t e r i o r approxExistence. we i n t r o d u c e a family V, of subspaces imation method of t h e " f i n i t e element" t y p e ;
of V with t h e p r o p e r t i e s : (1.28)
V i s finite-dimensional; m
(1.29)
V
(1.30)
TI
E V
,
t h e r e e x i s t s vm
IITI~-VII -. o
if m
there exists
v
Such a family e x i s t s ;
-L
€ V,
m
n
E
Vm such t h a t
;
K
vm
.
it i s s u f f i c i e n t f o r example t o choose vo
E
V
fl
K
194
OPTIMAL STOPPING PROBLEMS & V.I.’s
(assumed n o n a p t y ) and t o t a k e a sequence
vl,
yo,
of t h e
..., vk
Y.
yo,
( C W . 3)
..., vk, .... i n V
v1
such t h a t V k ,
a r e linearly independent and such t h a t t h e f i n i t e l i n e a r combinations we then t a k e for exrmrpze
a r e dense i n V ;
V, = space generated
by [vo, ...Y,-~].
f i n d um
We then consider “ t h e approximate problem“:
E
V,
a solution of
Such a u e x i s t s , i n view of a v a r i a n t of Brouwer’s theorem ( s e e LIONS [21, Lemma 4,3, b a p t e r 1). A p r i o r i estimates
We t a k e i n ( 1 . 3 1 ) v =
%-
a(um ,um-v o
(1.32)
+
we n o t e t h a t B(vo) = 0 so t h a t we have
yo;
(~(u,)
- ~ ( v ~ ) , u ~ - v ,=) (f,Um-vo)
t a k i n g account of (1.27),we deduce from (1.32) t h a t
a(um-v
u -v ) I m o
0’
(f,um-v ) 0
- a(v
u v )
0’
m- o
so t h a t allum-voll2 I cIlum-v0ll and t h e r e f o r e
We can t h e n e x t r a c t a sequence denoted by um such t h a t
(1.34)
um
+
f i E weakly i n
V, when m
We then deduce from (1.32) t h a t
hence
(*)
C depends n e i t h e r on
m
nor on E.
+ a.
;
(SEC. 1)
195
STATIONARY V.I.'s
We t h u s have ( 1 .36)
Furthermore,
(1.37)
If v (1.31);
-
(urn-+)+ E
2
B(um) i s bounded i n L (8) ( E f i x e d ) and we can t h e r f o r e assume t h a t
x
weakly i n L
V, a r b i t r a r y , we choose v we have
E
2
(6) I
we t a k e v = vm i n
V, s a t i s f y i n g ( 1 . 2 9 ) ;
We w i l l t h e r e f o r e have proved t h e theorem i f we can show t h a t
I n t h e p r e s e n t c a s e , we can g i v e two proofs of ( 1 . 4 0 ) . 2
Proof of ( 1 . 4 0 ) b y compactness. A s 6 i s bounded, t h e i n j e c t i o n from V + L (&) is compact, so t h a t u -L aE s t r o n g l y i n L2(@ and s i n c e J, J,' i s continuous -f
- (U",-J;)+
s t r o n g l y i n L ~ ( s ) we Eave:
(urn-+)+
strongly i n L
2
(s),
which t h u s g i v e s ( 1 . 4 0 ) .
We i n t r o d u c e
Proof of (1.40) by monotonicity.
xm
= a(u m-v m ,um-v m 1
+ (~(u,)
- B(v,),u~-v,)
vm s a t i s f y i n g ( 1 . 2 9 ) . We have:
xm
2 0.
Moreover, u s i n g (1.31) we have:
X, = ( f , um-v m )
- e(v m ,um-vm )
from which we deduce t h a t
(*)
C depends n e i t h e r on
m nor on
8.
- (p(vm),um-vm)
,
196
OPTIMAL STOPPING PROBLEMS & V . I . ' s
But u s i n g
(CHAP. 3 )
(1.39),we t h e n deduce t h a t :
- A?.
Taking v =
A
2
X > 0 , and
with a r b i t r a r y
1
+ A(~x-B(G~-h'p),'p)2
a(?,?)
V , we t h e n deduce t h a t
'p E
.
0
Dividing by h and t h e n making A + 0 , we t h e r e f o r e o b t a i n : 1
(y x-B(+)
(1.43) so t h a t
1
x
=
2 0
p(EE) ,
v
'p
E
i.e. (1.40)
.
m
Remark 1.6. The proof based on monotonicity c o n s t i t u t e s an a d a p t a t i o n t o t h e p r e s e n t c a s e of a method due t o G . MINTY L11.
1.4
Proof of e x i s t e n c e i n Theorem 1.1
We deduce from ( 1 . 3 3 ) and
~ I ~s ~c I, I
(1.44)
(1.36), t h a t (uE-
+I+
-
o
in
.
~ ~ ( 0 )
We can t h e r e f o r e e x t r a c t a sequence, also denoted by uE, such t h a t , when we have:
which g i v e s
-
u
(1.45)
-
I
u weakly i n
s o t h a t (ii-q)'
= 0.
-
(&)+
+ 0,
V,
e i t h e r by compactness, o r by monotonicity
(uE- $)+
E
weakly i n
L2((a)
-
(actually, strongly)
Hence
< € K , and i f we t a k e v
E
K , t h e n B(v) = 0 and we deduce from ( 1 . 2 4 ) t h a t :
a(uEJv-UE)
-
(f?v-uE) = (B(v)-B(u,),v-uE)
20
so that 8(uE,v) and hence
a(u",v)
- I f J v - u E )2 a ( u E J u E ) -
(f,v-u") 2 lim. i n f . a(uE,u ) E
so t h a t 6 i s a s o l u t i o n of (1,15),g i v i n g E = u;
z
- 6 "
a(u,u)
we t h u s have proved Theorem 1.1. 8
Remark 1.7. We have a l s o proved t h a t t h e s o l u t i o n u o f t h e generalised problem converges weakly in H1(8) t o t h e s o l u t i o n u of t h e V.14 ( 1 . 1 5 ) . We s h a l l now o b t a i n an e s t i m a t e of 1 ) ug -
uII .
STATIONARY V.I.'s
(SEC. 1)
1.5
197
Estimate of the 'penalisation error'
THEOREM 1.3. that
The assumptions are those o f Theorem 1.1. Suppose i n addition
We then have, if u, (resp. u) denotes the soZution of (1.24) (resp. (1.15)):
(1.48)
a(v,v) 2
2)
lie in a bounded subset of
aij, ai, a
CChl12
,
with the same
Writing
u-u
= u-+-(u,-+)
rE =
U-J,
+
(uc-+)-
= r,-(u,-+)+
a
>
0
L'(0)
with
.
,
,
we see that the problem amounts to showing that (*)
It is sufficient to have :
A$
is a measure and
(A$)-
E
L2(s)
.
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
198
1&.1 We t a k e v = r V-u
5
c
*
i n (1.24), and i n (1.15)we t a k e
= r& i . e . v
+-(ue-+)-s
+
v
(hence
v
defined by
c K
v = o
since
on
r)
.
Adding, we o b t a i n -a(u-uE,rL)
+ (u&-+)-)
+ -$(u&-+)+,u-+ 1
o
2
or
so t h a t t h e r e s u l t then follows from (1.50).
Remark 1.9.
w, a s o l u t i o n of
When JI s a t i s f i e s
2 (?,v-w) V T S
a(w,v-w)
(1.51)
(1.46), t h e
w 5 0
on
o
V.I.
on
(1.15) i s equivalent t o seeking
o,
v =
0 , f " = f - ~ + ,w = - +
4
on
r ,
o n r .
I n f a c t , it i s s u f f i c i e n t t o put w = u - $.
1.6
Monotonicity p r o p e r t i e s of t h e s o l u t i o n
THEOREM
1.4.
f,+ .-. u
u ( f , + > = solution of (1.15)
I
i s ( a l l other things being equal) an increasing function of uords, i f f,$ s a t i s f y
?rf , $ and i f
The function
The assumptions are those of Theorem 1.1.
= u(?,$)
f
and of
$;
i n other
2 a~ .e.
then
;2 u
a.e.
We t a k e i n (1.15)v-u = - (a-u)Proof. We s h a l l now show t h a t (U-u)- = 0. i . e . v = i n f . (u-a) 5 $, and i n t h e V.I. f o r U, we t a k e v defined by
STATIONARY V . I . ' s
(SEC. 1)
v
-
i; = -(6 - u)-
i.e.
a(&,
i.e.
v = sup. ( u , ? ~ )5
$.
199
We then deduce t h a t :
(L)-) 2 o z:
a((IkI)y(;-u)-)
0
so t h a t t h e r e s u l t then follows.
Remark 1.10. We a l s o note t h a t i f f 2 0 , Ji L 0 then u L 0. note t h a t i f we take ?={VIVEH'(d)
and i f
a
,vS$,vz:O
i s t h e corresponding s o l u t i o n , t h e n Q
(Take v = sup(u,a) i n t h e V . 1 . r e l a t i v e t o a). 8
8
Furthermore we
ontheboundary}
5 U.
r e l a t i n g t o u, and v = i n f ( u , f i ) i n t h e V.I.
Let X denote t h e s e t of sub-solutions, i . e . t h e s e t of t h e w E HL(S) such t h a t
s (f,cp) v w -JI s 0 a.e. w s 0 on r . a(w,cp)
Then t h e s o l u t i o n (1
.52)
'P E
1
H,(@)
, cp2
0
d ,
in
u of (1.15) s a t i s f i e s
u=sup.w
, W E X
.
Since u E X , it i s s u f f i c i e n t t o show t h a t , i f w E X , we have w S u. I n (1.)5) we t a k e v defined by v-u= (w-uy, which i s permissible, and t h e above 9 = (w-u) We then deduce t h a t
.
a(w-u,
(w-u)')
so
from which t h e r e s u l t follows. The property (3.52) w i l l play a very important r81e i n Section 3.7, i n connection with t h e i n v e s t i g a t i o n of o p e r a t o r s A which a r e not i n divergence form. 8
1.7.
"on-coercive'
case
THEOREM 1 . 5 . Suppose that L)e are i n the %on-coercive" case, i . e . t h a t (1.21, (1.4), (1.6) h o l d . AZso suppose t h a t
Then there e x i s t s one and only one function u
E
K n I,-(@),
the soZution o f (1.15). 8
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
200
Preliminary r e d u c t i o n : I t w i l l not r e s t r i c t t h e g e n e r a l i t y i f we assume t h a t
(1.55)
f> f
> o
8
a.e. in
I n f a c t , from Theorem 5.2, Chapter 2, t h e r e e x i s t s that
HA(o), 'p20,
A'p = c , 'p E
I f then
u i s a s o l u t i o n of ( 1 . 1 5 ) , u+'p = a(;,;-;)
;e
c a constant such t h a t
+
a(u,v-u)
u"
E HI(@)
n L"(0)
c + f 2 fo
s a t i s f i e s (v+'p =
z
(c,;-;)
'p
(f+c,;-u')
v TS $
>
P") f
t h u s have a prob_lem analogous t o t h e previous problem b u t with f+C ,and JI by $ = +q , which gives t h e r e s u l t .
.
0
:
= $t
f=
such
'p
;
replaced by
We s h a l l now prove uniqueness within t h e s e t of p o s i t i v e Proof of uniqueness. indeed i f u1,u2 a r e s o l u t i o n s . If V contains t h e c o n s t a n t s , t h i s i s s u f f i c i e n t ; two s o l u t i o n s , o f which a t l e a s t one i s not 2 0 , a . e . , we introduce ( a s i n Chapter 2 , Section 5 ) : F;
= min {idu,, i d u2)
,
5
o
and we n o t e t h a t
= a(Ui,v-ui)
a(uiq,v 2 .
E
H b ( 6 ) and
u
OPTIMAL STOPPING PROBLEMS & V.I.'s
206
(CHAP. 2 )
We adopt an assumption which i s somewhat s t r o n g e r than t h e c o e r c i v i t y of a ( v , v ) on H~(IY).
(p-l)C a . (x)C.C lj 1 j
(1.78)
+
C ai(x)CiCo
+ ao(X)502
2 0 a . e . i n 8.
We s h a l l now prove t h e following theorem:
Then if
u
is the solution of (1.15)we have:
Au E Lp(0)
(1.81
Proof.
.
We a g a i n s t a r t from t h e p e n a l i s e d equation
(1.82) and t a k e t h e
1
Aut + T ( u ~ - 4 ) '
= f
inner product with ((u
- +)+)"' .
We note t h a t
so t h a t from (1.78) we have
(1 3 3 )
a(q, (Cp+~~-2 l)
and hence
:;lr 1
Thus Au
o
(ut- +)+llLp(a)
.
5
c .
l i e s i n a bounded subset o f L p ( @ ) , from which t h e r e s u l t then follows. 8
STATIONARY V.I.'s
(SEC. 1)
If, under t h e assumptions of Theorem 1 . 9 , t h e c o e f f i c i e n t s a i j then t h e soZution u of (1.15) s a t i s f i e s :
COROLLARY 1 . 2 .
satisfy:
a.. 1J
E
(1.84)
Proof. u
207
C1(6),
.
w2q6)
u f
This i s achieved u s i n g Theorem
5 . 6 , Chapter 2 , w i t h (1.81) and
E Hi(0).
The assumptions are taken t o be those of Theorem 1 . 9 , wi t h
COROLLARY 1.3.
( a s (1.78) h o l d s
p finite).
u E
(1 235)
Proof.
v
We have
c'la(8)v
Then a
(1.84) V p
, ; t h e s o l u t i o n u of ( 1 . 1 5 ) We s h a l l now show t h a t t h i s s p e c i a l r e s u l t ( c o n t i n u i t y of u) can be e s t a b l i s h e d under much l e s s r e s t r i c t i v e assumptions than ( 1 . 7 9 ) on $.
Remark 1.14. i s continuous i n
2.
The f o l l o w i n g theorem p l a y s a fundamental p a r t i n t h i s connection: THEOREM 1.10. The assumptions are taken t o be (1.2), (l.$), ( 1 . 5 ) ( t h e r e g u l a r i t y assumptions a r e t h u s minimal). For f f b e d i n L (O),denote by are given i n L'"(CY), we have: the solution of (1.15). Assuming that $ and
( 1 .86)
Proof.
We denote by u
(resp.
h
) t h e s o l u t i o n o f t h e p e n a l i s e d problem
We w i l l have (1.86) i f we can show t h a t
~($1
208
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
(1.88)
We p u t
and we introduce w = u - C t - k .
w = -k i 0 and t h e r e f o r e
On ,'I
.
HL(0)
w+ E
By t a k i n g t h e i n n e r product o f (1.87) ( r e s p . adding, we t h u s o b t a i n
+
4uE- t t , w
But i f
GE
5
I$
d
then'
i
$,
((u
1+
I
x
- (at-
((us- 4J)'
then
o
- @)+- (ac- 0
.
The weight f u n c t i o n m
v
t h u s plays a
We can a g a i n d i s t i n g u i s h here between t h e "coercive case" and t h e 'Inon-coercive case".
Suppose that a ( u , v ) given by ( 1 . 0 0 ) satisfies:
THEOREM 1.13. (1 .1.06)
2 a ( v , v ) 2 al(vlIp
v
v
c vp , v
=
o
on
r
Then there a i s t s a unique u i n K
.
v such that (1.103)holds.
and (1.106) w i l l hold i f
c 2 ( ~ + p ) 2
0 such that
(1.128)
f(x) B c
Then the solution
u
for
x E 0 , 1x1 s u f f i c i e n t l y l a r g e .
i n K of (1.103) has compact support. !J
Before proving t h i s theorem, which i s due t o H. BREZIS C11, a few remarks should be made:
Remark 1.22. The s o l u t i o n s of e l l i p t i c p a r t i a l d i f f e r e n t i a l equaticns a r e functions "near t o " a n a l y t i c f u n c t i o n s , and which t h e r e f o r e do not i n generaz have compact support - and i n a l l cases, do not have compact support under assumptions of t h e t y p e ( 1 . 1 2 8 ) . There i s i n t h i s r e s p e c t a fundamental d i f f e r e n c e between t h e s o l u t i o n s of equations and i n e q u a l i t i e s . m We g i v e below t h e proof due t o H. BREZIS ( l o c . c i t . i n which a Remark 1.23. number of more general r e s u l t s w i l l a l s o 'oe found). A proof which i s , broadly speaking, j u s t a s t e c h n i c a l l y complicated a s t h e one which follows, but which is perhaps more i n s t r u c t i v e , w i l l be i n d i c a t e d l a t e r when we e s t a b l i s h an i n t e r p r e t m a t i o n of u i n terms of optimal c o n t r o l ( s e e Section 3 below).
Proof of Theorem 1.17.
(i) ws u
We s h a l l c o n s t r u c t a f u n c t i o n
w
such t h a t
STATIONARY V.I.'s
(SEC. 1)
223
( i i ) w has compact s u p p o r t . Since jl = 0 , we have: u 5 0 s o t h a t t h e theorem follows from ( i ) and ( i i ) .
To c o n s t r u c t (1.1 29)
w , we s h a l l seek a f u n c t i o n i n t h e form w ( x ) = S ( r ) , r = 1 x 1 , ( S t o be determined,)
such t h a t
and such t h a t ( i i ) h o l d s .
.
I t then foZZows from (1.117) t h a t we have ( i )
I t t h e r e f o r e now only remains t o c o n s t r u c t w i n accordance w i t h ( 1 . 1 3 0 ) . We choose r such t h a t f ( x ) t E f o r r 2 r We seek S ( r ) i n t h e form
.
(1.131)
-+(r-R)2
,
r o S r IR
,
where y i s a c o n s t a n t > 0 and is f r e e t o be chosen and where R > ro i s a l s o With t h i s s e l e c t i o n o f S , w 5 0 on open t o c h o i c e . down t o proving t h a t :
r
such t h a t t h e problem comes
it i s p o s s i b l e t o choose y and R such t h a t Aw 5 f .
(1.132)
Since t h e l a s t term i s 5 0 , we w i l l have (1.132) a f o r t i o r i i f we can show t h a t
(1..133)
I
y and R can be chosen such t h a t
We s h a l l f i r s t prove t h a t : (1 .I34)
I
y can be chosen s u f f i c i e n t l y s m a l l t h a t , f o r any s u i t a b l e R 2 Ro, we have X 5 f f o r 1x1 t r
I n f a c t i f 1x1 t r
0'
we have f Z
E,
and we have (1.134) i f X 5. E, i . e . i f
OPTIMAL STOPPING PROBLEE & V . I . ' s
224
xi x
---
y X aij
+
E
2
$-R)
a.
--axij i ) --i
y(r-R)E (ai
r
i.e. i f
x
ba
j
- cy(R-r)
-.cy 2 0
If R L R = r + t h e lower bound of a. ;(r-R)2 0 o a Cro,R1 and we have t h g r e q u i r e d i n e q u a l i t y i f (1
a. @r-Rl2
s
E
. -
cy(R-r) i s a t t a i n e d i n
.
2 E - c y - y Z O
.I351
(CHAP. 3 )
2a0
We t h e r e f o r e choose y such t h a t (1.135) h o l d s , R being a r b i t r a r y 2 Ro. remains t o show t h a t we can choose R such t h a t
36)
(1.1
X i f for 0 i
Now i n 0
5
r
5
r
5
It now
.
r
ro, we have:
x=-
2-r (R-r
2y(R-ro)
7' 0
- a.
'ij
'i 'j
r
+
(R-r y-+-(r;
- a.
+r0l2
x
0
(ai
-$I
ba j
xi
- r2)
and we thus have (1.136)i f
- CI(R-ro) - c 2 2 0
-(R-r0)2 aOY 2
which i s p o s s i b l e i f w e choose R s u f f i c i e n t l y l a r g e . 1.13
8
Unbounded open domain, unbounded c o e f f i c i e n t s
We now r e t u r n t o t h e s e t t i n g of Chapter 2, S e c t i o n 5.5.1, with i n Rn ( i n Chapter 2 , Section 5 . 5 . 1 , we had (1.137)
I
We assume t h a t
(1.138)
a (aij -xdxi
0 unbounded
Hence
,
A=Ao+A1+ao
=
8 =R").
d
3
,
aij E ~ " ( 0 )
,
(1.4)holds and t h a t
ao(x)
2 a.
> 0 a.e. in
8.
The functions ai s a t i s f y ( 5 . 9 5 ) , Chapter 2, which we r e c a l l s t a t e s t h a t :
(*)
Assumptions defined below.
225
STATIONARY V.I.'s
(SEC. 1)
39)
(1 .I
We note tha't the lastlcondition i n (1.139) i s unnecessaq i f m(x) = 1. We now wish t o s o l v e , i n a p p r o p r i a t e s e n s e , t h e V . I . : Au
-
f
0, u
5
-
JI
S 0,
(Au-f)(u-$) = 0 i n
8
r.
u = 0 on
The argwnents which have been used u n t i l now no longer apply, one o f t h e b a s i c d i f f i c u l t i e s being t h a t t h e term
J0 (A,u)vq(x) dx
supposes t h a t t h e assumptions on
u and v
, whatever
t h e weight q(x),
a r e asymmetric.
8
We a g a i n i n t r o d u c e , i n modified form, t h e concepts o f Chapter 2 , Section 5.5.1. We put (**) n ( x ) = (1+x2)-',
Lf(0) =
{ V I'Z
s > 0 fixed arbitrarily,
V
L2(0))
9
(f,d)?
-x fg dX
E
t
a l l equipped with t h e obvious H i l b e r t norms. We next introduce
F = ~ v l vE H:(o) For u
E
F,
'p
E
do),we
,v
vm E L ~ ( o,) v = o on
rI
.
put
We have
so t h a t
(*)
See
(5.95), Chapter
2 , f o r a somewhat more g e n e r a l assumption.
( * * ) We could in addition introduce t h e weights m
P
as i n Section 1.11.
OPTIMAL STOPPING PROBLEMS & V.I.'s
226
I
(1 .I421
2
,
E,(V,T) 2 a,llqIllF if
al
>
,v
0
'p
(CHAP. 3)
c a(@)
is sufficiently large.
tLo
We a l s o n o t e t h a t E T T ( u , v )i n g e n e r a l does not have a meaning i f u , v can d e f i n e E-(u,u), f o r u
K = {vlv
.I441
(1
E
E
E
F , b u t we
F , by
F , v s J, a . e . i n 0 )
K being assumed nonempty; we t h e n seek
(1 .I451
u satisfying
IUCK
- En(U,U)
Ex(u,cp)
2
'
v' '
(f,(P-U)x
qI
E 9,
(*I
.
We s h a l l now prove t h e following theorem:
TIBOREM f
W e take
1.18. We suppose that (1.137), j1.'4), (1.138), (1.139)hold.
2 LTI and J, w i t h
E
(1.146)
J,
2
L ~ K, being nonempty.
E
u, t h e solution of ( 1 . 1 4 5 ) .
Then there e x i s t s one and only one function
Proof of uniqueness.
Let Yl be a second p o s s i b l e s o l u t i o n .
We introduce a
function
0
E
B(Rn), 0
9 8 9
1, 0 = 1 i n a neighbourhood of 0 ,
We t h e n t a k e qI = 8 fi i n ( 1 . 1 4 5 ) and qI = BRu i n t h e e($). R r e l a t i n g t o fi ( t h e s e choices a r e permissible); thus V.I. and we p u t eR(x) =
(1 .I 47)
In
-
E ( u , e R ~ )+ E ~ ( u , ~ ~ u E,(u,u) )
cR =
But we s h a l l show t h a t as R
(1.1 48)
En(u,
-.o
(f,eRu-U+eRff-ff)
eRff) +
-L
-
R
--
.
we have
OR")
E,(Q,
if
- ~ ~ ( f f2, 5,~ )
-En(U,U)
- E,(fltfl)
-
-En(u-%u-u)
I f we assume f o r t h e moment t h a t t h i s i s t r u e , we deduce from
Ex(u-ff,u-ff) I
E,(u,u)
2 alllull:
Kc = s e t of t h e v E K w i t h compact support i n
,
B.
,
(1.147) t h a t
0
which, t o g e t h e r w i t h (1.143) and
(*)
t
shows t h a t u-13 = 0 .
STATIONARY V.I.'s
(SEC. 1)
227
To prove (1.148),it i s s u f f i c i e n t t o show ( t h e o t h e r terms being immediate)
that
ba
Y = 1
da i + 7cai+(u-U)2 1 an (5
c
dx
i
.
We t h u s have t o show t h a t i f
zR =
jo[ai n xi(eRii)
E
bU
+ ai
n
aff
i
eRu]dx
then
But
I
Since
a i ( x ) I 2 c ( l + l x l m(x)), we have:
bx
1
I ;~~1
(since
-
-f
Z as R +
m,
from Lebesgue's
We w i l l have t h u s e s t a b l i s h e d t h e r e s u l t i f we can show t h a t
theorem.
ZR2
(1.145 But
and t h e r e f o r e ZR1
S )*1
ax.
+
a eR
0 a . e . and 1-15 ax.
b9
ain
.
0
$u
Is
i,w i t h support
c(l+m(x))
i n 1x1 5 clR.
and
u ff E L1(S)
x
Hence
(1.149)i s a consequence of
m
Lebesgue's theorem.
Proof of existence We s h a l l u s e t h e method of Remark 5 . 5 , Chapter 2 .
(1.1
50)
Fo
e
[vlv
E F
, A,
V
6 L:(o)j
9
We i n t r o d u c e :
228
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
and f o r u,v
E
Fo, we put ( c f . ( 5 . 1 0 9 ) , Chapter 2)
(1.151)
&;(u,v)
= y(A1u,Alv),
,
+ En(u,v)
where
Y
>
0
.
We have: 2
2
v
8 ; h ) 2 Y IA1vI, + allIvIIF
( 1 . 1 52)
c Fo
We put
KO= K n F
( 1 .I531
If v
0
E
'
K has compact support, then v
there e x i s t s u Y ( 1 .I 54) Taking v = v
E
KO, and hence K
i s nonempty.
Hence
KO such that ( * )
E
By(u
n Y E
,v-u
Y
) 1 (f,v-u )
y n
v
P
E K
.
KO, we deduce from ( 1 . 1 5 4 ) t h a t
q u ,u 12 B;(uy,vo) Y
Y
- (f,vo-
Uyln
so t h a t , by using (1.152) we have: ( 1 . 1 55)
IbYll,
c
9
(1.1 56)
( 1 . 1 58)
En(Uy,v)
+ TY(Ty
A,Uy,AIV)
1 En(uy,uy)
.
- (f9v-uy n
and t h e r e f o r e ( * ) We apply LIONS-STAMPACCHIA c11 which uses an a d a p t a t i o n of t h e arguments given a t t h e start o f t h e present s e c t i o n .
(SEC. 1)
STATIONARY V . I . ' s
229
We can a l s o prove e x i s t e n c e f o r Theorem 1.18 by p e n a l i s a t i o n .
Remark 1 . 2 4 .
We s h a l l now give a regularity r e s u l t : THEOREM 1.19.
59)
(1.1
The assumptions are those of Theorem 1.18 and we asswne that
, A16
Q E F
Then the solution ( 1 . 1 60)
u
.
E L :(O)
of ( 1 . 1 4 5 ) satisfies
.
Aou € Lx(0) 2
Proof. For example, using We u s e p e n a l i s a t i o n (a s i n t h e previous Remark). t h e method by which Theorem 1.18 w a s proved, we can s e e t h a t t h e r e e x i s t s a unique uE E F , which i s t h e s o l u t i o n of ( 1 . 1 61)
+
A u
0 9
A , U ~
1 + aOuE+-$u9-
(L)+ =
f
.
Taking t h e inner product of (1.161)with (up - $)+a we o b t a i n 1
@)+) + F
En((ur- +)+,(u,-
I
I(uc-+
+ Ex(+t(U9-+
)+I2
I dx
= (ft(u9- +)f)
)+)
from which we deduce t h a t
We then deduce, p o s s i b l y by e x t r a c t i n g a subsequence, t h a t (1 .I 63)
2
+)+ -. x
+ur-
weakly i n L,(o)
and (1.161) g i v e s :
A u + A 1 ~ + a o u + ~I f 0
.
We then deduce t h e r e s u l t by applying Theorem 5 .6, Chapter 2 ( * )
1.14 Other i n e q u a l i t i e s We now r e t u r n t o t h e s i t u a t i o n of ( 1 . 1 5 ) but with another convex s e t K ; we cons i d e r two functions JI1 and JI given on 6 , and we t a k e 2
(1.1
64)
K=
(PI
v
1
E V = Ho(o)
, +l S v I4 ~ ~ .1
We adopt t h e assumption t h a t
( 1 . 1 65)
K # 0.
I f f o r example JI
(*)
1
E
H'(@),
Result v a l i d on 6 c Rn.
i = 1 , 2 , (1.165) holds i f
OPTIMAL STOPPING PROBLEMS
230
find u
(1 .1 67)
E
(CHAP
V.I.'s
. 3)
K with
a (u,v-u) 2 (f,v-u)
Remark 1 . 2 5 .
&
v
v
K.
E
We s h a l l g i v e l a t e r an i n t e r p r e t a t i o n o f
u, t h e solution of
( 1 . 1 6 7 ) , a s an optimal c o s t f u n c t i o n for a s t o c h a s t i c games problem.
8
Without e n t e r i n g i n t o d e t a i l s , we can s t a t e t h a t a l l t h e above r e s u l t s adapt
t o t h e case of ( 1 . 1 6 7 ) .
To prove t h e e x i s t e n c e of a s o l u t i o n (assuming t h e form a ( u , v ) t o b e c o e r c i v e ) we can u s e t h e penalisation method: we seek u E , t h e s o l u t i o n o f
(1.1 68) Since t h e operator
y
+
E
-+v-!+,)-
i s monotone, we can demonstrate,
e x i s t e n c e of a s o l u t i o n of (1.168).
as i n S e c t i o n 1 . 3 , t h e t h a t uc
1
--(v-$~)+
1 u weakly i n H (&, u being a s o l u t i o n o f ( 1 . 1 6 7 ) .
I t can be shown 8
We s h a l l now s t a t e t h e e x t e n s i o n of Theorem 1 . 3 which g i v e s a n e s t i m a t e o f the "penalisation error". THEOREM 1.20.
(1.1
69)
We suppose t h a t
+i f H 1 b )
and t h a t (1.166) holds. of ( 1 . 1 6 7 ) ) we have: (1 ,170)
Proof.
(1.171)
flu,-
UII
, AGi
E L2(S)
Then i f u
1c
3
E
( r e s p . u) is t h e solution of ( 1 . 1 6 8 ) (resp.
.
We n o t e t h a t ((V-G2)+
(v-@l)-)
D
0
-
We t a k e t h e i n n e r product of (1.168) w i t h ( u E - $,)+; gives:
using (1.171), t h i s
(SEC. 1)
231
STATIONARY V.I.'s
We then introduce rE= uc
( 1 .1.76)
- (uE-
It can be shown that r if it can be shown that (1 .I 77)
E
+
From (1.174)and (1.175),we shall have (1.170)
K.
.
c
\\u-rell I
.
(uc- +1)-
Taking, as is permissible, v = r in (1.167),and taking the inner product of (1.168)with -(r -u), we obtain by adding:
But
so that (1.178)gives a(u,-u
,r
-u) I
a(uE-u
,r
-u)
o
hence
s
a(rE-uE
, r -*L)
and therefore
11r,-u11 I c ]~rE-u,~\ 5 c
YE
from (1.174)and (1.175).
Remark 1 . 2 6 . ' We shall now describe another V.I. which also plays an interLet esting r81e in the theory of optimal control (see Section 3.10 below).
E
c
6 be a set with positive measure and let $
(1 .I79)
K = {v/ v
E
V, v s
E
2
L ( 0 ) ;we take
on E ) ,
K being assumed nonempty. Let u be the solution of the V . I . (1.167)corresponding to the convex set (1.179). We again have existence and uniqueness of the solutioc; a penalisation approximation is now 1 1 ( 1 .I801 AuE + ;(uE-$)+ xE = f, uE E H o ( 6 ) ,
OPTIMAL STOPPING PROBLEMS
232
where
xE
= characteristic function of E.
&
(CHAP. 3 )
V.I.'s
The a priori estimates are totally
analogous to the above. We do not know whether an analogue of (1.170)exists; we have been able to establish this only under the assumption that f 5 0 a.e. on E. We can construct a more convenient equivalent V . I . , when we assume that: E = open subset of 8 , E c e . 'p
Let F = 0 by
E;
we assume that aE is regular and that J,
A? = f in F,
'p
= $I on aE, 'p = 0 on
Y = 'p on
J,
on E.
E
1 H ( S ) . We introduce
r
and we define
We note that u
F,
satisfies
Av = f in F, u
on E and therefore on aE,
r,
u = 0 on so
5 J,
that, from the maximum principle, u
5
'p
on F.
So, if we introduce
IZ
(1.179) A
then u
= {vl v
E
V, v i Y one},
is also a solution of the equivalent V . I . a(u,v-u)
z (f,v-u) v v E f
.
In this case, a natural penalisation approximation is
Ad, +5(tiE-Y)+ 1
(1.180)-
=f
, fie
E Hi(0)
.
(This in fact shows that we do not have uniqueness of the penalisation approx2 mation ! ) . This time (1.170)is valid; we thus have, by assuming that AY E L (8): Hu-tiJJ s
1.15.
c yc
.
Estimates for A u .
Synopsis Let u be the solution of the V.I. (1.15) (the assumptions on the coefficients and the data are defined below). Then Au 5 f. Our objective is now to obtain a lower bound f o r Au. This will give, as a corollary, a number of regularity results which were previously obtained in Section 1.9 by different methods. THEOREM 1.21 The assumptions are (1.4),(1.5), (1.6). Let n(a) be the 2 space of Radon measures on 0. Suppose that f E L (0)and that JI satisfies ( 1 .I811
(SEC. 1)
STATIONARY V . I . ' s
(1 .182)
(1.15):
i s t h e s o l u t i o n of t h e V . I .
u
We t h e n have, i f
233
(*)
i n f ( A $ , f ) 6 Au 5 f .
Proof. 1) We reduce t o t h e c a s e where f = 0 I n f a c t we l e t w be t h e s o l u t i o n of Aw = f , w the V.I.
1
Ho(6).
E
Then u - w s a t i s f i e s
analogous t o ( 1 . 1 5 ) w i t h f r e p l a c e d by 0 and JI r e p l a c e d by $-w.
It i s t h u s r e q u i r e d t o show t h a t , w i t h f = 0 , we have:
inf(A+,O) S A u S 0
(1 .I831
.
2 ) We reduce t o t h e c a s e where JI = 0 on
r.
I n f a c t , (and t h i s same o b s e r v a t i o n has a l r e a d y been made i n t h e proof of Theorem l.ll), i f f = 0 , we have u 5 0 i n 6 and t h e r e f o r e t h e V . I . i s not changed i f we r e p l a c e $ by i n f ( $ , O ) = $ A 0 , and $ A 0 = 0 on r
.
Let us t h e r e f o r e assume f o r a moment t h a t we have e s t a b l i s h e d (1.183) f o r $
E
Hi(@);
thus i n f ( A ( $ A O),O) 5 Au.
(1.184) But
A($ A 0 )
L
(A$) A 0
so t h a t (1.184)i m p l i e s (A$) A 0 S Au, i . e . (1.183). 3)
We t h u s assume h e n c e f o r t h t h a t
+
( 1 .I851
In general i f
x
f HL(C)) E
,&
1 H (6), x 2 0 on
a k ( x ) , v 4 x ) )z
(1 .186)
8(x) i We n o t e t h a t i f X
We i n t r o d u c e
a,
r,
x
.
we denote by
ovv i x
S ( x ) f Hl(0)
H:(@)
E
g(x) =
(1 .187)
x
-
f
,v
.
$(x)
t h e s o l u t i o n of t h e V . I .
1
E H ~ ( o,)
w i t h AX 5 0 t h e n
t h i s being a s o l u t i o n of
(1.188) In point
4) below,
we s h a l l prove t h e following formula due t o J . L .
JOLY Ell:
(1.189) Then A(@+[&)) i 0 account of (1.188)
,
i . e . , reverting t o t h e notation
d+)= u
and t a k i n g
( * ) Obviously we know t h a t Au 5 f ; we t h e r e f o r e need t o prove only t h e f i r s t inequality.
(CHAP. 3 )
OPTIMAL STOPPING PROVLEMS & V.I.'s
234
(A$) A
4)
0 5 Au.
Proof of (1.189)
We t h e r e f o r e have by d e f i n i t i o n
(1.190)
ov
a(Y,cp-Y) 2
'p
1 ~ ~ (, 0 'ps) O-u
E v
.
Y s Q-u
I n o r d e r t o demonstrate t h a t Y = 0-u, it i s s u f f i c i e n t t o show t h a t
z=
(1.1911
, Y-O+U) s o
a(Y-D+u
.
But
z = z1 + z2 + z3' z1 = a(Y,Y-@+u), z2 =
a(-@,Y-@+u),
Z = a(u,Y-@+u). 3 We s h a l l show now t h a t Z. 5 0 v i. Next Z2 =(-AO,Y-O+u); - AO Z1 s 0
.
Taking 2
0 from
'p
= 0-u i n (1.190),we have
(1.188) and Y-@+U
5
0 from (1.190);
thus
z2 5
0.
F i n a l l y Z = -a(u,@-Y-u); now u s a t i s f i e s
3
a(u,v-u) 2
(1 .I921
ov
v 5 J,
, us J, , v
1
E ~ ~ ( 0 )
and we w i l l t h e r e f o r e have Z3 5 0 i f we can t a k e v = 0-Y i n ( 1 . 1 9 2 ) . 0-Y
E
$(&,
As
we o n l y have t o show t h a t 0-Y
5
$ in
that is
(1 .I931
Y 2 a-J,
We n o t e t h a t A(O-J,)
0-+
E
0,
. = (A+) A 0
S(Q-+)
.
- A+
0
,
a.e. in Q
OPTIMAL STOPPING PROBLEMS & V.I.'s
238
(2.24)
at f
(2.25)
dt
,
dt
,
E L'(o,T;H)
Suppose also t h a t (2.18) holds. admits a unique solution such t h a t
n
U E L'(o,T;v) at
(2.28)
( C H A P . 3)
Then the strong problem (2.10),
.
L=(o,T;H)
...
, (2.13)
m
I n t h e 'symmetric p r i n c i p a l p a r t ' c a s e we have t h e following theorem:
THEOREM 2.2. Suppose t h a t (2.l), (2.3) hold, with (2.23). have (2.6) together with (2.18) and t h a t
Then t h e strong problem (2.10), that
.
u E L=(o,T;v)
(2.31)
. . .,
Suppose t h a t we
(2.13) admits a unique soZution such
m
The The e x i s t e n c e w i l l be proved, for b o t h c a s e s , i n S e c t i o n 2.4 below. uniqueness of t h e strong solution can be proved immediately under assumption (2.23).
I n f a c t suppose t h a t u and tl denote two p o s s i b l e s o l u t i o n s . permissible,
v =
a(t) in
We t a k e , a s i s
(2.12)and v = u ( t ) i n t h e V . I . r e l a t i n g t o ii ;
if we p u t w = u - ii, t h e n , adding, we o b t a i n (2.32)
(2 - a(t;w,w) z o .
---,w)
However, from (2.231, t h e r e e x i s t s A such t h a t
(2.33)
a(t;v,v)
+ h1vl2>
a,l\v\12
, a, > o , v v
t h e n (2.32) g i v e s
I
d
-1T i t l w ( t ) l
2
+
a,1IW(t)ll
2
from which we deduce, s i n c e w ( T ) = 0 , t h a t
s
hlw(t)(
2
E
v
;
(SEC. 2 )
EVOLUTIONARY V . I . ' s
m
from which it follows t h a t w = 0.
Several proofs of e x i s t e n c e a r e p o s s i b l e . t h e p e n a l i s a t i o n method. 2.3.
239
We s h a l l s t a r t by i n v e s t i g a t i n g
Penalisation
We consider t h e equation
- $-+ BU
(2.34)
A(t)u, ++(u,-$)+
= f
with
(2.35)
which i s t h e penaZised equation a s s o c i a t e d with t h e V . I . variational f o m (2.34) i s writte,n BU
(2.36)
-($,TI
+ a ( t ; u , , v ) +1T ( ( u ~ - +=) (+f , ,v v ) v) v
Remark 2.4. It follows from ( 2 . 1 8 ) t h a t V v f u n c t i o n ( v - ~ i ) +i s i n L 2 ( a ) . I n f a c t t h e function (v-JI)+ i s measurable v-J,
= v-vo + vo-$
and hence t h e r e s u l t . meaningful.
( s e e Section 1 . 3 ) .
S v-v
5
E
v
.
2 2 L (0,T;H) = L (Q), t h e
0; i f v
so t h a t
E
E
K we have:
(~4)'s (v-vo)+
It follows from t h i s remark t h a t (2.34) or ( 2 . 3 6 ) a r e
We s t a r t By proving t h e following theorem: THEOREM 2.3. Suppose t h a t ( 2 . 1 8 ) holds along with ( 2 . j ' ) , ( 2 . 8 ) , ( 2 . 2 3 ) . There then e x i s t s a unique u such t h a t
(2.37)
Ur
E L'(o,T;v)
,
BU
E L~(o,T;P)
,
and ug s a t i s f i e s ( 2 . 3 4 ) and ( 2 . 3 5 ) . &OOf Of UnipeneSS. The uniquenes2 i s an immediate consequence of t h e monotonicity of t h e o p e r a t o r v -L (v-$) ; i f i n f a c t u and C. a r e two
p o s s i b l e s o l u t i o n s , we deduce from ( 2 . 3 6 ) , by p u t t i n g w = u - G c :
In
240
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
and t h e r e f o r e w = 0.
I
Proof of Existence. P r e l i m i n a r y Reduction:
We do n o t r e s t r i c t g e n e r a l i t y i f we assume t h a t
a ( t ; v , v ) 2 allv112
(2.38)
,
a
> o
,v
v
EV
.
I n f a c t i f we p u t
u
E
I
,-kt tlE
, where
k > 0 i s f r e e t o be chosen,
we o b t a i n
where
6 - e
kt
9 , i=e'f
. i,?
We t h u s have an analogous problem w i t h analogous p r o p e r t i e s f o r a ( t ; u , v ) r e p l a c e d by a ( t ; u , v ) + k ( u , v ) , which t h e r e f o r e g i v e s ( 2 . 3 8 ) .
and with I
Remark 2 . 5 . I n t h e l i g h t of t h e m a t e r i a l c o n t a i n e d in S e c t i o n 1, t h e most However, n a t u r a l method t o u s e for s o l v i n g ( 2 . 3 6 ) would be t h e Galerkin method. s i n c e JI depends on t , t h e analogue of ( 1 . 3 0 ) would suppose an a d d i t i o n a l a s s m p t i o n on JI We can avoid having t o make such an assumption by u s i n g t h e elliptic regularisation method which o f f e r s an i n t r i n s i c advantage. m
.
For y
0 , we seek u
EY'
a s o l u t i o n of
The problem ( 2 . 3 9 ) i s an elliptic problem (hence t h e t e r m i n o l o g y : ( 2 . 3 9 ) i s c a l l e d an "elliptic regularised equation" of ( 2 . 3 4 ) ) , and i s a simple v a r i a n t of t h e problems t r e a t e d i n S e c t i o n 1.
(SEC. 2 )
EVOLUTIONARY V . I . ' s b€J
, $f
f L2(0,T;V)
If we introduce
241
b
L2(0,T;H) with Y vanishes i n t h e neighbowhood of t = 0 , and i f we introduce t h e problem reduces t o t h e case when
-+
G
i n H, we have:
uY
E
O[we could choose
Y
(T) =
ii
dCY 3 u
Y
EY
and
- uY
TI
Y
i n such a way t h a t ,
Y
2 i n L (o,T;v),
By -. 'b bV 2 , G2 at at
2
i n L (0,T;V')I.
Let us t h e r e f o r e a s s m e t h a t
G
Y
= 0 ; t h e v a r i a t i o n a l formulation o f (2.39) is
then a s f o l l o w s : we d e f i n e t h e space V by
v
lvl 5
,
E L'(o,T;v)
E L'(o,T;H)
, TI
I
of
;
then
i s c o e r c i v e on V ( t h i s reduces t o t h e case ( 2 . 3 8 ) ) :
We t h u s have e x i s t e n c e and uniqueness f o r u
EY'
t h e s o l u t i o n of ( 2 . 4 0 ) , by t h e
It a l s o follows from ( 2 . 4 1 ) t h a t
same methods a s i n Section 1.
(2.42) We can t h e r e f o r e e x t r a c t a subsequence, a l s o denoted by u
u
EY
+
EY '
2
such t h a t
fi weakly i n L (0,T;V). E
I t i s p o s s i b l e t o e s t a b l i s h a supplementary e s t i m a t e : we deduce from ( 2 . 3 9 ) that 2 bU bU B U
$=
y*+
gy
bt
, gY
bounded i n
,
L2(0,T;V')
$(o)
If we put
1 E (t) =-exp Y
Y
--tY
we t h u s have bU
X
at
=
E
Y + gY
f o r t > 0, o r 0 f o r t
5
0,
0
OPTIMAL STOPPING PROBLEMS & V.I.'s
242
and s i n c e
s-
sm E ( t ) d t = 1 Y 2
,
bounded subset of L ( 0 , T ; V ' )
E 2 0 Y
when y
-+
(CHAP. 3 ) hU
, we
t h e n deduce t h a t
at
remains i n a
0.
We can t h e r e f o r e assume, p o s s i b l y by e x t r a c t i n g a subsequence, t h a t
Since t h e i n j e c t i o n from V -+ H i s compact, we t h u s have
u
EY
+
2 s t r o n g l y i n L (Q)
u
E
and we can immediately proceed t o t h e l i m i t i n y i n ( 2 . 4 0 ) ; we t h e r e f o r e o b t a i n t h a t uE i s a s o l u t i o n of
rn
f r o m which w e deduce ( 2 . 3 6 ) .
It i s p o s s i b l e t o g i v e another proof of t h e e x i s t e n c e o f uE, a s o l u t i o n of ( 2 . 3 4 ) , ( 2 . 3 5 ) , by t h e following
i t e r a t i o e procedure; w i t h
E
f i x e d , we p u t u
= w
and d e f i n e a sequence wn by
(2.43)
wo being d e f i n e d by
We put z = vn
-
and t a k e v = z i n ( 2 . 4 3 ) and v =
wn-l
analogous t o ( 2 . 4 3 ) for w n - l ;
-+&
(2.45) But ((wn-'
SO
that
(E
le(t)l 2
- $)+ -
(Wn-2
+
a(t;z,z)
- wn-'(t)12s c
Iwn(t)
from which it r e s u l t s t h a t + .
w i n Co(CO,Tl;H).
But ( 2 . 4 6 ) t h e n g i v e s
+3(wn-l- +I+ -
- +)+ISIwn-' - wnW21
being f i x e d )
wn
-
z i n t h e equation
t h i s gives:
1;
Iwn-'-
,
wn-212
(wn-2
- $)+,z)
=
o
so t h a t ( 2 . 4 5 ) i m p l i e s
ds
.
(SEC. 2 )
EVOLUTIONARY V.I.'s
243
2 a wn l i e s i n a and ( 2 . 4 3 ) shows t h a t wn l i e s i n a bounded s u b s e t of L (0,T;V) and 2 bounded s u b s e t of L ( 0 , T ; V ' ) . We t h e r e b y deduce t h a t w is a s o l u t i o n o f t h e
at
m
p e n a l i s e d equation.
Remark 2.6.
Since t h e c o n s t a n t C i n ( 2 . 4 2 ) i s independent of
E
(and of y ) ,
we have:
1,.1
(2.47)
L2( 0, T; V)
s
c
.
We s h a l l now o b t a i n some supplementary e s t i m a t e s on us.
2.4
I
Proofs of e x i s t e n c e i n Theorems 2 . 1 and 2.2
A p r i o r i e s t i m a t e (11 We f i r s t e s t a b l i s h an e s t i m a t e , w h i c h i s v a l i d b o t h w i t h i n t h e s e t t i n g of Theorem We t a k e vo E y and we t a k e t h e i n n e r product 2 . 1 and w i t h i n t h a t of Theorem 2 . 2 . of ( 2 . 3 4 ) w i t h uc - vo; t h i s g i v e s :
Since JI
-
vo 6 0 , we t h e n deduce t h a t
so t h a t (2.48)
Using ( 2 . 3 8 ) we can t h e r e b y reproduce r e l a t i o n ( 2 . 4 7 ) , and we can a l s o deduce that
(2.49)
244
OPTIMAL STOPPING PROBLEMS & V.I.'s
( C H A P . 3)
(2.50)
A p r i o r i estimate (111
The e s t i m a t e which follows i s v a l i d w i t h i n t h e s e t t i n g of Theorem 2 . 1 .
We p u t :
We d i f f e r e n t i a t e ( 2 . 3 4 ) with r e s p e c t t o t . (This d i f f e r e n t i a t i o n can be j u s t i f i e d , f o r example, on t h e b a s i s of t h e e l l i p t i c r e g u l a r i s a t i o n introduced i n t h e previous section). By p u t t i n g (2.53)
we o b t a i n
We take t h e i n n e r product of (2.54) with wE gives :
We deduce from t h i s t h a t
from which we then i n f e r t h a t
at
; since
$
=
0 on C, t h i s
EVOLUTIONARY V.I.'s
(SEC. 2 )
But we deduce from (2.34) and from t h e assumptions on (2.57)
wE(T)
I
A(T)C
u that
.
- f(T)
Furthermore
using (2.57),
(2.58) we conclude from ( 2 . 5 6 ) t h a t
Proof of e x i s t e n c e in Theorem 2.1.
I t r e s u l t s from ( 2 . 5 9 ) t h a t , when
- remains in a bounded subset of bUE
(2.60)
L2 ( 8 , T ; V ) n Lm(O,T;H)
E -t
0,
:
We can t h e r e f o r e e x t r a c t a subsequence, a l s o denoted by uE, such t h a t
-f
bt
-. &?
weakly i n
bt
L2(0,T;V)
and weakly s t a r i n
We then have
u and s i n c e (uE - $I)+
+
2
u strongly i n L ( Q )
+
0 s t r o n g l y i n L 2 ( Q ) (from ( 2 . 5 0 ) ) , we have:
(u-*)+ = 0 and t h e r e f o r e
u c y
(2.62) If v
E
y, bu
we r e p l a c e
. v
i n ( 2 . 3 6 ) by v-uE;
we have:
Lm(O,T;H).
2 46
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & V.I.'s
from which we deduce by i n t e g r a t i n g over ( s , t ) : bU
+ a(a;uE,v)
{:[-($,v-UE)
so t h a t we then deduce ( s i n c e us
+
-f
a(u;u,v)
(2.63)
s
J:[-(g,v-u)
l i m . inf.
f:
a(a;uE,uE)du
2 u s t r o n g l y i n L (Q)) :
- (f,v-u)]da> 2
hence, for any values of
- (f,v-uE)]do2
jz
l i m . inf.
a(o;u,u)da
;
and t , we have:
+
- (f,v-u)]do5:
a(a;u,v-u)
Dividing (2.63) by t - s and making m r e q u i r e d - i n e q u a l i t y a.e.
s
0
.
tend towards t , we then deduce t h e
A p r i o r i e s t i m a t e (111) o f Theorem 2.2.
In ( 2 . 3 6 ) we r e p l a c e v by
t h i s i s permissible s i n c e
(2.641
-1ulEl2
+ a(t;ue,ulE) + T 1 ((uE+)+,(uE+)t) +
a(t;uE,+l)
2
= 0 on Z.
= (f,ulE-
.
However, by v i r t u e of t h e symmetry of a (t;u,v),we have
$1)
-
We have:
- (ulE,
Fz ao(t;LE,uE) - F1a.o ( t ; u E , u E ) .
a O ( t ; ~ E t ~ '= E )i d
uIE
$ 8 )
$'
(SEC. 2 )
EVOLUTIONARY V . I . ' s
Hence ( 2 . 6 5 ) g i v e s
We t h e n deduce, by i n t e g r a t i n g over ( t , T ) and changing t h e s i g n s , t h a t
We n o t e t h a t
Is
l b ( ~ ; U p E )
c
llUE(S)Il
I
u
p1
so t h a t we deduce from ( 2 . 6 7 ) t h a t
We t h e r e b y deduce t h e e x i s t e n c e of a s o l u t i o n i n Theorem 2 . 2 , as b e f o r e .
2.5
m
Estimation o f t h e " p e n a l i s a t i o n e r r o r "
We s h a l l now prove t h e f o l l o w i n g r e s u l t s : Theorem 2 . 4 .
The assumptions are those of Theorem 2.1.
Suppose also t h a t
Then i f u (resp. u I denotes t h e solution of t h e V . I . obtained i n Theorem 2 . 1 Iresp. of t h e penalised equation) we have: (2.70)
248
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
where the constant C depends only on a i n ( 2 . 3 ) and on the norms i n L=(Q) of da
a
ij
, $,
( s i n c e (u,
-
ba
, $ , a. , $(f,+ $8
aj
$)+(T) =
u being f i x e d ) .
0).
We then deduce t h a t , when
-1 (uc-
(2.71
and
E
+.
0
Q)+ remains i n a bounded subset of
L2 (Q)
,
and t h a t
Since
u-u
E
=
,
r -(uc- Q)+ E
where
r
P
u-Q
+
(uE-
$1'
,
it follows from ( 2 . 7 1 ) t h a t , i n order t o prove (2.70), that
it i s s u f f i c i e n t t o show
(2.73)
I n ( 2 . 1 5 ) we choose t o define
v
by v-u =
- rc(i.e.
v
= Q-(uE- Q)-$
Q) and
EVOLUTIONARY V.I.'s
(SEC. 2 )
v = r
i n (2.36);
it then follows t h a t
and hence
We note t h a t r c ( T )
P
ii-+(T)
+ (E-+(T))-
=0
so t h a t
( 2 . 7 6 ) gives
But
and hence ( 2 . 7 7 ) gives
s o t h a t ( 2 . 7 3 ) then follows. In t h e case i n which a has "symmetric p r i n c i p a l p a r t " , we have, a f o r t i o r i , t h e same e r r o r e s t i m a t e ; however, t h e same proof then gives t h e same e r r o r estimate under l e s s r e s t r i c t i v e assumptions on t h e d a t a ; we can now s t a t e t h e following theorem:
THEOREM 2.5.
The assumptions are those o f Theorem 2.2, with ( 2 . 6 9 ) .
We have:
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
2 50
(2.78)
i n which t h e constant C depends only on
a
i n ( 2 . 2 3 ) and on t h e norms i n L m ( Q ) of
being f i x e d )
.
Remark 2.7. The above e s t i m a t e s can b e o f use i n Numerical A n a l y s i s , b u t t h e y have been e s t a b l i s h e d w i t h a view t o t h e asymptotic phenomena a s s o c i a t e d w i t h o p e r a t o r s w i t h very r a p i d l y o s c i l l a t i n g c o e f f i c i e n t s ; s e e BENSOUSSAN, LIONS and PAFANICOL4OU c11. 2.6
Maximum weak s o l u t i o n
-
We s h a l l now i n v e s t i g a t e t h e c a s e i n which t h e assumptions on f, a ( t ; u , v ) , u and e s p e c i a l l y on J, a r e minimal assumptions. We t h e n have t h e following fundamental r e s u l t due t o F. MIGNOT and J . P . PUEL [ll:
THEOREM 2.6.
coefficients
Suppose t h a t we have conditions (2.1),
(2.2),
(2.23)
on t h e
aiJ , a J . ao.
Suppose t h a t $ s a t i s f i e s ( 2 . 1 8 ) , (*), and t h a t we have ( 2 . 6 ) , ( 2 . 8 ) . I n the s e t o f weak solutions of ( 2 . 2 0 ) , (2.211, (2.22) there e x i s t s a m a x i m solution u; i n other words, i f w i s an arbitrary weak solution, we have: w 2 u.
Remark 2.8. By examining simple examples i n which V = H = R , it can q u i t e e a s i l y be seen t h a t t h e s e t of weak s o l u t i o n s does not i n general reduce t o a
single element.
Remark 2.9. We s h a l l show i n S e c t i o n 4 how t h e maximum s o l u t i o n i s t h e "good" s o l u t i o n , a t l e a s t f o r t h e a p p l i c a t i o n s which we have i n mind. Remark 2.10. I t can e a s i l y be seen from t h e e s t i m a t e s e s t a b l i s h e d e a r l i e r t h a t i n f a c t i f we s t a r t from u , t h e s o l u t i o n t h e s e t of weak s o l u t i o n s is non-empty; o f t h e p e n a l i s e d problem, we have, under t h e assumptions of Theorem 5.6, t h e e s t i m a t e s (2.471, (2.49), ( 2 . 5 0 ) . For v E K , we deduce from (2.36) t h a t
and we can proceed t o t h e limit ( n o t i n g t h a t
(*I
Assumption (2.7) i s redundant.
EVOLUTIONARY V.I.'s
(SEC. 2)
ji
lim. inf.
j'
a(t;uE,uc)dt 2
0
251
a(t;u,u)dt)
and hence we o b t a i n (2.22). Furthermore (u
-
/:
$I)'
-t
2 0 s t r o n g l y i n L (Q);
((uE- $)+
i n which v
E
- (v-+)+,uc-
v)dt 2 0
L2(Q) i s a r b i t r a r y , g i v e s i n t h e l i m i t (-(v-@)+,u-v)dt 2 0
Taking v = u-Aw, w i t h by A :
w
-t
.
a r b i t r a r y i n L2(Q),
X > 0, we o b t a i n a f t e r d i v i d i n g
2 0
(-(u-Aw-+)+,w)dt and making A
hence t h e i n e q u a l i t y
0 we t h e n o b t a i n
(-(u-+)+,w)dt
2 0
and t h e r e f o r e (u-Ji)' = 0 s o t h a t u
5
v Ji;
w E
u
2
L (Q)
9
a
i s a weak s o l u t i o n .
I t i s i n f a c t p o s s i b l e t o prove much more t h a n i s s t a t e d i n Remark 2.10:
THEOREM 2.7.
decreases.
The assumptions are those of Theorem 2.6.
When
E
decreases,
THEOREM 2.8. with the same assumptions a s i n Theorem 2 . 6 , if w weak solution and if E > 0 is arbitrary, we have:
(2.79)
W S U
uE
is an arbitrary
. 2
COROLLARY 2.1. When E 4 0, u decreases and converges weakly in L (0,T;V) t o the maximum solution of t h e weak broblem.
Proof of Theorem 2.7.
To s i m p l i f y t h e n o t a t i o n i n t h i s p r o o f , we p u t :
u = u, ue = a.
In (2.36) we t a k e We wish t o show t h a t i f c S e we have u 5 a. i n t h e p e n a l i s e d e q u a t i o n r e l a t i n g t o Q we t a k e v = -(u-a)+. Then
Y
= (u-a)'
and
252
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & V.I.'s
The f i r s t term i n t h e r e l a t i o n f o r X i s S 0. We s h a l l now show t h a t Y 2 0 ; on t h i s i n f a c t , t o c a l c u l a t e Y we i n t e g r a t e u-D. oyer t h e s e t f o r which u t Ci; s e t u-JI 2 Ci-J, and t h e r e f o r e (u-JI) 5 (O-J,) ; hence Y 2 0 and consequently X 2 0. We then deduce from (2.80) t h a t
and hence (u-a)'
= 0 and u I
a.
m
We f i r s t apply a number of transformations t o t h e s t a t e Proof of Theorem 2.8. ment of t h e Theorem, which a r e of a very simple a l g e b r a i c n a t u r e . LEMMA 2.1.
(2.81
(2.82)
We put z = w-u
9
jo[-(g,v-z) T
'Cr
= WE
+
a(t;z,v-z)
- $u,-
+)+,v-z)]dt
1
Proof.
We deduce from ( 2 . 3 6 ) t h a t , i f i.
E
K , we
have
We t a k e v = B i n ( 2 . 2 2 ) , where u = w , and we s u b t r a c t ( 2 . 8 3 ) ;
(*I
t h i s gives:
We assume once and f o r a l l t h a t we have reduced t o t h e form ( 2 . 3 8 ) .
(SEC. 2 )
EVOLUTIONABY V.I.'s
We p u t 5-u
= v ( a n a r b i t r a r y element o f
y
1
253
);
we n o t e t h a t
) = v-z
5-w = v-(w-u
so t h a t ( 2 . 8 4 ) i s i d e n t i c a l t o ( 2 . 8 2 ) .
m
We s h a l l deduce from ( 2 . 8 2 ) some i n f o r m a t i o n which w i l l be s u f f i c i e n t f o r our aims :
LEMMA 2.2.
Suppose w e have
Proof.
yo
Let
z
be an element o f
s a t i s f y i n g (2.82).
'(1 ('(1 4 $
since
of A > 0 , we have, i f 8 E
e
we choose
t o b e t h a t given by (2.87);
v
in
(2.82)
t h e c o e f f i c i e n t X of A 2 i s
and e ( o ) = 0.
so we o b t a i n
Then
yd
6)
; t h e n , f o r any value
:
zero;
we obtain
i n f a c t it i s equal t o
Hence (2.88) reduces t o
YX + Z
t 0
V A L 0 and t h e r e f o r e we have:
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
254
which then gives (2.85), in view of (2.87).
8
We will then have (2.79) if we can show that
of
We shall now show that t h i s follows from (2.85). We introduce 9
en
We have:
0 and therefore
2
The first term is
(all the terms being positive);
When n
-f
0, 0
But ((u uE
2 $
n
-
-f
z
+
erl E 8 and we can choose
0 = 9
n'
a
solution
n in (2.85)
hence
2 in L ( 0 , T ; V ) ;
therefore (2.92) gives
+ +
$ ) ,z ) = 0 since we are integrating over the set for which
and for which z
P
0 and therefore w
2
uE; since w 5 $, we have $ Hence (1.93) reduces to
and therefore uE = $, which gives the result.
a(t;z+,z*)dt
0
2 uE,
255
EVOLUTIONARY V.I.'s
(SEC. 2 ) so t h a t we have z+ = 0.
2.7.
W
Some p r o p e r t i e s of t h e maximum s o l u t i o n
We s h a l l now e s t a b l i s h some p r o p e r t i e s o f t h e mmimwn s o h t i o n mentioned i n N a t u r a l l y , t h e following p r o p e r t i e s a r e v a l i d when we have uniqueness Theorem 2.6. of t h e s o l u t i o n (which i s then t h e maximum s o l u t i o n ! ) . We denote by Y t h e s e t of functions $ such t h a t we have (2.18) f o r Notation: We s t a r t with t h e following theorem: t h e convex s e t y a s s o c i a t e d with Ji. THEOREM 2 . 9 . $,$
E
Y
with
The asswnptions are those of Theorem 2.6.
+c+
(2.9)
suppose we also have f, (2.95)
Let
f_c
u
Suppose we have
a . e . on 9 ; 2
3
E L (O,T;H),
.
B ,
c,
E H with
iis ii a . e .
( r e s p . Q) be t h e mmitnwn solution o f the weak V . I . re l a t i n g t o t h e t r i p l e t
if,%+l
(reap.
if,ii,ol).
We have (2.96)
Proof.
us6
.
We denote by u
( r e s p . QE) t h e s o l u t i o n of t h e p e n a l i s e d problem ( 2 . 3 5 ) ,
( 2 . 3 6 ) ( r e s p . o f ( 2 . 3 5 ) ^ , ( 2 . 3 6 ) * , t h e s e equations being analogous t o ( 2 . 3 5 ) , (2.36)
u, f , $ ) . ^
but with u,f,$ replaced by
^
^
From t h e arguments given above, we w i l l have
( 2 . 9 6 ) i f we can show t h a t
(2.97)
ucs be
.
We t a k e i n ( 2 . 3 6 ) v =(uE
-
QE)
+
+,
if w = u
+s
=' (fs*,w+)
= w
E
-
Q E and v = -w+ i n ( 2 . 3 6 ) ^ ;
adding, we o b t a i n :
(2.98)
-($,w+)
+ a(t;w,w+)
I n S , we i n t e g r a t e over t h e s e t for which uE
2 QE;
,
on t h i s set
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
256
(uE-
$1+ 1. (Qk-
hence S L 0 and t h e r e f o r e (2.98) implies
aw+ w+ + a(t;w+,w+)
-(zi
o
( s i n c e f - ? s 0)
so that
now w + ( T ) =
(G-z)+ ='
0 and t h e r e f o r e (2.99) shows t h a t w+ = 0.
8
We s h a l l now prove t h e following r e s u l t , which i s a l s o due t o F. MIGNOT and
J.P. PUEL, l o c . c i f .
THEOREM 2.10. The asswnptions are those of Theorem 2.6. Suppose t h a t t h e problem has been reduced t o the form (2.38) and t o t h e case a. 2 0. Let $,$I E Y w i t h $-$I E Lm(Q). Tuke f and i n L2(0,T,K) and in H. Let u ( re sp . a) be Then t h e solgtion of t h e weak V . I . r e l a t i v e t o { f , u , $ } ( re sp . {f,;,$}). u-a E L (Q) and
u
(2.1 00)
Proof.
We put
We introduce uE ( r e s p . h ) , a s o l u t i o n of t h e p e n a l i s e d equation r e l a t i v e t o f,G,$(resp,
f,G,$).
We s h a l l now show t h a t (2.1 01
w
=u
-
Qk-
k o S 0 a.e.
A proof i n a l l r e s p e c t s i d e n t i c a l t o t h e following, would show t h a t i3
E
-
u
E
-k
O
2 0 a.e.;
and hence t h a t
from which (2.100) then follows.
We t h e r e f o r e only need t o prove (2.101).
We take
(SEC. 2 )
EVOLUTIONARY V . I . ' s
257
i n (2.36) ( r e s p . ( 2 . 3 6 ) * ) v = w+(resp. Y = -w+) and add;
dw + -(x,w )
(2.1 02)
+ a(t;ue-GE,w+)
We s h a l l now show t h a t p
On t h e s e t f o r which
0.
2
,
= 0
+
which has t o be i n t e g r a t e d t o c a l c u l a t e p i s > 0. consider t h e expression
on t h e s e t f o r which
QE
- (fie-
$1'
u = ((ue-
-k
and uE - LIE
t
a = (u
Now u p 2 0,
-
t 0;
2 ko, so t h a t
QE
$)(ue- be- ko)
u -+-fie&>
5 0,
t h e function
hence u
2
Ji + ko
2
Ji
and
.
ko- JH & 2 0 and t h e r e f o r e a
2
0 , thus
s o t h a t ( 2 . 1 0 2 ) gives
dw -(n,w
(2.1 03) But
- +-fie+
ae - Ji
I t t h e r e f o r e remains only t o
- ko)+
&)+)(ue- fie
t h e r e f o r e on t h i s s e t we have:
t h i s gives:
a(t;u
- Pe,wc)
+1 + c:
+
a ( t ; u e - Qe,w ) I 0
.
a(t;w,w+) +(aoko,w+) 2 cillw+l12
so t h a t (2.103) gives
- 1l Ed l w+ I 2+ aIIw+II2 I o
(2.1 04)
Since w + ( T ) = (-ko)+ = 0 we deduce from (2.104) t h a t w+ = 0.
m
Remark 2.11. L a t e r on, we s h a l l deduce from Theorem 2.10 a c o n t i n u i t y r e s u l t f o r t h e maximum weak s o l u t i o n ( s e e Theorem 2.14). 2.8
Elliptic regularisation
I n Section 2 . 3 , we used e l l i p t i c regularisation t o approximate t h e s o l u t i o n uE We can use e l l i p t i c r e g u l a r i s a t i o n with regard t o t h e of t h e penalised probtem. V.I. Suppose t h a t we have performed We d e f i n e y C y by (2.105)
Yo =
( V ~ VE L2(0,T;V),
dV
t h e preliminary reduction t o t h e case ( 2 . 3 8 ) .
E L2(0,T;H)
,V
0 we seek u a s o l u t i o n of t h e ' s t a t i o n a r y ' V . I . Y'
}
OPTIMAL STOPPING PROBLEMS & V.I.'s
250
(CHAP. 3 )
(2.106) (2.107)
where, in
We see from (2.41) that we can apply the results of Section 1. We therefore have :
.
THEOREM 2.11. The conditions are those o f Theorem 2.6, w i t h y 6 For a l l there e x i s t s a unique u , the solution o f t h e e l l i p t i c reguZ8rrised V.I. Furthennorz, when y + 0 , we have: (2.106), (2.107).
y > 0,
(2.1 08)
Without any further assumptions on the coefficients and the data, we deduce from (2.108) that we can extract a sequence, also denoted by u such that Y' (2.109) u -. w weakly in L2 ( 0 , T ; V ) w being a weak so l u t i o n of the Y
evolutionary V.I. Open problem:
Do we have u
+
2
u = maximum solution, weakly in L ( O , T ; V ) ?
m
With the supplementary assumptions corresponding to Theorems 2.1 and 2.2, we can establish - f o r example by using (2.39) - further estimates on u corresponding -t u = to the properties of u in Theorems 2.1 and 2.2, and we then have "u Y the unique solution of the V.I. 2.9
Semi-discretisation
We shall now give some brief indications relating to the approximation of the stationary V . I . We introduce At > 0, of the form (2.1 10)
We put, f o r n
A t = T/N I
I?-1:
.
259
EVOLUTIONARY V.I.'s
(SEC. 2 )
(2.1 11
We d e f i n e
f=
(2.1 12) we assume t h a t
{vlv E
Kn # 0
v
n;
v , v s Qn
a . e . oil
0)
;
more p r e c i s e l y , we s h a l l assume h e r e t h a t :
t h e r e e x i s t s a sequence
v"
, vn
f
9" V ns
, such t h a t
N-1
(2.113)
where
11
= norm i n V'.
We s h a l l now d e f i n e t h e un s t e p by s t e p ;
(2.114)
UN=
we s t a r t from
ii
and we d e f i n e un, for n 5 N-1, by r e c u r r e n c e from
Un+l
-&-+(2.1 15)
unEP
n
,v-u")+
an(un,v-un)-(fn,v-un)
2
ov
v E
9"
.
We n o t e t h a t ( 2 . 1 1 5 ) i s a
stationary V . I . :
(2.1 16)
which admits a unique s o l u t i o n f o r A t s u f f i c i e n t l y s m a l l ( * ) .
(*)
And for any v a l u e o f A t , i f we have performed t h e p r e l i m i n a r y r e d u c t i o n t o t h e form ( 2 . 3 8 ) .
260
( C H A P . 3)
OPTIMAL STOPPING PROBLEMS & V.I.'S
The set (2.116) is an approximation by semi-discretisation of the evolutionary
V.I.
M
A priori estimates
If we take v = vn in (2.115), with vn satisfying (2.113), we obtain:
n
un+l
- u n n n n -(T,u + a (u ,u
un+l
At
Multiplying by 2At and summing over n
N-1
+ z lun-un+' 12s
-2
N N-1
N-l
=
-
+
N- 1 2 an(un,vn)At
2(u ,v
+c
N-1
2
+2Z
+ an(un,vn)
from q to N
N-I
z (un+l-un,vn)
n=q
n=q
N-1 - 2 z (fn ,v n -un )At n=.q
n
v,-
At
- 1, we obtain: N- 1 n n n + 2 z a (u ,v )At -q
n n-1 )At + 2(uq,vq)
w+l
/fn12
- (fn,vn- u")
- 2N-1Z (fn ,vn-u n >At +
C/Gl2
+
C
.
virtue of (2.113) and the discrete version of Gronwall's inequality, we then deduce:
-q
merefore, if we introduce UAt
=
n
in
In At,(n+l)At[
,
261
EVOLUTIONARY V.I.'s
(SEC. 2 ) we see that: (2.119)
2 u remains in a bounded subset of L ( 0 , T ; V ) n Lm(O,T;H) At
From this we can show that, as At by uAt, such that (2.120)
uM
-
w
-+
as
At
-
m
0.
0, we can extract a sequence, also denoted
2 m weakly in L (0,T;V) and weakly star in L (0,T;H) where w
is
a weak solution of the V.I. Open problem: Under the assumptions of Theorem 2.6, maximum solution? I
Remark 2.12.
We have given above one approximation procedure using semiIn fact, numerous other procedures exist; see GLOWINSKI-LIONSand the references given therein.
discretisation. TREMOLIERES [ll
2.10
does uAt converge to the
Regularity of the solution
First, we adopt the setting of Theorems 2 . 1 or 2.2, and we give some regularity properties of the solution. We shall then go on to consider the problem of the (possible) regularity of the maximum solution. I THEOREM 2.12.
hold.
The solution
(2.1 21 )
Suppose that the asswnptions of Theorem 2 . 1 or of Theorem 2.2 u of the V . I . s a t i s f i e s
A(t)u E
L2(d
(2.71),
we have:
Proof. From
from which it follows that
au at
E
--+dU bt
A ( t ) u E L2(Q).
Since we already know that
L 2 ( Q ) , it follows that we have (2.121).
COROLLARY 2.2.
Under the assumptions of Theorem 2.12, and aZso assuming that
(2.1 22)
we have the supplementary propertg
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
262
Proof.
We use ( 2 . 1 2 1 ) and Theorem 6.3,
Chapter 2.
m
We s h a l l now e s t a b l i s h some e s t i m a t e s i n t h e spaces L p , p > 2.
Suppose that t h e assumptions of Theorem 2 . 1 o r o f Theorem 2.2 Atso assume t h a t
THEOREM 2.13.
hold.
E L~(Q).
(2.1 24)
f
(2.1 25)
dC - rjz + A(t)
0 E Lp(P),
(2.1 26)
u
.
E LP(6)
J,
E LP(d
,
We then have
- $+
(2.1 27)
A(t)u E Lp(Q)
-
.
Proof. F i r s t , we n o t e t h a t v i a t h e same procedure as t h a t by which ( 2 . 3 8 ) was o b t a i n e d t h e g e n e r a l i t y w i l l n o t b e r e s t r i c t e d i f we assume t h a t
-
so t h a t
Using (2.128) we t h e n deduce t h a t
(SEC. 2 )
EVOLUTIONARY V.I.'s
263
from which we i n f e r t h a t
(2.1 29) Consequently
so t h a t ( 2 . 1 2 7 ) t h e n f o l l o w s .
8
The asswnptions are those o f Theorem 2 . 1 3 , with, in addition,
COROLLARY 2 . 3 .
( 2 . 1 2 2 ) and
E E w21P(0) n w~~P(o) (*)
(2.130)
.
We then have
u E LP(O,T, w2*P(0))
(2.131)
Proof.
,
We a p p l y ( 2 . 1 2 7 ) and Theorem
6.5, Chapter
m
2.
Remark 2.13. It f o l l o w s from ( 2 . 1 3 1 ) , ( 2 . 1 3 2 ) t h a t - p o s s i b l y a f t e r m o d i f i c a t i o n space on a s e t o f measure z e r o - u is continuous from C0,Tl + a(O) , where o f t h e t r a c e s a t t h e o r i g i n (for example) o f t h e f u n c t i o n s u s a t i s f y i n g ( 2 . 1 2 9 ) , T h i s s p a c e ( o f i n t e r p o l a t i o n between W2,P(&) and i s d e n o t e d by (2.130).
do)=
o(@))
&)
( s e e LIONS-MAGENES C11
-&1-l/P),qo)
-
LIONS-PEETRE
C11).
We have
(2.1
33)
if
n
p > ~ + l
.
We t h u s have
The assumptions are those o r Corollary 2 . 3 w i t h
COROLLARY 2 . 4 .
(2.134)
P
n+l > 2
Then the solution
(*I See
*
u o f t h e V . I . i s continuous i n
Remark 2.14 below
G.
8
OPTIMAL STOPPING PROBLEMS & V . I . ’ s
264
Remark 2.14.
I n (2.128) it i s s u f f i c i e n t t o t a k e
We now proceed t o t h e maximum s o l u t i o n ;
u
(CHAP. 3 )
E
B2-1’p’p(a.
we have t h e f o l l o w i n g theorem:
Also assume t h a t we The conditions are those o f Theorem 2 . 6 . THEOREM 2.14. have ( 2 . 2 4 ) and (2.122) on t h e c o e f f i c i e n t s of A ( t ) , and t h a t
Then t h e maximum s o l u t i o n of t h e weak V.I. i s continuous i n Proof. 1)
5.
We s h a l l use Theorem 2.10.
F i r s t , we n o t e t h a t t h e g e n e r a l i t y w i l l not be r e s t r i c t e d i f we assume that
-
f = O , u = O . I n f a c t , i f we i n t r o d u c e 0 , a s o l u t i o n o f
t h e n t h e f u n c t i o n 0 E Co($ u - 0 = 6, v - 0 = 0, v E
v 2)
( b y v i r t u e of ( 2 . 1 3 6 ) , ( 2 . 1 3 7 ) ) and i f we put see t h a t
K , we
-
d f y =y
-Q
,
from which t h e r e s u l t follows.
-
We t h e r e f o r e assume t h a t f = 0 , u = 0 and we i n t r o d u c e a sequence Jlj w i t h
Then t h e maximum s o l u t i o n u
3
uj
E
COG)
from C o r o l l a r y 2 . 4 . From Theorem 2.10, we have:
of t h e problem r e l a t i v e t o
iji s a regular s o l u t i o n :
(SEC. 2 )
EVOLUTIONARY V. I.
from which it follows, using (2.139), t h a t u
3
2.11
+
u i n Co(Q).
A free-boundary problem and a one-phase S t e f a n problem
When t h e s o l u t i o n u of t h e s t r o n g V . I . s a t i s f i e s (2.123), then ( 2 . 9 ) , (2.10). In t h e s e t Q we t h e r e f o r e have two regions: (2.140)
265
s
u
satisfies
$1
{x,tl u < I continuation s e t (*) and t h e s e t which i s t h e complement of C - t h e "stopping s e t " . The i n t e r f a c e S between C a n d i t s complement i s a f r e e surface.
C=
If t h i s s u r f a c e i s s u f f i c i e n t l y r e g u l a r , then (as i n Section 1 . 1 0 ) we have (2.141) w i t h , i n t h e s e t e(which i s open, under t h e conditions of Corollary 2 . 4 ) , equation (2.142)
- bU
+
A(t)U
= f
the
.
The conditions ( 2 . 1 4 1 ) a r e t h e c o n d i t i o n s on t h e f r e e surface.
m
We s h a l l now e s t a b l i s h t h e connections which e x i s t between t h e above problem and t h e S t e f a n problem, one of t h e a s p e c t s of which we r e c a l l below. Let 0 denote an open domain with boundary r = r ' u r", r ' = i n t e r i o r p a r t of t h e boundary and r" = e x t e r i o r p a r t of t h e boundary. We assume t h a t @ i s a ( t h r e e dimensional) enclosure f i l l e d with a mixture of water and ice a t Oo(one-phase problem); we assume t h a t I" i s maintained a t a given temperature b 0 and t h a t r" i s maintained at 0'. Let C ( x , t ) denote t h e temperature of t h e water a t t h e t h u s we have point x , at t h e i n s t a n t t ; (2.143)
C(x,t) > 0 i n t h e domain
C containing a neighbourhood of
r'.
C occupied by t h e w a t e r ,
I n C , we have
(2.144)
on t h e f r e e surface S, t h e w a t e r / i c e i n t e r f a c e , we have: (2.145)
(*)
C.. 0
z , -i = bn
- LV.n
See t h e next s e c t i o n f o r t h e explanation o f t h i s terminology.
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
266
where n = normal t o S ( a t t h e i n s t a n t t ) and where L i s a constant > 0 ( l a t e n t We also s p e c i f y t h e i n i t i a l h e a t of f u s i o n ) and V = speed of displacment of S . temperature (2.146)
and t h e boundary condition
Remark 2.15. Boundary conditions of types o t h e r than (2.147) also a r i s e i n physical problems; for example a condition of t h e type
t c
;j;;+ b3
D
h0 on one p a r t of t h e boundary;
t h i s comes i n t o t h e category of problems t r e a t e d i n Volume 2; i n a l l cases, we shall obtain an interpretation of C ( x , t ) i n terms of an optimal cost f o r a stop-
ping-time problem.
8
Remark 2.16. So far, we have been working with V . I . ' s for which t h e time t i s " r e t r o g r e s s i v e " s i n c e t h i s i s t h e way i n which t h e problem i s encountered i n t h e Theory of Control. However, by changing t i n t o T - t , t h e problem i s immediately reduced t o t h e V . I . ' s : xBU+ A ( t ) n - f s 0 U-+S 0
,
( 2.1 48)
dU (z+ A(t)u-f)
(u4)
,
0
P
in Q
,
with (2.149)
u=OonX
and (2.1 50)
u ( x , o ) = ;(x) on 8,
Obviously, a l l t h e preceding r e s u l t s a r e v a l i d .
I
Remark 2.17.
The methods considered h i t h e r t o can immediately b e adapted t o t h e case i n which t h e convex s e t u S $, i s replaced by u L $, t h e problem then being as follows: (2.1 51)
I =+
A(t)u-f 2 0
bu
,
u-+Z 0
,
with (2.149) and ( 2 . 1 5 0 ) . None of t h e r e s u l t s i s changed as far as t h e strong sohitions a r e concerned. For t h e weak solutions, t h e method of Theorem 2 .6 proves t h e e x i s t e n c e o f a minimum s o l u t i o n . Let us now s t a r t from u , a s o l u t i o n of problem ( 2 . 1 5 ) , ( 2 . 1 4 9 ) , (2.150) in the
following special case: (2.1 52)
A(t) =
(2.153)
$ = O
-
A,
EVOLUTIONARY V.I.'s
(SEC. 2 )
(2.154)
267
f = - L .
We then have i n C:
(2.1 55)
g - A u = f ,
We introduce w
(2.1 56)
all
I-
bt
which, i n C , s a t i s f i e s :
.
a~ w - A w = O
(2.1 57)
On S we have (from ( 2 . 1 4 1 ) ) : U E O
.
, ;ibU ;;=o
Let us assume, a t l e a s t l o c a l l y , t h a t we can r e p r e s e n t S by t h e s e t t - S ( x ) = 0 , S being a "regular" function. Hence
u(x,s(r))
= 0
n(x,s(x)) dU
I
,qx,s(x))
c
0
v
i
.
Consequently
0
'*
so t h a t w = 0 on S.
(2.1 58) Next we have
and hence
-
AU c 2 -
aw
"s
bXi dXi
Using (2.155),which on S
gives
On
-
s. Au = f,we t h e r e f o r e have by using (2.154)
I f we d e f i n e t h e o r i e n t a t i o n o f n , t h e u n i t normal t o S, by
++
we c a l c u l a t e V.n as follows: t h e displacement p along n during t h e i n t e r v a l of time A t i s such t h a t
t + A t = S(x+pn) s o t h a t t h e leading terms i n a s e r i e s expansion a r e :
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
268
bS + .. bxi
At =
P
PI
sI+
.*
and t h e r e f o r e ++
V.n
.
=+q
With t h i s n o t a t i o n , t h e second c o n d i t i o n i n ( 2 . 1 4 5 ) i s w r i t t e n :
Comparing w i t h (2.159) we t h u s s e e t h a t w and
a
same conditions on t h e f r e e boundary.
C both s a t i s f y (2.157) and t h e
We now argue i n t h e r e v e r s e sense: we s t a r t from C having t o s a t i s f y ( 2 . 1 4 4 ) , ( 2 . 1 4 5 ) and we associate with t h i s a f u n c t i o n which i s a s o l u t i o n o f a V.I. We can p r e s e n t t h i s c a l c u l a t i o n i n a s y s t e m a t i c manner as f o l l o w s . We d e f i n e :
-
= e x
x
= c h a r a c t e r i s t i c f u n c t i o n i n Q o f t h e s e t C where
C = e x t e n s i o n of C t o Q
10,TC by 0 ,
-
c > 0.
I
We c a l c u l a t e 'p
--ba t
s E -
"
and
f B(Q) , we have:
Let N denote t h e normal t o S i n R Z
Generally speaking, i f distribution
(2.1 60)
js
g
g 'p dS
i n t h e sense of d i s t r i b u t i o n s i n Q.
YE
X
Rt,
We have
i s a given f u n c t i o n on S, we denote by { g I s t h e
c -&{"O) at
.
NX
We t h u s have
S
and t h e r e f o r e , from ( 2 . 1 4 5 ) , we have
aa t s- LC.. * - LIV.Nx) . S Furthermore
p o i n t i n g outwards from C .
If
(SEC. 2) and since V.N. (2.1 61) so
EVOLUTIONARY V.I.'s =
-
269
Nt:
%= {V.NxfS
that (2.160) may be written:
(2.1 62)
*
2at - G p-L-at ax .
All information relating to the conditions on the free boundary is contained in (2.162).
We are naturally led to integrate (2.162) with respect to t; we therefore define:
ul. 0
u - Audt (2.1 67)
,
U(--
on
u 1 0
on
u(x,o) = 0
on @ ,
u
So t
go> 0
g(x,s)ds
BU
- g) = 0
a~
AU
r'
x
(O,T),
r"
x
(O,T),
in Q
,
OPTIMAL STOPPING PROBLEMS & V . I . ' s
270
(CHAP. 3 )
We s h a l l give i n Section 4.12 a r e p r e s e n t a t i o n of e ( x , t ) i n terms o f optimal control. 2.12
Further discussion on r e g u l a r i t y
I n t h i s s e c t i o n we s h a l l give some f u r t h e r r e g u l a r i t y r e s u l t s for t h e s o l u t i o n . The following r e s u l t s a r e not t h e most g e n e r a l p o s s i b l e . THEOREM 2.15. Suppose that we are under the conditions of Theorem 2.10, A having coefficients independent of t (for simpZicityl. Suppose that
,
(2.1 68)
f E dt
L=(Q)
69)
2 E at
L=(Q)
.
2 E dt
L=(Q)
.
(2.1
Then (2.170)
Proof.
Let
w
~ i Ei
LYS) ,
b e t h e s o l u t i o n of
%(t)
=
u(t+h) i f t 5 T-h 0
from which we deduce t h e r e s u l t .
i f T-h 5 t 5 T , h > 0 ,
m
EVOLUTIONARY V.I.'s
(SEC. 2 )
271
THEOREM 2.16. The conditions are those of Theorem 2.15, A having constant c o e f f i c i e n t s (for s i m p l i c i t y ) . Suppose that
We then have bU axi
(2.1 72)
.
E L~(Q) v i
Proof. 1)
A s i n t h e proof of Theorem 2.15, we reduce t h e problem t o t h e case where
f = 0, U = 0, J, = 2)
o
on
x
f o r t = T.
We introduce t h e f u n c t i o n 8 , a s o l u t i o n of
I -g+Ae='
(2.1 7 3 )
8 =
We have:
o
u + kB
(2.1 74)
on Z, 8 ( x , T ) = 0.
'
5 0
f o r ' k > 0 sufficiently large.
In f a c t we introduce uE, a s o l u t i o n of t h e p e n a l i s e d problem: bU
1 -at2+ AU, +-(u,+)+ o u = o on X, u~(x,T) o ;
(2.1 75)
in
Q,
It i s s u f f i c i e n t t o prove t h e analogue of (2.174) with u
i n s t e a d of u; we
m u l t i p l y (2.175) by (uE+k8)- and equation (2.173) by k(u + k8)-; adding, we obtain
However, i f we choose
- k8 we have (ue
-
0 arbitrary.
Proof.
-
M/fo
By v i r t u e o f ( 2 . 2 1 2 ) , (2.213) we can apply Section 2.15 with
rn
We consider a p r i o r i t h e f u n c t i o n
We have - = b+W A w e
SO
that bW -at+ AwS
(2.21 6) We choose (2.217)
c
.
f
such t h a t w(x,T) = c
-
f T =
-
M (5
L)
and hence c = f
T - M .
We then have, from Theorem 2.9 (extended t o t h e s e t t i n g of Section 2 . 1 5 ) :
This s o l u t i o n i s defined for a l l t < T i f t h e d a t a a r e defined f o r a l l t < T ( t h i s remark i s , moreover, a b s o l u t e l y g e n e r a l ) . Here we have a Strong s o l u t i o n . But t h e following r e s u l t extends t o t h e maximum weak s o l u t i o n s , with u 5 J1 (and by r e p l a c i n g "support" by " s e t f o r which u = $ " ) . (*)
2 80
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ’ s
w
(2.218)
u.
5
Since w = 0 f o r t
0
an i n c r e a s i n g f u n c t i o n of 1x1 (for example)
1J bX
~i
-(mV-u'U)n BU
v
281
j
a (x,t) a u
dx
O
- EA(t;u,u) - (f,v-u),
v i J, a.e.
i
;S
10
,
(CHAP. 3)
OPTIitAL STOPPING PROBLEMS & V.I.'s
282
"Weak formulation" We introduce
( 2.23 2)
2
y = ~ v l vE L (o,T;F) , v
5 )I
,
E L ~ ( o , T ; F ~(*I )
a . e . i n Q};
we assume t h a t
(2.233)
Y46
*
We introduce
(2.234)
,
yC = (v(v E y
v with support i n K
x
CO,Tl,
= compact s e t i n &};
xc
1 I f v E ?( and if 8 E C ( B ) , with compact support i n 6,we have & E # 0 i f we have (2.233).
We say t h a t
u
vc, so
that
i s a "weak solution" of t h e evolutionary V.I. if
( 2.235)
u
(2.236)
u I JI a . e . i n Q
E
L~(o,T;F),
and
(2.237)
T
av [-(-,v-u) o a t VVEYc
x
+ E (t;u,v)- E (t;u,u)- (f,v-u)
.
x
v
1
]dt +T1v(T)-c1f2
m
F i r s t , we s h a l l e s t a b l i s h an e x i s t e n c e and uniqueness theorem f o r t h e strong 8 solution. For s t r o n g s o l u t i o n s , we s h a l l adopt t h e following assumptions;
(hence Ji = 0 on Z ) ,
We then "nave t h e following theorem: (*)
Dual of F when L2 i s i d e n t i f i e d with i t s dual.
0
EVOLUTIONARY V . I . ' s
(SEC. 2 )
283
. . .,
THEOREM 2.21. Suppose that the assumptions ( 2 . 2 2 3 ) , ( 2 . 2 2 4 ) , ( 2 . 2 3 8 ) , (2.242) are satisfied. There then exists one and o n l y one solution u of the problem ( 2 . 2 2 8 ) , (2.231). Furthermore
. . .,
- E L'(o,T;F)
& E L~(o,T;L~(o)) .
,
dU
(2.243)
at
Proof of iiniqueness. Let eR(x) =
u and 6 be two p o s s i b l e s o l u t i o n s and l e t w = u-ti.
e(:)
Also l e t
(as i n t h e proof of Theorem 1 . 1 7 , S e c t i o n 1 . 1 3 ) ; we t a k e i n (2.230)
( r e s p . i n t h e analogous V . I . obtain
r e l a t i n g t o 6) v = 6 ti ( r e s p . v = 8 u); adding, w e R R
It can be shown (as i n t h e proof o f Theorem 1.17) t h a t , i f R
ji[En(t;u,ERu-u)
+
En(t;ff,eRff-ff)]dt
-
0
+
-,
,
s o t h a t (2.244) g i v e s
so t h a t
and hence w = 0.
m
Proof of existence. We i n t r o d u c e t h e penalised equation
I n o r d e r t o s o l v e ( 2 . 2 4 5 ) , we i n t r o d u c e "eZZiptic regularisation in the space as d e f i n e d by (1.150),and f o r y > 0 we
VariabZes"; we i n t r o d u c e t h e space F denote by u t h e s o l u t i o n of EY
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
284
Since t h e form
8: ( t ; u , v )
(2.248)
I
y(A1 ( t ) u , A , ( t ) v ) R
+ ER(t;u,v)
(with i s continuous and c o e r c i v e on Fo, we have e x i s t e n c e and uniqueness of u (2.246), (2.247)). EY
I
A u r i o r i estimate
We t a k e v = u
EY
-
v
0
i n ( 2 . 2 4 6 ) , vo
EX,.
We n o t e t h a t
1
so t h a t (2.246) g i v e s :
--I2
-dt
d lu
1 2 + ~ I A , ( t ) u 2, ~+l ER(t;UEYtUey) ~ +
EY R
1~ / ( u $1 ~ +~ 2-
From t h i s we deduce, a f t e r a few c a l c u l a t i o n s , t h a t :
( C = c o n s t a n t independent o f E and of y),
A p r i o r i e s t i m a t e s I1 We d i f f e r e n t i a t e (2.246) w i t h r e s p e c t t o t .
dll +=
w
.
We denote by i , ( t ) t h e o p e r a t o r
To s i m p l i f y t h e n o t a t i o n we p u t
+x
b
(xai(Xtt))-
i T ( t ; u , v ) t h e form analogous t o E T ( t ; u , v ) w i t h a i J , ai, a.
ba ba, bao ij at , at,t
.
We o b t a i n
We t a k e , i n ( 2 . 2 5 2 ) , v = w
+
; we n o t e t h a t
bV
a= i
r e p l a c e d by
and by
(SEC. 2)
EXOLUTIONARY V.I.'s
so that
(2.254)
We deduce
(2.255) (2.256) (2.257) Proceeding to the limit in y , we establish the existence (the uniqueness being proved as for the uniqueness of the V.I.) of u E , the solution of (2.245) with
and
The theorem is then established by proceeding to the limit in
Remark 2.24. obtain
If we take v = (u
EY
- I)'
E.
8
in (2.246) (as is permissible), we
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
286
We t h e n deduce, i f we make t h e assumption:
that
Proceeding t o t h e l i m i t i n y, we have:
Suppose t h a t we have t h e conditions of Theorem 2.21 w i t h Then, when E +. 0
THEOREM 2.22.
(2.261).
(2.263)
-(uE1 +)+ remains i n a bounded subset of L ~ ( o , T ; L ~ ( ~ ) .
COROLLARY 2.5. Under the conditions of Theorem 2.22, t h e solution strong V . I . also s a t i s f i e s (2.264)
--+d U
A(t)u E L2(0,T;L:(0))
at
.
u of t h e
m
I n r e l a t i o n t o t h e weak s o l u t i o n s , we may conjecture t h a t , under t h e assumpt i o n s (2.223), (2.224) and (2.233), t h e weak problem admits a m a x h solution. I n comparison w i t h S e c t i o n 2.6, t h e e x t r a d i f f i c d t y d e r i v e s from t h e f a c t t h a t E T ( t ; u , v ) i s d e f i n e d f o r u and v belonging t o d i f f e r e n t spaces ( O r f o r
v = u).
A s f a r as t h e p a a l i s e d problems a r e concerned, we c l e a r l y have t h e o r d e r i n g relation
(2.265)
if
E
5
e
we have u d u-. E
E
I n f a c t , we p u t u = u, u. = u, and i n t r o d u c e
(2.266) to
v = ( ~ - a ) + eR
,
w i t h BR as i n t h e proof of Theorem 2.21.
Taking t h e i n n e r product o f (2.245) ( r e s p . o f t h e p e n a l i s e d e q u s t i o n r e l a t i n g i) w i t h v given by (2.266) ( r e s p . by -v) ( t h e i n n e r product being t a k e n i n t h e
sense o f L E ( o ) ) , we o b t a i n :
(2.267)
- ( ~ ( U - G ) ,(u-fi)+e R )n
+ En(t;u-G,(u-fi)+BR) + X
= 0
,
It can b e shown t h a t each o f t h e terms of X i s 2 0 , so t h a t (2.267) g i v e s
(SEC. 2 )
287
EVOLUTIONARY V . I . ' s
We can make R +
-I
-, and d
I
i n t h e l i m i t we have:
;it (U-0)
In + En(t; (u-fi)',
+ 2
(u-fi)') s 0
from which it follows t h a t ( u - a ) + = 0.
8
Remark 2.25. We again have t h e monotonicity r e l a t i o n s (as i n S e c t i o n 2 . 7 ) f o r t h e s t r o n g s o l u t i o n s (and, probably, f o r "the" maximum s o l u t i o n ) . We t h u s a l s o have a theorem analogous t o Theorem 2.19 f o r t h e s u p p o r t , i n t h e v a r i a b l e t , o f t h e s o l u t i o n . We a l s o have a theorem analogous t o 2.20, t h e growth p r o p e r t i e s o f a . not b e i n g involved. 8 J We can a l s o i n t r o d u c e t h e weights exp -ulxl
Remark 2.26.
and we can work w i t h
A s s e r t i o n s analogous t o t h o s e given i n t h e p r e s e n t s e c t i o n a g a i n v = 0 on r}. hold i n t h i s context. 8 2.18.
Other i n e q u a l i t i e s
We s h a l l now consider t h e e v o l u t i o n a r y analogue o f t h e problem i n v e s t i g a t e d i n We t h e r e f o r e t a k e two measurable f u n c t i o n s J1 1, J1, i n Q w i t h S e c t i o n 1.lL.
4~~ (x,t) s G2(x,t)
(2.268) We seek
-
a.e.
i n t h e s t r o n g formulation
-
a function
u
such t h a t ( * )
(2.269)
(2.271)
- (mv-u)+
(2.272)
U(X,T) =
dU
is
solutions is solution).
:(I)
,
a.e.
in
2
d
o
.
The weak formulation i s given e a r l i e r . The s e t of t h e weak nonempty The investigation of the structure of the set of weak an open problem. ( t h e r e does not appear t o be a maximum o r minimum
Remark 2.27.
solutions
a(t;u,v-u)-(f,v-u)
.
8
The penalised problem ( c f . ( 1 . 1 6 8 ) ) a s s o c i a t e d w i t h t h e preceding V . I .
i s as
~~
( * ) We c o n s i d e r t h e c a s e w i t h
'I Obounded" o r a l t e r n a t i v e l y w i t h Ounbounded b u t w i t h t h e o p e r a t o r having bounded c o e f f i c i e n t s .
OPTIMAL STOPPING PROBLEMS
288
&
V.I.'s
(CHAP.
3)
follows : Find u
s a t i s f y i n g t h e analogue of ( 2 . 2 6 9 ) , such t h a t
au
( 2.273)
-2 at +
(2.274)
u ( x , T ) =
0
,
The f u n c t i o n uc e x i s t s and i s unique. We s h a l l now prove t h e following theorem:
THEOREM 2.23.
aZso t h a t
Suppose t h a t ( 2 . 1 ) , ( 2 . 2 ) , ( 2 . 2 3 ) , ( 2 . 2 4 ) , ( 2 . 2 5 ) h o l d .
Suppose
(2.275)
ci> o
on
I:
,
$-=
o
and
--b 2
on
I:, i = 1,2
and t h a t ( * ) (2.276)
- zb( + ~ - + ~ ) 2 0
at2
(+
-+
')
Z 0 a.e. i n Q,
and t h a t (2.277)
There then e x i s t s one and onZy one function u, the soZution o f (2.2691,
(2.272) ; furthermore
(2.278)
U at E
L'(G,T;v)
n
...
LYG,T;H).
Proof. The uniqueness i s proved as i n Theorem 2.1. The f i r s t a p r i o r i e s t i m a t e i s o b t a i n e d by t a k i n g t h e i n n e r product of (2.273) w i t h
It i s w i t h t h e second a p r i o r i e s t i m a t e s t h a t t h e r e a r i s e s a t e c h n i c a l d i f f i c we d i f f e r e n t i a t e (2.273) w i t h r e s p e c t t o t ; u l t y , which r e q u i r e s ( ? ) ( 2 . 2 7 6 ) ;
( * ) We do n o t know whether t h e assumption (2.276) can be suppressed.
EVOLUTIONARY V.I.'S
(SEC. 2 )
au
we put (2.279)
at
= w;
dW
- L+ at
and we multiply by
this gives:
A(t)w
a
+
(up
;(t)uE
-
q2).
l a +--.(uEE a t
G2)+
-1 E e(u a t E-
We obtain (note that
Gl)-
df =i F
at
where
from which we deduce, by virtue of (2.276) (in which we have created precisely what was needed f o r this purpose!), that (2.281
:'Ids20
.
OFTIbNL STOFFING PROBLEMS & V. I . ' s
290
(CHAF. 3 )
However, (2.2731, f o r t = T , g i v e s
w(T) = A ( T ) i i
- f(T)
2
E L
(a)
by h y p o t h e s i s ,
s o t h a t (2.282) i m plie s:
m e theorem t h e n f o l l o w s .
2.19
Problems p e r i o d i c w i t h r e s p e c t t o t
We adopt t h e c o n d i t i o n s o f S e c t i o n 2 . 1 and we c o n s i d e r t h e f o l l o w i n g problem: f i n d u such t h a t , i n some s u i t a b l e s e n s e , we have t
I
U = O
on
z
and t h e "periodicrt condition w i t h respect t o t
Remark 2.28. t h e operator
I n t h e c a s e where ( 2 . 2 8 5 ) h o l d s we c o u l d e q u a l l y w e l l c o n s i d e r
-dmu + A(t)u
i n s t e a d of t h e operator
+
dU
+
A(t)u
We s t a r t w i t h t h e weak formulation. We i n t r o d u c e
I t can immediately b e shown t h a t
so t h a t we have t h e f o l l o w i n g f o r m u l a t i o n :
we s e e k
u
with
i n (2.284).
I
(SEC. 2)
EVOLUTIONARY V.I.'s
,
E L~(o,T;v)
(2.287)
us
(2.288)
$
29 1
a.e. i n
4
and
:j [-
(%,Y-U)
y
+
a(t;u,v-u)
1
(f,v-u)]dt 5 0 V v
uE E L*(o,T;v) ,
au
$E
~ ~ (=0U c)( T )
for
> 0 , we s e e k u
E
E ?(
,
a
b e i n g assumed nonempty.
The penalised problem is a s f o l l o w s :
(2.290)
-
with
L~(o,T;v~),
.
We s h a l l now prove t h e f o l l o w i n g theorem: THEOREM 2.24.
(2.291
Suppose t h a t we have (2.1)w i t h f
, >
a ( t ; v , v ) 2 altv11~ a
o
,vv
E
L2 (o,T;H) and
E
.
v
Suppose t h a t Y h 6 and t h a t f i s given i n L2 (o,T;H). There then e x i s t s a solution u of the problem (2.287), (2.288), (2.289)which i s maximum i n t h e s e t of the weak solutions.
Proof. 1) We s t a r t by p r o v i n g t h a t (2.290)a d m i t s a unique s o l u t i o n ; t h e uniqueness is an immediate consequence o f (2.291) and o f t h e m o n o t o n i c i t y o f t h e o p e r a t o r v + (v-q)'. A s r e g a r d s t h e e x i s t e n c e , we c a n , f o r example, approximate u by u E
t h e s o l u t i o n o f t h e e l l i p t i c problem:
2) Taking t h e i n n e r p r o d u c t o f (2.290)w i t h w i t h t h e c a s e s given e a r l i e r , t h a t
u
-
vo, vo
E y , we
deduce, as
CY'
OPTIMAL STOPPING PROBLEMS
292
(2.295)
uE
u
weakly i n
L'(o,T;v)
&
(CHAP. 3 )
V.I.'s
,
u b e i n g one weak s o l u t i o n o f t h e problem.
3)
I t can be shown t h a t
This i s a s t r a i g h t f o r w a r d v a r i a n t of Theorem 2.7.
4) I t t h e r e f o r e remains f o r us t o prove t h e f o l l o w i n g : l e t w b e an a r b i t rary weak s o l u t i o n ; t h u s we have t h e e q u i v a l e n t of (2.2871, ( 2 . 2 8 8 ) and (2.297)
Ji[-(g,v-w)
+ a(t;w,v-w)
l e t uE b e a s o l u t i o n of ( 2 . 2 9 0 ) ;
(2.298)
wsu
- (f,v-w)]dt
2 0 V v E
Y :
then
.
For t h i s , we f i r s t show, a s i n Lemma 2 . 1 ( S e c t i o n 2 . 6 ) , t h a t i f we p u t
(2.299)
=
W'UE
, y1
I
y - uc
,
t h e n we have
We t h e n deduce (as i n Lemma 2 . 2 ) t h e following:
then
i f we d e f i n e 8 by
(SEC. 2 )
EVOLUTIONARY V.I.'s
29 3
( 2.302)
8
n
I t now remains t o show t h a t (2.302) implies z hy means of
We have:
Brl 2 0 We can t a k e 8 = 0
1:
rl
,
8 11
-
z*
i n (2.302).
We can make rl + 0 i n (2.304);
L2(0,T;V)
We introduce ( c f . ( 2 . 9 1 ) )
0.
when
rl
*0
.
Tnis g i v e s :
t h i s gives
[ a ( t ; z , e + ) -T((uE1
- $)+,z+)
However, ((u
in
5
$1+, z +) ] d t S
0
.
= 0 , s o t h a t t h e r e s u l t then follows.
m
With regard t o t h e "strong" s o l u t i o n s , we can prove t h e following theorem: THEOREM 2.25.
1 and
$(O)
Suppose t h a t t h e conditions o f Theorem 2.24 hold, with
= +(TI
.
There e x i s t s one and only one function
u
such t h a t we have (2.287), (2.288)
OPTIMAL STOPPING PROBLEMS & V.I.'s
294
(CHAP. 3)
Consequently, by virtue of (2.305), we have:
Since 1(u EY
(2.31 0)
11
C , we deduce from (2.292) and (2.309)that when
E,Y
+
0:
L2(0,T;V)
a 2u
+
y
3
bu
+
9
We deduce from (2.310) that
remains within a bounded subset of L2(G,T;vt)
29 5
EVOLUTIONARY V.I.'s
(SEC. 2 )
We can then proceed t o t h e l i m i t i n E and y, which proves t h e e x i s t e n c e of a n solution. The uniqueness i s proved by t h e usual procedure. Another p o i n t of view concerning t h e p e r i o d i c s o l u t i o n s With t h e d a t a t h e same as i n Theorem 2.25, we extend a ( t ; u , v ) , f , q onto R, A
,
.
p e r i o d i c a l l y i n t , with p e r i o d T ; l e t I ( t ; u , v ) , f , $ denote t h e s e extensions. Let u b e t h e s o l u t i o n given by Theorem 2.25 and l e t fl be i t s "periodic extension" t o R,. Then
fl i s a s o l u t i o n of t h e V . I . on R,:
054
,
-(x, v-6) + b(t;G,v-C1) - (f,v-6) 2 0 v bb
V S (L
,
and s a t i s f i e s
aE
L~(IL+:v) = L ~ ( V ),
E L~(VI)
.
Since we have seen ( f o o t n o t e ( * * ) , page 275 ) t h a t t h e r e i s uniqueness o f t h e s o l u t i o n i n t h e c l a s s of f u n c t i o n s fl such t h a t
e-ytfi E L'(v),
,-yt
E
~
~
(
~
8y 1
0 ) . Then it i s e a s i l y shown t h a t it i s p e r m i s s i b l e t o t a k e
(3.143).
c1
= 0 i n ( 3 . 1 4 2 ) and
The f u n c t i o n u We t h e n c o n s i d e r t h e p a r t i c u l a r c a s e i n which f = 0 , a = 0. + i n (3.143)),so u p o s s e s s e s , amongst o t h e r s , t h e properties
i s 5 0 ( f o r we can t a k e f3 =
OPTIMAL STOPPING PROBLEMS & V . I . ' s
378
(CHAP. 3 )
(3.145)
I n t h e terminology of p o t e n t i a l t h e o r y ( s e e e . g . BLUMENTHAL-GETOOR [11) t h e f u n c t i o n -u i s excessive
(-u 2 0
, @(h)(-u)5
, Q(h)u(x)
-u
-
u(x) V x)
.
Furthermore -u t -$I and i f w i s an e x c e s s i v e f u n c t i o n such t h a t w L -$I, t h e n n e c e s s a r i l y , as a consequence o f p r o p e r t i e s a l r e a d y proved, -u i s t h e smaZZest e x c e s s i v e f u n c t i o n which i s g r e a t e r t h a n o r equal t o -$. This r e s u l t has been p r e v i o u s l y demonstrated ( u s i n g very d i f f e r e n t t e c h n i q u e s ) by DYNKIN [ 2 1 , t h e n by GRIGELIONIS-SHIRYAEV C11. 3.8.
P r o b a b i l i s t i c proof o f c e r t a i n p r o p e r t i e s o f v a r i a t i o n a l i n e q u a l i t i e s
I n t h i s s e c t i o n , we r e t u r n t o o p e r a t o r s " i n divergence form". Our o b j e c t i s t o g i v e p r o b a b i l i s t i c p r o o f s o f c e r t a i n p r o p e r t i e s o f V . I . ' s which were demonstrated i n S e c t i o n 2. Obviously we s h a l l be making u s e o f t h e i n t e r p r e t a t i o n of t h e s o l u t i o n o f t h e V . I . a s a lower bound of an optimal stopping-time problem. At t h i s p o i n t we a r e not seeking t h e widest p o s s i b l e g e n e r a l i t y , and we concern ours e l v e s above a l l with t h e b a s i s o f t h e p r o o f . I n o r d e r t o e s t a b l i s h t h e framework of our approach, we c o n s i d e r t h e V . I .
(3.146)
u s i n g t h e f o l l o w i n g assumptions
(3.147)
f
s a t i s f i e s (3.24).
s a t i s f y (3.25) ;
9 €
The c o e f f i c i e n t s of t h e form
Co(6),
J,lr2 0
(*I
We know from Theorem 3.7 t h a t we have
u ( x ) = Inf Jx(e)
e
( * ) We r e c a l l t h a t @ i s s t i l l assumed t o be o f c l a s s C
2
.
a
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
379
To begin w i t h , we have t h e r e s u l t of Theorem 1 . 4 . Let u = u i f , J I ) be t h e s o l u t i o n o f d ( 3 . 1 4 6 ) ; then i f f L 0 , JI L 0 , we have u L 0 and i f ? L f, 5 L JI then u ( f , q ) 2 u ( f , J I ) , p r o p e r t i e s which a r e evident i n t h e formula (3.148). We a r e now concerned with p r o p e r t i e s o f t h e s o l u t i o n with r e s p e c t t o t h e domain also o f c l a s s C 2 , 6 bounded and 8 c 0 c 6 . (Theorem 1 . 6 ) . Thus suppose we have 0' We have THEOREM 3.17.
further assume that
We make the assumptions (3.147) (on d i n place o f
Then U'
Proof.
We denote by Ji(e) t h e analogue of
N a t u r a l l y T;
Since T; that
.
0 (*)
on
2 u
L T ~ .
0). We
Jx(e);
t h u s ( i n an obvious n o t a t i o n )
Thus
5 'cX we have
xeCT,
2
x ~ < ~ Since ~ .
f
and
JI
a r e 2 0 on
s' , we
see
X
J;(e)
- J,(e)
and hence t h e r e s u l t .
2 0 V 8 n
THEOREM 3.18. ( P e n a l i s a t i o n of t h e domain). We make t h e asswnptions (3.147) 0,and t h e characteristic function o f (on @ i n place o f 0). We denote by 0 on 8 , and i f f E L (dwe have we l e t uE be the solution of ( 1 . 7 2 ) . Then i f $ u,(x) +. u ( x ) a t any point o f b , uniformly i n x.
g-
(*IWe
do not assume t h a t t h e forms a 6 and a d a r e coercive ( s e e Theorem 1 . 6 ) .
OPTIMAL STOPPING PROBLQMS
380
Proof.
&
v.1.1~
(CHAP. 3 )
We put
We p u t
G
I
8'-
d
G i s open).
(thus
With any s t o p p i n g time 8 we a s s o c i a t e
0.
B3'01
B4> 0
We have f u r t h e r
x e-a'P4t ,bPlCl
e-
h 1 < - 2tx ( 2
P
dc
and i f c > 1, t h e n applying H6lder's i n e q u a l i t y , once more it follows t h a t
(where c ' denotes t h e conjugate of c ) .
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP. 3 )
We assume t h a t
(3.1 59)
,
ca'tp
cb=p
;
I n view o f t h e f a c t t h a t
-a2 (14) 2t
r
n -
dF; 5 C
independent o f
t2 we o b t a i n
and t h e r e f o r e i f
(3.1 60)
alp4
>
2 2
we can l e t h t e n d t o 0 and
( c 8 - 1 ) A
(n+2) (K+48,
48~
(SEC. 3 )
389
OPTIMAL STOPPING : STATIONARY CASE
We can then l e t T tend t o + -; using an argument similar t o t h a t given e a r l i e r , we can show t h a t t h e i n t e g r a l s ( o r d i n a r y and s t o c h a s t i c ) converge. Moreover t h e We t h u s o b t a i n term r e l a t i n g t o I$ a l s o converges because I$ i s bounded.
The term r e l a t i n g We now t a k e t h e mathematical expectation ( w i t h respect t o P ) . From t h e i n e q u a l i t y Au 5 f, and t o t h e s t o c h a s t i c i n t e g r a l has expectation 0 . reasoning a s i n t h e proof of Theorem 3.1, we can show t h a t ub) I
Jx(e)
.
We now consider t h e stopping time
ex,
i n (3.162).
Since
Now
But t h e proof of Lemma 3.8 shows t h a t glu(yx(T)) /exp
- BT
-L
0
when
T*
+m
.
since Thus we can pass t o t h e expectation i n ( 3 . 1 6 4 ) and l e t T tend t o + -; jl i s bounded, t h e n , using Lebesgue's theorem and t h e preceding arguments, we obtain
OPTIMAL STOPPING PROBLEMS & V . I . ’ s
390
(CHAP. 3 )
A s i n Theorem 3 . 1 we can show t h a t
i n o t h e r words, s i n c e t h e opposite i n e q u a l i t y i s t r u e ,
It may be shown t h a t t h e method of proof of Theorem 1 . 1 2 (approxRemark 3.10. imation by bounded domains) d e f i n e s a convergent sequence. Suppose
OR =
{XI
I X I 4
we approximate (3.155) by
Thus uR i s given by t h e formula
n
e
where T~ denotes
We have
:
(SEC. 3 )
When R
391
OPTIMAL STOPPING : STATIONARY CASE
-
+=
, F{wl.rg 0 such that f(x) t y for 1x1 2 ro and suppose that R > r , We take x such t8at 1x1 2 R and we denote by T = T the exit time from Furthermore, as J, = O,x we have the :pen domain 151 161 > ro}.
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
S i n c e f i s bounded, we have f 2 - c .
J,(@) 2
E”
1:
Moreover s i n c e a
393 C M , we have
- M t dt .
f exp
We p u t
We have
We have
c
+-E
M
e
-Mt
(e-
1 -(t-QAt) E
-(Mt
- x t s e1
+ -1( t - e A t ) ) E
dt 2
-
YE
.
OPTIMAL STOPPING PROBLEMS & V.I.’s
394
( C W . 3)
I
1
Since T - AT i s also t h e e x i t time from t h e open domain(c 15 > r )corresponding t o t h e i n i t i a l s t a t e BAT), it follows from t h e s t r o n g Markov pro$erty t h a t we have
0 t h e open domain
O6 = If
(*)
~f i s
k
E6 1 d(S,aa) > 61
t h e e x i t time from
0 , we
The open domain 0 i s of c l a s s C
2
.
put for a l l 8
.
(SEC. 3 )
Now if
OPTIMAL STOPPING : STATIONARY CASE
~f
0 where f3 i s s u f f i c i e n t l y l a r g e ;
(3.206) We now have t o e s t i m a t e t h e right-hand s i d e of ( 3 . 2 0 6 ) . from I t o ' s formula
F i r s t l y , we have,
from which we deduce, i f f3 i s s u f f i c i e n t l y l a r g e r e l a t i v e t o t h e L i p s c h i t z c o n s t a n t s of g and u, t h a t we have
and we t h u s s e e t h a t
(3.208)
(*)
sup
tz 0
IY,(t)-yxl
This i s not a l i m i t a t i o n .
(t)
I 2 exp
- 2gt s c0Ix-x1 I 2
(SEC. 4 )
so
A s f o r (3.207), we see t h a t we have, f o r
(Yx(t)-Yxl(t) I4eXP
(3.209)
where
B"
411
OPTIMAL STOPPING : EVOLUTIONARY CASE
- 48t + 8"
B
sufficiently large
t /Y,(S)-y,,
(s) j4exp
- 48s d s s C2 1x-x' I4
> 0.
We w r i t e (3.209) f o r t h e R (where BR i s t h e f i r s t i n s t a n t a t which one o f t h e processes y x ( s ) , y x r ( s ) leaves t h e b a l l with c e n t r e 0 and r a d i u s R).
We can
-
then t a k e t h e mathematical expectation and make u s e of t h e f a c t t h a t t h e expecta t i o n of t h e s t o c h a s t i c i n t e g r a l i s zero. We next l e t R + -, t h u s B R + + a.s. Using F a t o u ' s theorem we o b t a i n
Furthermore, we have f o r T > 0 f i x e d
where t h e constant C/, does not depend on T ( s e e Chapter 2 , Theorem 2.3). We can then l e t T + + -, By v i r t u e of (3.210) and making use of t h e CauchySchwarz i n e q u a l i t y , we deduce from (3.208) t h a t
But t h e n , r e t u r n i n g t o (3.206), we r e a d i l y deduce t h a t lu(x) - u(x')I
5
c
Ix-x'
1.
We have thus shown t h a t under t h e assumptions (3.204), (3.205) t h e s o l u t i o n of t h e V . I . i s Lipschitz continuous ( i n t h e c a s e 8 1 R"). m
u
e#
I n t h e case R n , we surmise t h a t t h e r e s u l t (3.198) remains t r u e . The d i f f i c u l t y i s t h a t (3.203) i s no longer t r u e because t h e t r a j e c t o r i e s may be d i s t a n t (with a low p r o b a b i l i t y ) . I n t h e estimates ( s e e ( 3 . 7 3 ) , ( 3 . 7 4 ) ) , an a d d i t i o n a l term i n introduced of t h e form IX-X'I., t h i s term estimates t h e p r o b a b i l i t y t h a t
Ix-x'~~,
Choosing 6 = o f order
COntinuOuS
3.
we can t h e n show t h a t t h e f u n c t i o n
4.
OPTIMAL STOPPING-TIME PROBLEMS
4.1
SYNOPSIS
-
u
i s HUZder
EVOLUTIONARY CASE.
We now e n t e r upon t h e s u b j e c t of evolutionary problems.
Our aim here i s t o
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.’s
412
i n t e r p r e t t h e e v o l u t i o n a r y V.I.‘s i n v e s t i g a t e d i n S e c t i o n 2 . Since c e r t a i n r e s u l t s a r e v e r y s i m i l a r t o t h o s e o f t h e s t a t i o n a r y c a s e , we s h a l l o f t e n r e f e r Except f o r c e r t a i n c a s e s , we f o l l o w a p l a n back t o S e c t i o n 3 for t h e p r o o f s . s i m i l a r t o t h a t o f S e c t i o n s 2 and 3. 4.2
ReRular c a s e - bounded open domain
Let O b e a bounded open domain o f R n , w i t h
Q = d x ]o,T[,
C
=
r
x
-.
r
= a e o f class C
T
< +
lo,T[,
2
We c o n s i d e r f u n c t i o n s a i j ( x , t ) , a . ( x , t ) , a ( x , t ) s a t i s f y i n g
We p u t
(4.3)
u E b2”” US
+
on
(u-+)(-=+ dU
(Q)
(*) ( t h u s u E
’e, -E+ A(t) dt A ( t ) u-f)
uIC = 0, u(x,T) = E(x)
= 0
,
Co(q))
U S f a.e.
a.e.
on
Q
on
.
(4.4)
(*)
We r e c a l l t h a t
bJ2’’’’(Q)
, vxi
= {vlv,vt,vx
i
vx
E
j
Lpl.
Q
,
.
We p u t
4)
(SEC.
413
OPTIMAL STOPPING : EVOLUTIONARY CASE
lg(x,t)
(4.5)
Igl
I
- g(x',t)I
la1 s
c
+
la(x,t)
- a(xl,t)I
s
CIX-xfI
9
;
t h u s we can d e f i n e t h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n ( i n t h e s t r o n g s e n s e )
(4.6)
t h e s o l u t i o n of which i s denoted by y x t ( s ) or y ( s ) (when no c o n f u s i o n i s possible).
We d e n o t e by T
8 satisfying 8
E
xt
Z
T
For any s t o p p i n g t i m e
t h e e x i t t i m e from
[ t , T I , we p u t
(4.7)
We t h e n have t h e f o l l o w i n g theorem:
THEOREM 4.1, Under t h e assumptions (4.1), ( 4 . 2 ) , ( 4 . 3 ) is given by
problem
u ( x , t ) = I n f Jxt(e)
(4.8)
e
u of
t h e solution
.
Moreover there e x i s t s an optimal stopping time given by
Qxt = i d
(4.9)
ITZ
ST
,
t)u(yxt(s),s) = +(Y~~(S),B)~
where by convention we have put U(S,S)
= 0
for
F; $ 8 ,
E [o,T]
and where $ ( S , s ) i s extended by continuation outside 0, so that $ 5 do, s E [ o , T l .
2 0
for
T h i s i s similar t o t h e p r o o f o f Theorem 3.1, t h r o u g h a p p l i c a t i o n o f Proof, Actually, t h e t h e g e n e r a l i s e d I t o formula ( 8 . 4 8 ) , Theorem 8 . 3 , Chapter 2. proof h e r e i s s i m p l i f i e d by t h e f a c t t h a t we a r e working on a bounded h o r i z o n . rn We now g i v e t h e "non-homogeneous" a n a l o g u e o f t h e p r e c e d i n g theorem. b e a f u n c t i o n on 1, t h e t r a c e o f a r e g u l a r f u n c t i o n 0 , i . e .
(4.1 0)
h = GIz,
(E E W"P(0)) (*)
291 IP
0 E II,
where
(Q), h S
,
AS u s u a l we s p e c i f y a system
t
(Q,o,P,~ ,w(t)).
+Iz,
h(x,T)/ = E l r
r
Let h
9
414
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
and l e t
u
be a s o l u t i o n o f t h e V . I .
(4.1 1
which e x i s t s and i s unique (because u-0 i s a s o l u t i o n of a homogeneous problem We have t h e n t h e same r e s u l t s a s i n meorem 4 . 1 , t a k i n g of t h e type ( 4 . 3 ) ) .
B y way of ezuJi?ple, l e t u s w r i t e down t h e analogue o f formula ( 3 . 2 3 ) , ( t h i s being u s e f u l f o r t h e S t e f a n problem, s e e S e c t i o n 2 . 1 1 ) . !Fhus @ i s a r i n g , i t s boundary i s made up of two p a r t s r',"' and we suppose t h a t @; i s t h e open domain contained w i t h i n r ' ( t h e i n t e r i o r boundary),
.
6" = 6' u 8 w r ' We denote by rxt t h e e n t r y t i m e i n t o 6', and by r2 t h e xt exit time from 6". I f h = h, on Z' = I"x
(4.8) w i t h
]o,T[,
h = h 2 on
xt < T
T'
r2 < xt
T
Z" = P I x ]o,T[
, we
t h u s have
(SEC.
4)
OPTIMAL STOPPING : EVOLUTIONARY CASE
415
If a(x,t) denotes the temperature in the Stefan problem considered in Section 2.11, we then obtain
&,T-t)
- inf
= d
Jxt(e)
, &,o)
= io(x)
e
where J ( 8 ) is given by (4.13) with, referring back to the concepts presented x,t in 2.11:
f ( x , t ) = -go(X,T-t), h t ( x s t ) = h2(x,t)
I
0
.
dy = y2 dw 4.3. Extension I.
C”,
on
- j :-t
g(x,s)ds
on
,
Z’
,
9 = 0, ii = 0
Weakening of the assumptions concerning the coefficients.
We now assume that the coefficients of A(t) satisfy
(4.14)
Z aij Ci E j 2 ‘ij
= ‘ji
a Z
Ei2
,a >
0
v c,...c, ,
*
We use Girsanov‘s transformation as in Section 3.4.
(4.1 5 )
We define successively
dy = u(y(s),s)dw(s), s > t, y ( t ) = x,
(4.1 6)
4x t (s) = e(s) =
(4.17)
d T t = d p = dP exp[
w(8)
-
J’t
a’(yxt(h),A)g(y,,(h),h)dh
-+ J’T 1 0
o-~g(y(A),A).dw
,
-1 gl 2 ds]
,
We have THEOREM 4.2. (4.11)s a t i s f i e s
Under t h e assumptions (4.13)and (4.19) t h e solution u of u(x,t) = Inf Jxt(e)
e
416
(CHAP.
OPTIMAL STOPPING PROBLEMS & V . I . ' s
where Jxt(e) i s defined by (4.18). time given by (4.9) Proof.
4.4
Moreover there e x i s t s an optima2 stopping m
S i m i l a r t o t h a t o f Theorem 3.4.
Extension 11.
3)
Weakening of t h e assumptions concerning I$and 0
and
i n t e r p r e t a t i o n of t h e Denalised problem. We now assume t h a t t h e d a t a v a l u e s f ,
i, +I,0
satisfy
(4.20)
and h = 01
z*
I n S e c t i o n 2 we i n t r o d u c e d t h e concept o f a weak V . I . t h a t two equioalent f o r m u l a t i o n s were p o s s i b l e .
I n f a c t , we have seen
We put
We assume t h a t we have
(4.23)
y
nonempty.
We s h a l l term t h e weak soZution o f t h e V . I .
u (4.24)
-
-
2 E L (o,T;v) (g,v-u)dt
1
, us 4 +
s
a function
u
satisfying
a.e.
a(t,u,v-u)dt
V v E Y . We have seen i n Theorem 2.6 t h a t i f i n p a r t i c u l a r t h e assumptions ( 4 . 1 4 ) , ( 4 . 2 0 ) and ( 4 . 2 3 ) a r e s a t i s f i e d t h e n t h e r e e x i s t s a rnax~mwn weak s o l u t i o n , a g a i n denoted by u , t o save on n o t a t i o n . The second formulation i s as f o l l o w s : we c o n s i d e r t h e s e t o f f u n c t i o n s satisfying
(4.25)
w
(SEC. 4 )
417
OPTIMAL STOPPING : EVOLUTIONARY CASE
We have s e e n i n Theorem 2.27 ( * ) t h a t t h e s e t of f u n c t i o n s ( 4 . 2 5 ) a l s o p o s s e s s e s a maximwn element which coincides wi t h
w
satisfying
U.
Remark 4 . 1 Any weak s o l u t i o n of t h e V . I . s a t i s f i e s ( 4 . 2 5 ) ; however t h e conTo a c e r t a i n e x t e n t , i n f a c t , ( 4 . 2 4 ) t a k e s a c c o u n t of t h e verse i s not t r u e . " z e r o p r o d u c t ' ' c o n d i t i o n , w h i l s t ( 4 . 2 5 ) c e r t a i n l y does n o t t a k e t h i s i n t o 8 account. We have seen i n Theorem 2.14 t h a t t h e m a x i m u m s o l u t i o n b e l o n g s t o C o ( G ) ( s t i l l w i t h t h e assumptions ( 4 . 1 4 ) , ( 4 . 2 0 ) , ( 4 . 2 3 ) ) . As i n t h e s t a t i o n a r y c a s e , we can p r o v e t h i s r e s u l t i n a p r o b a b i l i s t i c manner and we c a n a l s o g i v e t h e i n t e r p r e t a t i o n o f t h e maximum s o l u t i o n .
THEOREM 4 . 3 . Under the asswrrptions ( 4 . 1 4 ) , ( 4 . 2 0 ) , ( 4 . 2 3 ) t h e u of the V . I . belongs t o C o ( Q ) . Moreover, we have
s o l u tion
maximwn weak
u ( x , t ) = Inf Jxt(9)
9
where J x t ( e ) is defined by (4.18) Proof.
T h i s i s similar t o t h e p r o o f of Theorem 3.5.
We n e x t i n t r o d u c e t h e p e n a l i s e d problem: f i n d u
8
satisfying
(4.26)
F o r any p r o c e s s v ( s ) , s (4.27)
J:t(~) =
E"
E
Ct,T1 adapted such t h a t v ( s )
J*rxtAT(f t
+a
+v)(yXt(s),s)(exp
-
E
S"
[0,11, we Put 1 (aO+Tv)dh)ds
We have THEOREM
4.4.
Under the asswnptions of Theorem 4 . 3 , we have
Moreover u
+
Inf ~
e
~
~ i n( c0(G) 9 )
( * ) We a r e u s i n g h e r e t h e r e s u l t o f t h e "nonhomogeneous" problem; t h e proofs are identical.
-~
( * * ) We r e c a l l t h a t P = PXt, which t h u s depends on x , t .
418
(4.29)
OPTIMAL STOPPING PROBLEMS & V.I.'s
f E LP(Q) @
,+
E la21'1P(Q)
, c E ~'(0) , Q I z S +Iz, p >%+ 1, E
( C H A P . 3)
cO(Q)
IJ, (x,T)
a.e-
4)
(SEC.
419
OPTIMAL STOPPING : EVOLUTIONARY CASE
Interpretation of the penalised problem. We begin by i n t e r p r e t i n g t h e p e n a l i s e d problem. We cannot a p p l y Theorem 4 . 4 , s i n c e t h e assumptions concerning U a r e not s a t i s f i e d . However, we have nevertheless
(4.33)
u ( x , t ) = Inf JEt(v) V
where
Ct(v) =
(4.34)
E”
s
TAT
t
+ E” h(dz))x,,T +
UN
E”
v)(exp
(fcr $
exp
J‘
-
s
s
t
1
(a + - v ) d h ) d s o r
1 (ao+ TV) dh
- j5
- ‘ exp
ii(Y(T))xT
t , we t a k e 6 > 0 such
( n o t e t h a t 6 depends on w ) .
-
,
The f u n c t i o n 6 b e l o n g s t o Co(b [ o , T a ] ) , which, due t o t h e uniform convergw on Q , a l l o w s u s t o show t h a t f o r € s u f f i c i e n t l y small we have ence o f w E
6
9's
-6
hence ( 4 . 5 0 ) . We now p r o v e t h a t
ext
i s optimal.
We may t a k e t < T and
w(x,t)
0 such t h a t Let w < $ - 6 ( x , t f i x e d , 6 i s n o t random b u t depends on x , t ) .
8
ext
= inf is E
lt,T] I
w(Y(s)) 2
(+-B)(Y(S))
8
- T1
.
OPTIMAL STOPPING PROBLEMS & V.I.'s
426
By virtue of the convergence of w
el',
6 ext
2
(CHAP
.
3)
to w in C o ( c ) , we have, as in Theorem 3.7
cs c 6
A T~~ for
Consequently, we have
Letting€
-+
0, and utilising the uniform convergence of w
wG(e6A
w(x,t) =
(4.52)
The sequence e 6 4 when 6
+
r))exp
t
y(t) =
and we w r i t e t h e Girsanov t r a n s f o r m a t i o n ( f o r a l l T )
d
P = dP exp
[
1;
dw(X)
a-’g(y(A),h)
which d e f i n e s a p r o b a b i l i t y measure on il
am( c f .
-Gjt
2
la-lgldh]
(3.28)).
We t a k e f u n c t i o n s f , JI s a t i s f y i n g
(4.75)
(4.77)
I
e-yt f E Lp(Q) ; J, E Co(p)
d U , v-u) - (=
v g
(*)
(I a . e .
+ a(t;u,v-u)
, e-Yt
J,
bounded,
2 (f,v-u) V v E V
in t.
We assume t h a t u and $ a r e extended by 0 o u t s i d e
8.
such t h a t
436
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
Proof.
ingful.
We show t h a t t h e i n t e g r a l on t h e r i g h t - h a n d s i d e o f We have Let I d e n o t e t h i s i n t e g r a l .
/I1S
E"
lTxtlf t
I
We d e n o t e by 'p t h e f u n c t i o n If
(yxt(s),s) exp
1
-
extended by 0 o u t s i d e
&
Thus
(4.79) From Lemma 3 . 1 we have
Thus i f we w r i t e y
J S E
= y1 + y, we have
1;
c;
n/a n'a
'p(s-t),
(8-t)
and from HBlder's formula we o b t a i n
Now
Thus
-y, (s-t)
-u(s-t) e
ds
( 4 . 7 6 ) i s mean-
(SEC. 4)
and if
OPTIMAL STOPPING
is such that
y
1
y1
>
:
EVOLUTIONARY CASE
2
(4.80) For the same reason
is well defined and, from Ito's formula, we have
Since
$Iz
= 0, we see that we may write
and from (4.80),we see that we have, for all 8 (4.82)
We put ?=f+$-AJ,
.
We consider the sequence of V.I.'s on a finite horizon
(4.83)
437
438
OPTIMAL STOPPING PROBLEMS & V . I . ' s
(CHAP. 3 )
We can apply Theorem 4.2 and we t h e r e f o r e have
It i s c l e a r t h a t zn C ( w i t h x , t f i x e d ) .
Furthermore a s i n ( 4 . 8 2 ) we have
(4.84) and t h u s zn J. z. We now show t h a t n e c e s s a r i l y we have
In fact i f 8
2
t , 8An i s a d m i s s i b l e f o r t h e problem on t h e horizon [ o , n l ; t h u s
But
which i s i n t e g r a b l e ( w i t h r e s p e c t t o P ) . obtain (since 8 i s arbitrary)
(4.86)
lim sup z
Furthermore T~~ t h e r e e x i s t s 8n
(*)
This V . I .
n
I5
S
Inf E
e 2t
Thus t a k i n g t h e l i m i t when n
f (exp
-
1;
a. dh) d s
-f
-, we
.
i s a . s . f i n i t e ( s i n c e B i s bounded) ( * * ) and from Theorem 4.2, such t h a t
i s a v a r i a n t of t h e V . I .
considered d u r i n g t h e proof o f Theorem 2.16.
( * * ) The proof i s s i m i l a r t o t h a t i n t h e s t a t i o n a r y c a s e , c f . Theorem 7.1.
(SEC. 4)
439
OPTIMAL STOPPING : EVOLUTIONARY CASE
(4.87)
Moreover
Since z
zn (yx t
I
n C
(T),T)
= 0 and s i n c e we can always d e f i n e z
= o and t h u s en
= o f o r s > n , we have
2 T.
Moreover t h e sequence 8 i s i n c r e a s i n g , s i n c e z is decreasing. ye can *thus proceed t o t h e l i m i t i n (4P87) by v i r t u e o f L e b e s g t e ' s theorem. I f 8= l i m e n , we obtain
z(x,t) =
;1
s^
(exp
1;
-
a. d h ) ds
and t h u s
which along with
(4.86) proves ( 4 . 8 5 ) .
We n o t e t h a t 8 5
T.
We now show t h a t we have
z E
(4.88)
CO(P)
.
F i r s t we s h a l l prove ( 4 . 8 8 ) asswning t h a t f E . & ( Q ) , Let
z c ( x , t ) = Inf
s"
P
We know t h a t z z En
E
Co( y. I n f a c t , i f we r e t u r n t o t h e e s t i m a t e (4.79) we have
We can improve (4.78) by u t i l i s i n g an i t e r a t i v e method a s i n t h e Remark 4.7. s t a t i o n a r y case, i . e . we s o l v e , f o r a s u i t a b l y chosen y > 0 ,
442
OPTIMAL STOPPING PROBLEMS
&
(CHAP. 3 )
V.I.'s
The sequence un i s monotonically convergent. We l e a v e t o t h e r e a d e r t h e t a s k of d e f i n i n g t h e p r e c i s e c o n d i t i o n s o f v a l i d i t y .
4.9.
Stopping-time problems i n Rn - bounded c o e f f i c i e n t s . We assume t h a t t h e c o e f f i c i e n t s s a t i s f y
We t a k e Q = Rn x lo,T[.
(4.92)
a E ij
c' (Q),
a . . bounded, 1J
da
ba..
2 bounded, 2 Bxk dt
bounded ;
/ a i measurable and bounded, a. measurable and bounded; We t a k e f u n c t i o n s
(4.93)
U,
f , JI s a t i s f y i n g
f E LP(O,T ;
J, bounded,
L~'~(R")) ;
JI E
Co(a,- $+
b(t)J, E LP(o,T ; L p P p ( R n ) )
4 E L P ( ~ , T ; w " P ' ~ ( R ~ ) ) ; ii E &PPP(R~) ; iis
A s i n S e c t i o n 3.9.1, we have
where
dai , Bt
+
(TI
.
,
da
2 bounded. at
(SEC. 4 )
OPTIMAL STOPPING : EVOLUTIONARY CASE
443
We can also g i v e a n i n t e r p r e t a t i o n o f t h e weak s o l u t i o n s . L e t u s now c o n s i d e r t h e problem o f t h e s u p p o r t s ( s e e Theorem 2 . 1 8 ) . c o n s i d e r t h e p a r t i c u l a r c a s e i n which
(4.97)
We
.
A ( t ) = - A , $ = O
u
We t h u s c o n s i d e r t h e s o l u t i o n
of
U S O + rd U+ A U + f 2 0
(4 .gel
dU
,
u + f) = 0
u (=+A
.
u(x,T) =
t
a Z C
i jx [o,T]) =
ai j E C2"(Rn
dXkdXL
C 5
3)
Is
[=bI f
c
(1
Ic (1
+ +
Then, t h e i n i t i a l c o n d i t i o n
Ix
1")
1x1")
u
sa
For t h e s t u d y of t h e s t r o n g V.I., we s p e c i f y r e g u l a r i t y a s s u m p t i o n s c o n c e r n i n g t h e o b s t a c l e III. namelv
c o n t i n u o u s and bounded on
I
.
;(x) S 4J ( x , T ) We f i r s t c o n s i d e r t h e p e n a l i s e d problem, i . e .
G,
(SEC.
4)
(4.1 20)
u E L2 ( bu
-C bt
L
2
0 , ;~) 'H ( 0 , ;~
n
LP(O,T ; w21P,P)
n
L):
bU
u S f
(-=+ au
A ( t ) u-f)(u-J,)
--+A(t) bt
We have
451
OPTIMAL STOPPING : EVOLUTIONARY CASE
9
LP(O,T ; ~p9I.r)
, ,
USJ,
= 0, u ( x , T ) =
.
OPTIMAL STOPPING PROBLEMS & V.I.'s
452
(CHAP.
3)
Naturally we have a l s o
-
We now weaken t h e r e g u l a r i t y assumptions concerning J I , f , u. We assume
(4.1 24)
JI uniformly continuous and bounded on Q f
E
f bounded,
Co(G),
f Co(Rn),
bounded.
By r e g u l a r i s a t i o n , as i n t h e proof of Theorem 3.23, we can show t h a t we again have ( 4 . 1 1 7 ) , t h i s time with
Furthermore, i f we put
(4.1 26)
w(x,t) = Inf
e
~~,(e)
we have, as i n t h e proof of Theorem 3.7,
(4.1 27)
u
+.
w uniformly on
61
During t h e proof of Theorem 2.20, we showed t h a t u 2 domain of L (o,T ; ) 'H so t h a t w E L2 (o,T ; H 1a ) and
u
+
remains w i t h i n a bounded
2 1 w weakly i n L (o,T ; H a ) .
A s has already been s a i d i n Section 2.17, we do not know i f w = u, t h e p o s s i b l e maximum s o l u t i o n of t h e weak V . I . (2.235), ( 2 . 2 3 6 ) , ( 2 . 2 3 7 ) . Nonethel e s s , as a consequence of t h e proofs a l r e a d y given, t h e r e e x i s t s an optimal stopping time f o r t h e problem (4.126)
4.11.
Problems which a r e p e r i o d i c i n
t.
We adopt t h e conditions used i n Section 2.19, and i n o r d e r t o c l a r i f y t h e conc e p t s we consider more p a r t i c u l a r l y t h e case of s t r o n g s o l u t i o n s , i . e . Theorem We extend t h e c o e f f i c i e n t s and t h e d a t a values p e r i o d i c a l l y over 2.24. Let ( 0 , + m), with a period T , as w a s done a t t h e end of Section 2.19.
i,?, $
b e t h e s e p e r i o d i c extensions.
Let u be t h e s o l u t i o n of t h e V . I . on ( 0 , ce) ( i . e . with i n f i n i t e horizon corresponding t o 8 , $ such t h a t
a,
(SEC. 4 )
OPTIMAL STOPPING : EVOLUTIONARY CASE
453
(4.1 28) We have:
(4.129)
the solution t o (o,T).
with period T coincides with t h e r e s t r i c t i o n of
u
There immediately follows from t h i s a probabilist i c i n t e rp re t a t i o n of t h e s o l u t i o n of t h e p e r i o d i c V . I . , i n t h e form of an optimal stopping-time problem on an i n f i n i t e horizon, ( s e e Section 4 . 8 ).
4.12.
The p r i n c i p l e of s e p a r a t i o n f o r stopping-time problems.
4.12.1.
Introduction.
Up t o now, we have always assumed t h a t t h e evolution of t h e system could be A s i n d i c a t e d i n Remark 3.1, we s h a l l now conobserved without r e s t r i c t i o n s . We begin with s i d e r cases where t h e observation of t h e system i s incomplete. some review m a t e r i a l on t h e f i l t e r i n g of l i n e a r systems. We consider t h e m a t r i c e s :
(4.1 30)
F(t) E L(Rn ; Rn), G ( t ) E &(Rm
We assume t h a t f o r t
E
;
Rn), H ( t ) E &(Rn ; # )
.
Co,TI, we have
We denote by y ( s ) t h e s o l u t i o n of t h e l i n e a r Ito equation
(4.1 32)
d y ( s ) = G(s)
Y(t) =
Y(S) d s
+
G(s)
dW(S)
9
5
> t
= +5
where w i s a standardised Wiener process and 5 i s a centred Gaussian R.V. and w ( s ) w ( t ) i s independent of 5. covariance matrix P 0’
-
with
A s u s u a l , y ( s ) i s t h e system s t a t e a t t h e i n s t a n t s .
However, i n c o n t r a s t t o what we have always assumed h i t h e r t o , we suppose now We consider an observation t h a t we can no longer observe t h e evolution o f y ( s ) . process, z ( s ) = z ( s ) defined by 0 .t
(4.1 33)
dz(s) = H(5) Y(S) ds
+
dq(s)
,
s> t
z(t) = 0
( * ) This assumption of t h e independence o f but it s i m p l i f i e s t h e p r e s e n t a t i o n .
and w i s not s t r i c t l y necessary,
454
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ’ s
We d e n o t e by : Z t h e a - a l g e b r a g e n e r a t e d by z(A),
X
E
“c,sl.
We p u t
3Sxt(~)
(4.1 35)
= E[Y(~)
I
zij
r
S S
t
We n o t e t h a t 9 ( t ) = x, s o t h a t t h e n o t a t i o n i s c o n s i s t e n t . xt Equation (4.138) i s a linear I t o e q u a t i o n ; t h u s E ( s ) i s a Gaussian p r o c e s s . Furthermore, u s i n g t h e w e l l known p r o p e r t i e s o f c o n d i t i o n a l e x p e c t a t i o n s (See Chapter 2 , (l.g)), we have
(4.136)
S(t)
+ P(s)B,(s) R-l ( s ) ( d z ( s )
= F(s)f(s)ds
dy’(s)
= x
I
.
- H(s)f(s)ds)
The matrix P(s) is t h e s o l u t i o n of t h e Riccati equation
" Equation ( 4 . 1 3 7 ) p o s s e s s e s one and only one s o l u t i o n . It i s convenient t o i n t r o d u c e h e r e t h e p r o c e s s E ( s ) = E E
The p r o c e s s E ( s ) i s termed the estimation error. (4.136) we s e e t h a t E i s a s o l u t i o n of
= 5
,t
( s ) s p e c i f i e d by
- P(s).
( s ) = Y(S)
E(t)
0
By d i f f e r e n c i n g (4.132) and
.
Equation (4.138) i s a linear I t o e q u a t i o n ; t h u s € ( s ) i s a Gaussian p r o c e s s . Furthermore, u s i n g t h e w e l l known p r o p e r t i e s o f c o n d i t i o n a l e x p e c t a t i o n s ( s e e Chapter 2 , (l.g)), we have
(4.139) In addition,
(4.140)
We have t h e f o l l o w i n g p r o p e r t y :
(4.1 41
v ( s ) is
a standardised Wiener process and
a :Z martingale. I n f a c t , from ( 4 . 1 4 0 ) we f i r s t have
v(s) =
1;
3
and s i n c e ?(A)
X
1:
R3(h)
R3(A) dz(h)
(A)dh
B
-
+
1
R3(A)
R 3( A ) dq (A) H (A) f(h) dh
i s Z -measurable, it i s c l e a r t h a t v ( s ) i s a n a d a p t e d p r o c e s s . t
(SEC.
4)
455
OPTIMAL STOPPING : EVOLUTIONARY CASE
F u r t h e r m o r e , we have
A1 f o r A1 2 X 2 ,
which i s independent o f Zt S L X
for all
s i n c e € ( s ) i s independent o f Z;‘,
1’
Thus v ( s ) i s c l e a r l y a 2 s m a r t i n g a l e .
t
We now show t h a t u ( s ) i s a s t a n d a r d i s e d Wiener p r o c e s s . Gaussian p r o c e s s , v ( s ) i s a l s o a Gaussian p r o c e s s .
S i n c e e is a
It i s c l e a r t h a t E v(s) = 0. Thus it now remains t o c a l c u l a t e t h e c o v a r i a n c e matrix ?f v ( s ) . To do t h i s we d e f i n e c ( s ) by
We o b s e r v e t h a t r , ( s ) i s a s,tandardised Wiener p r o c e s s w i t h v a l u e s i n
Rn x Rp. From ( 4 . 1 3 8 ) , ( 4 . 1 4 0 ) we s e e t h a t t h e p a i r (e ( s ) , v ( s ) ) i s a s o l u t i o n o f t h e I t o equation
(4.142)
= Q ( 5 ) ds
+Y
,
dS
with
=(R3F
- HP F R-l
H
0 ),L(Gg - p p RI 4 ) .
If we d e n o t e by n(s) t h e c o v a r i a n c e m a t r i x o f t h e v e c t o r t h e unique s o l u t i o n o f t h e e q u a t i o n
t)
,
then
li
.
(4.143)
ff=~a+xw+YiY,LX(O)=O
But we can w r i t e
*
Now v ( s ) i s Zs-measurable
t
If we expand e q u a t i o n
ff 1 = (F 1
,
and
E
( s ) i s independent of Z,:
(4.143) we o b t a i n
- P P R-’
L (0) = 0
and
[
H) L,
+ x1
(P
- F R-lQ)
+
so t h a t v12 = 0.
GG*
+PP
R-l
EF
,
is
OPTIMAL STOPPING PROBLEMS & V.I.'s
t q= I hence we deduce t h a t
, X&O) 'TI
1
,
= 0
= P (which we a l r e a d y know) and ' T I ~ ( =s )I
completes t h e proof of ( 4 . 1 4 1 ) . The process
I(s) = z(s)
m
It"
-
H(A) 9 ( A ) dA
i s c a l l e d t h e Innouatkm process.
I(s)
I
j;' R
which
s,
Thus we have, from (4.140)
.
(A) dv ( A )
Moreover, t h e f i l t e r g ( s ) i s , according t o ( 4 . 1 3 6 ) , a s o l u t i o n of t h e I t o equation dy(s) = F ( s )
(4.1 44)
S(t) = x 4.12.2.
.
9 ( s ) ds
+ P(s)
F ( s ) R 3( 8 ) d v b )
Optimal stopping-time problem.
We put Q = Rn x l0,TC and we t a k e f , $,
s a t i s f y i n g t h e assumptions (4.113),
( 4 . 1 1 4 ) , (4.115).
Let 8 be a stopping time w i t h respect t o the f a m i l y ,Z: W e put
(4.145)
,
Ie
Jxt(e) = E
+E
r~ (Y(e,e)
+E
u
(y(T))
f(y(s),s)
xeCT
xeXT
exp
E
Ct,Tl.
ds
- e(8-t)
exp exp
- B(s-t)
8
- B(T-t)
A s u s u a l , we endeavour t o c h a r a c t e r i s e t h e f u n c t i o n
(4.146)
w(x,t) = Inf Jxt(e).
e
The ( e s s e n t i a l )difference from the problems considered up t o now i s t h a t adapted t o the observation and no longer t o the system s t a t e . We put
(4.147)
-
f ( x , s ) = E f(x
+
E(s),s)
+(X,S) = E +(x
+
E ( ~ ) , s )
I I
9
We t h e n consider t h e d i f f e r e n t i a l o p e r a t o r
where
= E u(x
+
E(T))
.
e
is
(SEC. 4 )
457
OPTIMAL STOPPING : EVOLUTIONARY CASE
We s h a l l l a t e r g i v e s u f f i c i e n t
so t h a t A(s) i s a non-degenerate o p e r a t o r . conditions f o r (4.150) t o be s a t i s f i e d . We introduce t h e V . I .
( s e e (4.120))
u E L 2 ( t , T ; Hi)
(4.1 51)
n
-dS 2+ Z(s)
us P
du
u(T)
u +p
*
+ A(s) u + p u
.
I
,
LP(t,T ; $lppir)
d t E L 2 ( t , T :):L
(-
-".=
n
I
-.-,-
ii -
LE(t,T ; Lptp) for
,
,
s > t, u 5
- :)(uG)= 0 ,
s
J, > t
-
Since f , $, u possess t h e same p r o p e r t i e s as f,$,z, t h e V.I. (4.151) f a l l s i n t o t h e category of V . I . ' s with unbounded c o e f f i c i e n t s (with l i n e a r growth for t h e f i r s t - o r d e r c o e f f i c i e n t s ) considered i n Section 4.10. Thus (4.151) possesses one and only one s o l u t i o n . We put
(4.1 52) We s h a l l now prove t h e following theorem: THEOREM 4.12. (4.150) we have
(4.1 53)
Under the a s s m p t i u n s ( 4 , 1 1 3 ) , (4.114), u(x,t)
3
w(x,t)
, vx
where w ( x , t ) i s defined by (4.146). optimal stopping t h e f o r (4,146).
E Rn
Moreover
(4.115) a n d ( 4 . 1 3 1 ) ,
, ext as defined by
( 4 . 1 5 2 ) i s an
-
Remark 4.12. The V . I . (4.151) is posed on Ct,Tl, because t h e o p e r a t o r N a t u r a l l y t h e choice of A(s) depends on P(s) which is only defined on C t , T I . t is arbitrary. 8 Remark 4.13. The assumption (4.150) i s very r e s t r i c t i v e as r e g a r d s a p p l i c ations. I n f a c t i f (4.150) i s s a t i s f i e d , then n e c e s s a r i l y H(s) i s i n v e r t i b l e . Now i f we r e f e r back t o (4.133) t h i s imposes a severe r e s t r i c t i o n on t h e c l a s s of observation processes. I f we want t o weaken t h e assumption (4.150), we s h a l l These w i l l be considered i n need t o be a b l e t o handle degenerate V . I . ' s . Volume 2 . a Remark 4.14. Theorem 4.12 expresses t h e principle of separation f o r stoppingtime problems. I n g e n e r a l , when we have a s t o c h a s t i c c o n t r o l problem ( e i t h e r with continuous decision v a r i a b l e , or of t h e stopping-time v a r i e t y ) where t h e observation i s incomplete ( i . e . we cannot observe t h e evolution of t h e s t a t e ) , we say t h a t t h e p r i n c i p l e of s e p a r a t i o n i s s a t i s f i e d i f t h e optimal d e c i s i o n a t each i n s t a n t can be obtained s o l e l y from a knowledge o f t h e b e s t e s t i m a t e f(s) of t h e Thus i n our problem, t h e d e c i s i o n a t each i n s t a n t i s t o s t a t e at that instant. Theorem 4.12 says t h a t we s t o p i f f(s) i s know whether or not we a r e t o stop. The term separation s i g n i f i e s t h a t we have o u t s i d e a c e r t a i n continuation s e t .
458
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
broken down t h e p r o c e s s o f o b t a i n i n g t h e optimal d e c i s i o n i n t o two d i s t i n c t ope r a t i o n s ; on t h e one hand an e s t i m a t i o n o p e r a t i o n which p r o v i d e s t h e f i l t e r , and on t h e o t h e r hand an o p t i m i s a t i o n o p e r a t i o n f o r a problem w i t h complete informa t i o n (where t h e system s t a t e i s t h e f i l t e r ) . N a t u r a l l y t h i s p o s s i b i l i t y of s e p a r a t i o n i s very f a r from e x i s t i n g g e n e r a l l y .
- . - -
Proof of Theorem 4.12.
We f i r s t assume t h a t f , a r e bounded. Then f , $I, I t t h e n f o l l o w s t h a t u i s bounded and continuous on
a r e a l s o bounded.
Then i f 8 i s a s t o p p i n g time w i t h r e s p e c t t o ,Z: Rn x [ t , T 1 . formula g i v e s
8
E
E
[t,TI, Ito's
t h u s from ( 4 . 1 5 1 ) we s e e t h a t
(4.1 54)
But we have
and
since
xe ( s )
i s Zs-measurable.
Furthermore we have
t
Y(S)
=
f(s) +
E(S)
thus
E(f(f(s)
+ &(s),s)
1
=
2:
f"
(f(s),s)
s i n c e Q(s) is Zs-measurable a n d € ( s ) i s independent o f Z:
t
Consequently we have
(4.155)
E
J
f " ( f ( s ) , s )e-8(s-t)ds
P
E
We s h a l l now show t h a t we have
(4.156)
E
y(Y(e),e)
For kN = T-t
(N t 'k0
-@(e-t) = E
x ~ e( ~
+-a)
1
I
f(y(s),s)
( s e e Chapter 2 , ( 1 . 2 5 ) ) .
8-
$(s-t)
\Y"(St(e),e) xe 0, arbitrary.
-
Furthermore, t h e process y ( s ) i s a Gaussian process with expectation y x t ( s ) d7 where y ( s ) i s a s o l u t i o n of -= F(s) 7 , y ( t ) = x and with covariance matrix C(s) ds where dZ -= ds
F Z
+
Z F*
+
GG*,
C(t)
= Po
.
Thus
( * ) Since Z(s) i s i n v e r t i b l e f o r a l l s 2 t , t h e i n t e g r a l i s not s i n g u l a r . Here t h e r e i s an important d i f f e r e n c e from t h e case o f complete observation, where there is a singularity for s = t . We again encounter t h i s s i n g u l a r i t y i f Po = 0 , a case which we have avoided.
(SEC. 4 )
w i t h a similar upper bound f o r t h e term i n
.
fJ,
.
uJ which
461
OPTIMAL STOPPING : EVOLUTIONARY CASE
u.
-
We approximate f , u by f u n c t i o n s
a r e r e g u l a r and with compact support such t h a t
fJ
-
f
in
LF”(Q)
and
CJ
-u
in
LF”(Q)
.
Observing t h a t ( w i t h an obvious n o t a t i o n )
and t h a t
(from t h e theorem concerning and t h a t uJ + u , f o r example, weakly i n L p ” ( Q ) s t a b i l i t y , with r e s p e c t t o t h e d a t a v a l u e s , of s o l u t i o n s o f V . I . ’ s ( s e e S e c t i o n 2 . 2 2 ) , we o b t a i n ( 4 . 1 5 3 ) . I t remains t o prove t h a t (4.152) a g a i n d e f i n e s an optimal s t o p p i n g time. However reasoning as f o r Theorem 3.19, we s e e t h a t we a g a i n have t h e r e l a t i o n ship
u(x,t) = E
f ’ f” ($(s),s)
ds
-
I n a d d i t i g n , t h e r e l a t i o n s ( 4 . 1 5 5 ) , (4.157) which a r e t r u e for fj, uj, a r e again t r u e f o r f , u by proceeding t o t h e l i m i t . We t h e n show a s above t h a t we a g a i n have
which completes t h e proof o f t h e theorem.
8
To end t h i s s e c t i o n we g i v e a s u f f i c i e n t c o n d i t i o n f o r (4.150) t o be s a t i s f i e d . We assume
(4.1 61
H invertible,
t h e n (4.150) i s s a t i s f i e d . ible.
Po i n v e r t i b l e ;
It i s a c t u a l l y s u f f i c i e n t t o show t h a t P ( s ) i s i n v e r t -
However we can re-express (4.137) i n t h e form
dP -=
ds
P(t)
(F t
- PH+ R - ~ H ) P+ PW-
Po
.
I f we put
f’ = F -
PHX R-’
H
HX R-~EP)+
GC*+ PHX R-’
HP
,
462
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
s 2 ) i s t h e fundamental matrix o f F , i . e .
and i f A (sl,
di(s,t)
= F(s) x(s,t)
ds then
P(s) = 2
x
Po X*(s,t)
(S,t)
,s>
1(s,a)(GGI+
+
ii (s,t) Po i* b , t ) 2
t , n"(t,t) = I
c 1
,c>
P F R-'HP)(a)
0
b e c a u s e A ( s , t ) i s i n v e r t i b l e as w e l l as P o , hence ( 4 . 1 5 0 ) .
5.
I* (s,o)da
m
STOCHASTIC DIFFERENTIAL GAMES W I T H STOPPING TIMES Synopsis
Here we c o n s i d e r o p t i m a l stopping-time problems w i t h two p l a y e r s . The game i s This b r i n g s us t o t h e s u b j e c t o f V.I.'s w i t h b i l a t e r a l c o n s t r a i n t s , The g e n e r a l c a s e o f non-cooperative a l r e a d y t o u c h e d on i n S e c t i o n s 1 . 1 4 and 2 .1 8 . games w i t h N p l a y e r s , N > 2 , does n o t l e a d t o v a r i a t i o n a l i n e q u a l i t i e s , b u t t o q u a s i - v a r i a t i o n a l i n e q u a l i t i e s . We s h a l l s t u d y t h e s e i n Volume 2.
a zero-sum game.
5.1 5.1.1
The s t a t i o n a r y c a s e
Assumptions
- notation - statement
of t h e problem
We t a k e c o e f f i c i e t s a . . ( x ) , g ( x ) d e f i n e d on sukd t h a t
s e t of Rn o f c l a s s C',
e,where B i s a bounded open sub-
We assume t h a t a and g a r e extended a p p r o p r i a t e l y over R n , i n such a way a s t o p r e s e r v e t h e i r p r o p e r t i e s . We w r i t e
(5.3)
2
a =
2
.
We d e f i n e t h e second-order d i f f e r e n t i a l o p e r a t o r
(5.4)
Let t h e r e be
A = - Z
a2 a..--Zgi
ij
a 2 0
We t a k e f u n c t i o n s f ,
1J b X i d X
J
i
. q1,
$,
on O s u c h t h a t
b b.r i
(SEC. 5 )
(5.5)
STOCHASTIC DIFFERENTIAL GAMES
+z
+1 L f
,
E LP@)
f
-A+,
on
,
Gi E co(8) 0 ,
- W2S C
,
+, s
,
ACi E LP(b)
+' lr I0 5 +*1r f -A+l
+,
463
p
n
> T;
(r =do) ,
-a+, 2
-C
u ,
,
ulr = o
,
We s h a l l prove t h e e x i s t e n c e and u n i q u e n e s s o f t h e s o l u t i o n
u
(5.6)
u E w2,P(0)
,
A u c a u ~ f if
US
JI1 0 . E Co(&),
we put S ( y ) = z where z i s t h e unique s o l u t i o n of
Az + az
(5.1 2 )
+ -22
z/r = 0
= f
,
z
- &CJ+
+
.
E W2*P(S)
+-+, 1- + 1
2 '9
9
The e x i s t e n c e and uniqueness of t h e s o l u t i o n of ( 5 . 1 2 ) follow from (3.133). have a l s o seen ( c f . ( 3 . 1 3 4 ) ) t h a t we have
for a l l x
E
&.
We have t h u s defined a mapping S from C o ( & )
+
?"(S).
We now show t h a t S i s a c o n t r a c t i o n on Co(&). (pl, p ',
E
We
C0(&
and z1,z2
corresponding t o
Q 1, cp2 .
I n f a c t , suppose we have We have, from (5.13)
We denote by X t h e q u a n t i t y i n b r a c k e t s on t h e right-hand s i d e of have
(5.14). We
STOCHASTIC DIFFERENTIAL GAMES
and t h u s
Consequently we deduce from
(5.14) t h a t
and s i n c e a > 0 , it can be seen t h a t S i s a c o n t r a c t i o n . But i f u
i s a f i x e d point o f , S , it i s c l e a r t h a t uE i s a s o l u t i o n of ( 5 . 7 ) .
We have t h u s proved t h e e x i s t e n c e of a s o l u t i o n of ( 5 . 7 ) . We s h a l l now prove t h a t any s o l u t i o n o f ( 5 . 7 ) s a t i s f i e s t h e r e l a t i o n ( 5 . 1 1 ) . This w i l l prove and supplement t h e proof of t h e theorem. We put
(5.
(5.
We s h a l l now prove t h a t we have
f o r all vl, v2.
It i s c l e a r t h a t t h i s e s t a b l i s h e s t h e d e s i r e d r e s u l t .
Suppose t h a t vl, v2 a r e a c t u a l l y two a r b i t r a r y c o n t r o l s . we have, s i n c e us ( Y ( T ) ) = 0 ,
(*) J, 1
From t h e I t o formula,
We put u = 0 o u t s i d e @ a n d we also have $, extended by c o n t i n u a t i o n with 0 outs?de 6 . S i m i l a r l y f o r G2 with J, 2 0 2
.
466
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP.
3)
From e q u a t i o n ( 5 . 7 ) it t h e n follows t h a t
and f u r t h e r
From t h i s l a t t e r r e l a t i o n s h i p it i s easy t o deduce ( 5 . 1 7 ) . c l e a r l y proved t h e theorem under t h e assumption cr > 0. We now c o n s i d e r t h e g e n e r a l c a s e . s o l u t i o n of problem ( 5 . 7 ) w i t h
c1
But t h e e q u a t i o n
I
changed t o
a+n.
n
> 0 and l e t u:(x)
We t h u s have
J
from which it follows t h a t
lu;(x)
Suppose we have
We have t h e r e f o r e
C
,
a c o n s t a n t independent o f
n.
be t h e
(SEC. 5 )
467
STOCHASTIC DIFFERENTIAL GAMES
i m p l i e s , s i n c e t h e r i g h t - h a n d s i d e i s bounded i n d e p e n d e n t l y o f
-
n,
that
L e t t i n g rl t e n d t o 0 , we e a s i l y s e e t h a t
'u where u solutio:
E
u
E
weakly i n
W2'p(0)
and s t r o n g l y i n C " ( 6 )
i s a s o l u t i o n o f ( 5 . 7 ) . It can t h e n b e shown as p r e v i o u s l y t h a t any of ( 5 . 7 ) s a t i s f i e s ( 5 . 1 1 ) , hence t h e r e s u l t .
Remark 5 . 1 Theorem 5 . 1 i s v a l i d under l e s s r e s t r i c t i v e assumptions on t h e The o t h e r assumptions f u n c t i o n s $. ; f o r i n s t a n c e , t o c l a r i f y i d e a s , Gi € Co(3) o f ( 5 . 5 ) w i h o n l y come i n t o p l a y when E + 0.
.
5.1.3
Solution of the i n e q u a l i t y .
We s h a l l now l e t E t e n d t o 0 , prove t h e e x i s t e n c e and u n i q u e n e s s o f t h e s o l u t i o n o f ( 5 . 6 ) , and g i v e a n i n t e r p r e t a t i o n o f t h e problem s o l v e d . We b e g i n by g i v i n g a v a r i a t i o n o f t h e f o r m u l a ( 5 . 1 1 ) , which i s u s e f u l i n what f o l l o w s . By v i r t u e o f t h e r e g u l a r i t y o f t h e f u n c t i o n s Jli ( a s s u m p t i o n ( 5 . 5 ) ) , we can w r i t e
We t h e n p u t
(5.18)
We t h e n have
(5.19)
460
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP. 3)
(5.20)
Then we have, s i n c e
S Q2 and
J,
lr
2 0
- ( a t +T lt (vl+ v 2 ) d e ) ) d t v2 1 Max Exl g2{exp - ( a t +.$ exp v,ds]dt 1 C Max EX {: (exp - ( a t ++ exp - 4 J: v l d s ) d t
Min
ax E"
5;
V
V
s
J"
g2iexp
1
C - 5 1 a +-
C L .
We t h u s o b t a i n 5 C
(5.22)
independent of
L
Furthermore, s i n c e
and t h u s
J122J,, and
JI
lr
0
,
x
and
E
.
(SEC.
5)
469
STOCHASTIC DIFFERENTIAL GAMES
(5.23)
-= -
E
C
independent o f
x
and
E
But from t h e e s t i m a t e s ( 5 . 2 2 ) , (5.23) and t h e e q u a t i o n ( 5 . 7 ) we e a s i l y deduce
(5.24)
1 C
IIUEII
W2,
independent o f
S i n c e t h e i n j e c t i o n mapping from W 2 3 p ( ~ ) i n t o
4'p(e)c Co(b),f o r p > $, we s e e t h a t , u From (5.221,
-
E
(0)
.
W' "(19) i s compact, and s i n c e
a f t e r e x t r a c t i n g a subsequence, we have
u weakly i n W2"(0)
and s t r o n g l y i n
co(8).
( 5 . 2 3 ) we deduce
X
X
and t h u s
so t h a t J, S
1
V S G2 a . e . on 8.
We have ( A U ~+
auE)(v-uE) = f(v-u ) --(uE1 E
+ 3 U E - JI,
s.
I n t e g r a t i n g over
E
)'(V-Uc)
6 and p r o c e e d i n g t o t h e l i m i t i n (AU
+ au-f)(v-u)dx z o
From t h i s we e a s i l y deduce t h a t
J,~)+(V-U~)
.
E,
we o b t a i n
OPTIMAL STOPPING PROBLEMS & V.I.'s
470
a.e.
(AU
+ au-f)(v-u) 4Jl ( X I 5
v
v
2 0
s
v
(CHAP. 3 )
r e a l such t h a t
4J2(x)
hence a l s o t h e r e l a t i o n s ( 5 . 6 ) . We have t h u s demonstrated the existence of a s o l u t i o n o f ( 5 . 6 ) . Now l e t S1, S2 be two stopping times.
(5.25)
f(Sl,S2)
We put
= EX {
+ +l ( y ( s l 1) x s <s 1-
f(y(t))exp
2
'
+ ( L ~ ( Y ( s ~xSSSlnr ))
s1< exp
exp
-a
- art
dt
s1
- a s2 I .
We s h a l l prove t h a t any s o l u t i o n of ( 5 . 6 ) s a t i s f i e s
(5.26)
I n f a c t , applying I t o ' s formula, we o b t a i n
- a(-cASlhS2) = u p ) Au - a u ) ( y ( t ) ) e x p - a t
Ex u(y(rAS1AS2))exp
(5.27)
+ Ex
rhS AS
'(-
0 A
.
dt
*
Now l e t S1, S2 be defined by
5,
(5.28)
i2
= i d Is 2 O~U(Y(S)) = +1
id I s = olu(y(s)) = + 2 ( ~ ( s ) ) J (*)
( * ) We put u = 0 o u t s i d e @ a n d Q1, $,
G1
s
0 and ( L a =
0
.
(Y(~))J
.
a r e defined o u t s i d e 8 i n such a way t h a t
471
STOCHASTIC DIFFERENTIAL GAMES
1
,
.
However t h e d e f i n i t i o n of S1, S2 immediately shows t h a t we have
(5.29)
u(x) =
J%, ,i2)
.
We s h a l l now prove t h a t we have
(5.30)
u(x)
For t E [0,zA$AS2[
s
f(i,,S2)
Y
we have u ( y ( t ) )
v s2 >
.
+ , ( y ( t ) ) so t h a t
(Au + a d ( y ( t ) ) s f ( y ( t ) ) hence, from ( 5 . 2 7 ) it follows t h a t
u(x) 5
E”
[‘A’1As2
hence ( 5 . 3 0 ) . We next prove t h a t we have
(5.31) Now f o r t
E [0,zAS,Ai2[ U(Y(t))
so t h a t
we have
, sij ~ n ~ s O , st ) n islo > t ) ~3~
clE ) we
(SEC. 5 )
477
STOCHASTIC DIFFERENTIAL GAMES
- at
V2)ds)exp
+
EX(
J
xs1~S;’slE< T %AS, AS X
1
0
E
2
-4
S;
s;
- at exp
t-s;
, so2
T
.
.
We then see that we can combine all cases into the same formula, namely
+ Let
t-s,
(S, E+h)AT
- -dt S1 LA^ a be such that & . If lexp
E
>
Az
+
(exp
We have
-3
s'
( S1 =+h)AT
If Idt
.
-
at dt ,
(SEC. 5 )
STOCHASTIC DIFFERENTIAL GAMES
from which we deduce
479
a
(5.54) We shall now estimate the quantity I1 + 111.
11
+
111 =
We can thus assume S A Si I€ We then consider two cases: Case (11 + 111)~:
o
if
slE A s 2o ->
slEs so2 and s
LIE'
7
.
< T.
10
-T
h ~ 4~~ l (exp
4~~ ( y b l
.
+b t
bW
w
A(t)Wc
= 0
,
+
w
B))+
-1
i3))'
wc(x,T) = 0
and we have ( c f . proof of Theorem
(5.1 08)
1
E b29l9P(Q)
4.5)
,
WE
E
C0(?j)
.
Next we put
We t h e n have ( c f . Theorem
(5.1 10)
sT%,,S2)
4.5) = B(x,t) + LX't(S1,S2)
We s h a l l now show t h a t we have
.
= 0
(SEC. 5 )
STOCHASTIC DIFFERENTIAL GAMES
(5.1 12)
w
+
w in
493
~‘(6).
F o r t h i s purpose, we approximate Ji 1, Ji,,
N
by a sequence Jil
N
EN such
that
(5.113)
(5.114)
Such a sequence e x i s t s . If
N
through s o l u t i o n o f (5.106) we have B corresponds t o iN,
bN
+
B in c o ( ~ ) .
5 . 4 , we s e e t h a t ( 5 . 1 1 1 ) and ( 5 . 1 1 2 ) w i l l be t r u e for N N -N J i l , Q2, u , from t h e moment when t h e y a r e t r u e for Ql, Ji,, u But we can apply Theorem 5 . 6 , hence t h e r e s u l t . Arguing a s i n Theorem
-
.
It f o l l o w s from (5.111) and (5.112) t h a t
Inf sup s2 and
u Thus, s i n c e u
(5.1 15)
+
-
ft (4,s2)
= sup Inf f t ( S , , s2) s2
s1 ~nfsup
f t ( s , ,s2)
in
2
u i n L (6) f o r example, we o b t a i n
u ( x , t ) = Inf sup
ft (s,,s2)
I
s2 (5.11 6 )
.
COW
u,(x,t) + u(x,t) in
?,
sup I&
ft(S,,S2)
s2
~‘(6).
-2
We t h e n show t h a t S i s a s a d d l e p o i n t through a c o n s i d e r a t i o n o f t h e f u n c t i o n w,, a s i n t h e p r o o f s o f Theorems 4 . 5 and 5 . 4 . m 5.2.3.
Principle of separation.
We can extend t h e p r i n c i p l e of s e p a r a t i o n i n v e s t i g a t e d i n S e c t i o n 4.12, i n t h e c a s e o f a s i n g l e p l a y e r , t o t h e p r e s e n t c a s e , provided t h a t t h e o b s e r v a t i o n It follows from t h i s t h a t t h e f i l t e r i s process i s identical f o r both p l a y e r s . t h e same for t h e two p l a y e r s and t h a t t h e stopping t i m e s S1, S r e l a t e t o t h e 2 same family o f a - a l g e b r a s . By analogy with ( 4 . 1 5 1 ) we can e a s i l y w r i t e t h e V . I . o f t h e problem and we can adapt without d i f f i c u l t y t h e proof o f Theorem 4.12 t o t h e games s i t u a t i o n , by f o l l o w i n g t h e approach used i n t h e l a s t p a r t of t h e proof o f Theorem 5.2. ’
The c a s e where t h e p l a y e r s have different o b s e r v a t i o n s i s an open q u e s t i o n .
This Page Intentionally Left Blank
CHAPTER 4
S T O P P I N G - T I M E AND S T O C H A S T I C O P T I M A L CONTROL PROBLEMS
INTRODUCTION I n t h i s chapter we d i s c u s s s t o c h a s t i c optimisation problems i n which t h e r e A s i n Chapter occur t o g e t h e r both a stopping time and a continuous c o n t r o l . 111, we can a s s o c i a t e with t h e s e questions an a n a l y t i c problem which i s again a When t h e stopping time i s not V . I . , but t h i s time with a n o n l i n e a r o p e r a t o r . p r e s e n t , t h e nonlinear V . I . reduces t o t h e Hamilton-Jacobi equation of s t o c h a s t i c c o n t r o l , for which t h e recent book by FLEMING-RISHEL [11 i s one of t h e most comprehensive r e f e r e n c e s . To avoid r e p e t i t i o n , we s h a l l move d i r e c t l y t o t h e case of evolutionary d i f f e r e n t i a l games, s t a t i n g c l e a r l y t h e f u r t h e r r e s u l t s which a r e only t r u e i n t h e c o n t r o l case. We follow a plan s i m i l a r t o t h a t adopted i n Chapter 111. I n Section 1, we d i s c u s s t h e purely a n a l y t i c problem of s t a t i o n a r y and evolutionary V . I . ' s i n which t h e o p e r a t o r contains a nonlinear term i n t h e f i r s t order d e r i v a t i v e s . I n Section 2 , we review m a t e r i a l on t h e Hamilton-Jacobi equation and s t o c h a s t i c c o n t r o l problems (without stopping t i m e s ) . Sections 3 and 4 d i s c u s s Hamilton-Jacobi V . I . ' s and problems o f s t o c h a s t i c d i f f e r e n t i a l games with c o n t r o l and stopping time simultaneously.
1. 1.1.
CONTROL BY "CONTINUOUS VARIABLE" AND BY STOPPING TIME
Synopsis.
We begin by studying d i r e c t l y , without reference t o optimal c o n t r o l t h e o r y , some evolutionary V . I . ' s , t h e n some s t a t i o n a r y ones, i n which t h e p a r t i a l d i f f e r The methods a r e c l o s e l y r e l a t e d t o e n t i a l o p e r a t o r contains a nonlinear term. However, some of t h e t e c h n i c a l d e t a i l s t h o s e o f Sections 1 and 2 , Chapter 3. a r e more complicated and a number of new d i f f i c u l t i e s a r i s e ; t h e s e a r e i n d i c a t e d in the text. 1.2.
The case "0 bounded" with bounded c o e f f i c i e n t s .
The operator A ( t ) is defined as i n Section 2 . 1 , Chapter 3 ; t h e assumptions on t h e c o e f f i c i e n t s a . . ( x , t ) , a . ( x , t ) , a o ( x , t ) a r e 1J (1.1)
t h e a . . being e i t h e r symmetric o r non-symmetric ( t h e y w i l l 1J n a t u r a l l y be t h u s i n a p p l i c a t i o n s t o c o n t r o l ) ,
495
496
STOPPING TIMES MD STOCHASTIC OPTIMAL CONTROL
(CHAP.
4)
(1.3) The f u n c t i o n H(x,t,u,Du) We s h a l l put
(1.4)
c ,..., - 1 . dxl d=n dU
DJ =
dU
We t a k e a f u n c t i o n x , t , u , p + H ( x , t , u , p ) measurable from & having t h e following p r o p e r t i e s :
x
lO,T[
x
R x Rn
(1.5)
( 1 .6)
@x,t,U,P)
15 C(h(x,t)+
(1.7)
t&,t,u,p) dH
I +c
lul+
/PI)
9
Is c l~i(x,ttu,P) dH
.
9
The problem considered corresponds t o games; i n o r d e r t o o b t a i n a c o n t r o l problem, it i s s u f f i c i e n t t o t a k e J, = - m i n what f o l l o w s , i n which case t h e 1 occur t o g e t h e r , become assumptions i n which J, occurs, o r i n which J,, and J,,
redundant. (1
1
We t h u s t a k e two f u n c t i o n s J , , ( x , t ) , J,,(x,t) with
.a
s
4J2
and with t h e supplementary assumptions given below.
( 1 . 1 0)
P(u) =
-z+ A(t)u
+
H(x,t,u,Du)
if
u = G2
then
bus 0
if
u =
then
i>uS 0
,
-f=
O
(*)
-f
R,
497
CONTINUOUS VARIABLE AND STOPPING TIME CONTROL
(SEC. 1)
with ( 1 . 9 ) , (1.12).
R m a r k 1.1. ( 1 .I
5)
Under t h e assumption (1.5), we have
H(x,t,u,Du) E L2(Q)
if
2
u E L (0,T;V)
so t h a t ( 1 . 1 4 ) is meaningful.
R m a r k 1.2. The “weak” formulation i s immediate, from what had been g i v e n i n We s h a l l say t h a t u i s a weak s o l u t i o n if S e c t i o n 2 , Chapter 3.
and
( I .I 7 )
a(t;u,v-u)+(H(x,t,u,Du) 1
+T Iv(T)
-
2
with
( 1 .18)
K=
s
(VI+~
v s
2 0
,
b‘
v E
,v-u)-(f,v-u)]dt
v E K
L2 (O,T;V)
,
$E
L~(O,T;H)I
.
We s h a l l i n v e s t i g a t e weak s o l u t i o n s i n S e c t i o n 1.11 below. We s h a l l prove i n t h e following s e c t i o n s
THEOREM 1.1. Suppose that the c o e f f i c i e n t s a i j , a i , a. s a t i s f y (1.2), (1.31, avd t h a t H ( x , t , u , p ) s a t i s f i e s (1.5), (1.6), (1.7). Suppose t h a t
a 4Ji
GI(X,T)
u
C
C
4J2(X,T)
9
I[.
E 5
-d1 t z
= o
Under these conditions, there e x i s t s one and only one function u i s a solution of (1.13), ( 1 . 1 4 ) , ( l . g ) , (1.12) and such t h a t
.
such t h a t
(1.22)
( * ) This assumption i s perhaps unnecessary. i f ,$I = - a.
I n any c a s e it i s meaningless
498
(CHAP. 4 )
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
1.3.
Proof of uniqueness
Let u and 6 be two p o s s i b l e s o l u t i o n s . Taking v = ti i n (1.14)( r e s p e c t i v e l y v = u i n t h e analogous V . I . r e l a t i n g t o a) and p u t t i n g w = u - 3 , we have: (1 -23)
(c bt
.
w ) -a( t ;w ,w ) -( H(x ,t ,u, Eu)-H( x ,t ,5,DiT) ,u-U) 1 0
But by v i r t u e of ( 1 . 7 ) we have:
(1.24)
I(H(x,t,u,Du)-H(x,t,Q,W),u-ff)I
Thus (1.23) implies 1 d
-sElw(t)l 2+ aIlw(t)Il 2 s
5 Clwl
Clw(t)
I
2
+
Clwl lDHl
Ilw(t)ll+ Clw(t)I
2
9
which, t o g e t h e r with w ( T ) = 0 and Gronwall's lemma, proves t h a t w = 0.
1.4.
Penalisation
We now introduce the penalised equation:
(1.25)
( 1 .27)
(monotonicity of the operator u (A(t)u-A(t)v,u-v)
v u,v E v
where ( 1 .30)
.
+ (H(x,t,u,Du)
-
A(t)u+H(x,t,u,Du))
- H(x,t,v,h),u-v)
:
10
,
(SEC. 1)
CONTINUOUS VARIABLE AND STOPPING TIME CONTROL
(1 .31)
ji(x,t) = e -k(T-t) Gi(x,t),
(1 .32)
1-
.
(1.7) t h a t
It follows from (1.30) and
d i
i = 1,2
+ dir
C = constant independent of
dP
2) We s h a l l o b t a i n (1.26) i f we t a k e k a r b i t r a r y with k 2 k such t h a t 2
a(t;v,v)
(1.33) 3)
+
499
k
.
, where
k
is
koIvI 1 allv112
Next we note t h a t i f we put
X = (A(t)u
- A(t)v,u-v)
2 + klu-vl + (g(x,t,u,Ixl)
t h e n , from ( 1 . 3 2 ) and (1.33) we have:
2
X 2 a(Iu-v(I + (k-ko) Iu-vI
- CIu-v12-
2
- i(x,t,v,I)V),u-v)
Cllu-vll
IU-VI
from which it can be deduced t h a t X 2 0 f o r k s u f f i c i e n t l y l a r g e , 2 ko.
8
Thus from now on we s h a l l assume t h a t (1.26), ( 1 . 2 7 ) hold good. In t h i s c a s e it follows from t h e a p r i o r i e s t i m a t e s which appear below and from t h e monot o n i c i t y method ( a v a r i a n t of t h e method followed f o r (1.40), Chapter 3; see a g e n e r a l account, e.g. i n LIONS C21, Chapter 2, and t h e r e f e r e n c e s of t h i s work) D t h a t probZem (1.25) possesses a unique solution.
We s h a l l now e s t a b l i s h some a p r i o r i e s t i m a t e s . A p r i o r i estimates ( I )
We choose v
E
K
and we t a k e t h e i n n e r product of (1.25) with vE
We note t h a t
(1.34) (1.35) so t h a t (1.25) g i v e s :
-
vo.
500
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
We i n t e g r a t e (1.36) from t t o T 2nd we note t h a t
Thus (1.36) implies
(1.37)
11~,11
+
L~(o,T;v)
IbEIl
L"( 0,T;H)
sc
I
A p r i o r i estimates (111
We deduce from (1.25), f o r t = T , t h a t dU
-€(x,T)
= A(T)z
6U $(x,T)
E H 2nd
dt
so t h a t
(1.39)
+
H(x,T,c,S) 6U IK~(X,T)
We d i f f e r e n t i a t e ( 1 . 2 5 ) with r e s p e c t t o t .
- f(x,T)
I Ic
.
We o b t a i n , p u t t i n g
dU
(1.40)
2 = w dt a
( t h e r e can be no confusion with ( 1 . 2 8 ) )
(1.41
1
where we have w r i t t e n :
a We now t a k e t h e i n n e r product of ( 1 . 4 1 ) with =(uE-q2).
We note t h a t
(CHAP.
4)
(SEC. 1)
501
CONTINUOUS VARIABLE AND STOPPING TIME CONTROL
We t h u s o b t a i n
By i n t e g r a t i n g and t a k i n g i n t o account ( 1 . 4 2 ) , it follows t h a t :
where m
E
2 L (0,T).
so t h a t it can be deduced from ( 1 . 4 3 ) t h a t
(1.44)
1.5.
P r o o f o f e x i s t e n c e i n theorem 1.1.
It follows from (1.37) and ( 1 . 5 ) t h a t
(1.45)
X E = H(x,t,uE,Du,)
remains w i t h i n a bounded domain of L
2
(9).
We t h e n e x t r a c t a sub-sequence, a g a i n denoted by up, such t h a t
u ( 1 .46)
E
bU -
4
dt
(1.47)
weakly s t a r i n
4 u -
X, - X
au
dt
L~(o,T;v) ,
weakly i n L 2 ( 0 , T ; V ) 2
weakly i n L (Q).
It remains t o show t h a t X = H(x,t,u,Du) and t h a t problem. It follows from ( 1 . 4 6 ) t h a t , i n p a r t i c u l a r
u
-t
OD
and weakly s t a r i n L (O,T;H),
2
u strongly i n L (Q)
u
is a solution of t h e
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
502
(1.48)
= 0
(u+,)+
,
(U-q-
= 0
(CHAP.
4)
.
I f we choose v i n y , we deduce from (1.25) t h a t
(1.49)
Jt
bU
) + a(t;uE,v-uC)+(XE,v-uc)-(f,v-u
[-(+v-u,
€) ]
,
dt 2 0
and we can w r i t e
y.
V v E
Since from
(1.46), (1.48) u
i s i n y , it may be deduced t h a t
hence we deduce t h a t
But t h e o p e r a t o r v + A ( t ) v + H ( x , t , v , h ) i s bounded and monotone from V t h u s it i s pseudomonotone ( * ) ( s e e H. BREZIS [21 and t h e account given i n LIONS C21, Chapter 2, S e c t i o n 2 . 4 ) ; we t h e n deduce from (1.51) t h a t
-f
V',
But i f we choose v i n r a n d i f we make use of (1.50) we have:
(1.53)
I'
[-(bu
=
0
:1
and (1.52),(1.53)
( 1 .54)
1.6.
(f,v-u)]dt
H(x,t,u,Du)
-
-? 1 1u(o) l2
Iu,(o)
- (f,v-u)]dt
,,V-U) dU
l2
2 0
.
Regularity of t h e s o l u t i o n .
THEOREM 1.2.
(1.55) Then t h e solution
(*) the
XV
-
imply
[(dth +
that
(1 .56)
)
--12
We adopt t h e conditions of Theorem 1.1.
,
A(t) G2 E L2(Q) u
of the V . I .
a(t)u E
i = 1,2
We asswne i n addition
.
(1.13), (1.14),(lag), ( 1 . 1 2 ) s a t i s f i e s
.
L~(Q)
The method of Section 1 . 6 below g i v e s , by means o f a d d i t i o n a l assumptions on a s t r o n g e r r e s u l t without u s i n g pseudo-monotone o p e r a t o r s .
IJI~,
(SEC. 1)
CONTINUOUS VARIABLE AND STOPPING TIME CONTROL
Under t h e asswnptions of Theorem 1.2 and i f , . b e s i d e s ,
COROLLARY 1.1.
a. . E W"~(Q)
(1.57)
13
,
then (1
.58)
503
2
u E L (0,T;H2(3))
.
m
First proof of Theorem 1.2. We m u l t i p l y (1.25) by (uE-$,)+; we note t h a t follows t h a t
((U;$~)-,(U~-$~)+)
a (yJ2)
1
(- 7+ A ( t ) (u,-+~),(u,-+~)+)+ 7
(1.59)
= (f-H(x,t,u,,h,),
+
(r dJ12 - d t ) J12,
(u,-+,)+)
= 0 ; it t h u s
I (u,-+~)I
-
(U,'G2)+)
But from ( 1 . 4 5 ) we may deduce t h a t
hence
(1 .60) Similarly ( 1 .61)
so t h a t (1.25) and ( 1 . 4 3 ) imply t h a t
We may t h u s deduce (1.56)
Second proof of Theorem 1 . 2 .
(*)
We s h a l l now provide a longer proof (without g i v i n g a l l t h e d e t a i l s ) which does not make use o f Theorem 1.1 (and does not t h e r e f o r e employ pseudolnonotone operators). We introduce
( 1 .62)
VP = {
(1 .63)
Vi
where i n ( 1 . 6 3 ) x (*)
E
U(X)
vi u , I
1 7 [U(Xl
i = 1
,...,u]
..,xi-,
,
..
, x ~ + ~ , ,x ~ .,Xn) + ~
O a n d u i s ecctended by 0 outside o f B .
With t h e assumptions of Corollary 1.1.
-
U(X)
I
9
(CHAP. 4 )
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
504
We t h e n c o n s i d e r , i n p l a c e of (1.25), t h e penalised and p a r t i a l l y discretised equation ( * )
In ( 1 . 6 4 ) t h e o p e r a t o r v
+
2
2
H ( x , t , v , v v ) i s continuous from L (Q) + L (Q).
We can a l s o g e t down t o t h e c a s e i n which
+
(A(t)uo- A(t)v,u-v)
( 1 .65)
(H(x,t,u, vu)- H(X,t,v, vv),u-v) 2 0
However t h i s t i m e we o b t a i n t h e e x i s t e n c e of u , t h e s o l u t i o n of d i r e c t l y "by compactness" ( s e e LIONS, [el, Chapte? 1 ) .
.
(1.64),
Then we e s t a b l i s h a p r i o r i estimates on u , t h e s o l u t i o n o f ( 1 . 6 4 ) . We can v e r i f y , e x a c t l y as b e f o r e , t h a t u s a t i s f i e s E ( * * ) ( 1 . 3 6 ) , ( 1 . 3 7 ) , ( 1 . 4 3 ) , ( 1 . 4 4 ) , (1.60), ( 1 . 6 1 ) , so t h a t
and under t h e assumptions of C o r o l l a r y (1
4.1,
.66)
It t h e n f o l l o w s t h a t we can e x t r a c t a sub-sequence, a l s o denoted by u E , such t h a t when E + 0 , w i t h 6 f i x e q w e have ( 1 .67)
u
( 1 .68)
u
u6 weakly i n L~(o,T;E~(s)) u6 strongly i n L'(o,T;v)
( 1 . 4 6 ) (where u i s r e p l a c e d by u ). 6
and
Since u
(1
-
.69)
+
2
u6 s t r o n g l y i n L ( O , T ; V ) ,
H(X,t,uE, v u E )
-
H(x,t,u6, vu6)
and it can e a s i l y be deduced from
du
( 1 -70)
we have:
L2(Q)
(1.64) t h a t u 6 s a t i s f i e s
(--+ 6 A(t)u6+ H(x,t,u6, vu6) bt
in
- f, v-u 6 ) 2
0
with
(*)
(**)
N a t u r a l l y we a r e no l o n g e r r e f e r r i n g t o t h e same f u n c t i o n uE a s i n ( 1 . 2 5 ) .
The c o n s t a n t s o c c u r r i n g i n t h e s e e s t i m a t e s being independent of E and of 6 .
( S E C . 1)
CONTINUOUS VARIABLE
AND S T O P P I N G T I M E CONTROL
505
( 1 .71)
u (x,T) = z(x)
(1 .72)
6
We can now l e t 6 fact that
+
0 (*).
The theorem can b e deduced from t h i s , u s i n g t h e
H(x, t , u 6 , Vus)
(1.73)
.
-
2
H(x, t ,u,Du) i n L ( Q ) .
We now i n v e s t i g a t e t h e r e g u l a r i t y i n t h e s p a c e s Lp. We s h a l l now prove THEOXEM 1 . 3 . We adopt the conditions f o r Theorem 1.2. t h a t the function h which occurs i n ( 1 . 5 ) s a t i s f i e s
(1.74)
11 E
LP(Q)
.
We asswne that
Proof.
(*)
From ( 1 . 7 ) we have:
We c o u l d a l s o l e t
E
and 6 t e n d s i m u l t a n e o u s l y t o 0.
We a s s m e i n addition
506
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
(CHAP.
4)
Taking account of (1.80) we can t h u s deduce from (1.81) t h a t
from which it can be deduced t h a t
I n t h e same way (1 3 4 )
and we conclude t h e proof without f u r t h e r d i f f i c u l t y . COROLLARY 1 . 2 .
Under t h e conditions of Theorem 1 . 3 and Corollary 1.1, we have :
.
u E LP(o,T;w~*~(o))
(1 3 5 )
1.7.
m
Monotonicityproperties of t h e s o l u t i o n
THEOREM 1.4. We adopt t h e conditions assumed for Theorem 1.1. Let ( r e s p . fi) be t h e solution of (1.13), (1.14), (1.9), (1.12) reZating t o
-
{f,$l,$2,u}
( r e s p . of t h e identicaZ problem but r e l a t i n g t o {f,v1,$
assume t h a t f , .
. . have
t h e same r e g u l a r i t y properties as f , .
Gi ,
(1 3 6 )
f Si
(1.87)
i i ii~ a.e. i n O .
i
t
. . and
1 , 2 a . e . i n Q,
We then have ( 1 .88)
Proof.
U S Q a . e . i n Q. Let uE ( r e s p . u ) be t h e s o l u t i o n of (1.15),( r e s p . of
(1.89)
It w i l l s u f f i c e t o show t h a t
-
2
;GI
that
u ; We
(SEC. 1)
507
CONTINUOUS VARIABLE AND STOPPING TIME CONTROL
But from ( 1 . 7 ) we have:
We i n t e g r a t e over t h e subset of 0 i n which
We now show t h a t Y E t 0.
u E ( x , t ) t G E ( x , t ) ( t f i x e d ) , i . e . i n which
(uE-JI2)+t
(c€-$ 2 )'
and a l s o i n which u -$,
t h e r e f o r e every term of Y (1.921, ( 1 . 9 1 ) gives
hence (u -d )+ E
E
I
0
.
u € - $ ~t
t
us-$,
fiC-$Jl,
i s 2.0, hence t h e r e s u l t .
thus
i . e . i n which 5
(GE-$l)-;
U t i l i s i n g t h i s r e s u l t and
a
We now g i v e a property regarding t h e dependence of u with respect t o t h e domain 0.. The n o t a t i o n i s t h a t of Section ( 1 . 6 ) . We have THEOREM 1 . 5 . We t a k e f,$Ji,; i n 6' x lO,T[ as we22 as H ( x , t , u , p ) with t h e same p r o p e r t i e s as i n B. Let u ( r e s p . be t h e soZution of t h e V . I . (1.13), ( 1 . 1 4 ) , ( 1 . 9 ) , ( 1 . 1 2 ) i n Q = 6 x 10,TC (resp. i n Q' = 6' x 10,TC). We assume t h a t 0' c 8 a n d
u)
(1.93)
fi2 0 a . e . i n 9'.
W e then have
(1.94)
us
fi a . e . i n Q.
We can i n t e r p r e t (1.94) a s a growth resuZt on t h e d a t a vaZues Remark 1 . 3 . withAnon-homogeneous c o n d i t i o n s on X ; i n f a c t u 2 0 on Z, and u = 0 on Z, t h u s u 5 u on Z, t h e d a t a values being t h e same elsewhere, hence (1.94) ( i f we have proved t h e growth of t h e s o l u t i o n with respect t o t h e d a t a on X). 8 Remark 1 . 4 .
(1 .95) (1
.96)
I f we assume t h a t +*Z 0
a.e. i n
jH(x,t,u,h)
I I(
9, C(/ul+lDuI)
,
508
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
(1.97)
f 2 0, iil 0 a . e . i n Q o r i n
(CHAP. 4 )
6,
then
(1.98)
u 2 0
( t h u s we have - applying t h e same r e s u l t i n 0' x l0,TC- a s u f f i c i e n t c o n d i t i o n t o obtain (1.93)).
+ .
I n o r d e r t o prove ( 1 . 9 8 ) , ye choose v v i a v-u =u-, i . e . v = u i n ( 1 . 1 4 ) ; t h i s i s p e r m i s s i b l e because $l 5 u S Ji2 i f Ji2 5 0 ; t h e r e t h e n f o l l o w s
d - I =Iu-l 2 (from
(1.96))
2
+ a(t;u-,u-) +
C lu-1
2
+
(f,u-) 5 (H(xlt,u,Dd,u-)
C Iu-1 11U-11
hence u- = 0. We choose v v i a v-u = - (u-6)' i n ( 1 . 1 4 ) , so t h i s choice i s p e r m i s s i b l e because v = 0 on Z and $,I 5 v 5 Ji
Proof of Theorem 1.5.
v = i n f (u,;);
2
a . e . i n Q. We choose v v i a v-6 = (ii-&)+ i n Q', where 6 i s t h e e x t e n s i o n of u t o 8' by 0 o u t s i d e a, i n t h e V . I . analogous t o ( 1 . 1 4 ) f o r 6. Since (0-0)'
= 0 i n (6'= @ )
x
lO,TC, from (1.93), we o b t a i n :
(&(u-t~), (u-a)+)
- a(t;u-B,(u-fi)+)
- H(x,t,u,Du) - H(x,t,ii,DQ),
(u-tI)+) 2 0
and we hence deduce (u-a)+ = 0 j u s t as p r e v i o u s l y (u
E
1.8.
-a E ) +
= 0 i n Theorem
1.4.
The c a s e "Ounbounded" w i t h bounded c o e f f i c i e n t s .
We now suppose t h a t t h e open domain @is not bounded, w i t h t h e p o s s i b i l i t y Chapter 3. We assume We u s e t h e n o t a t i o n of S e c t i o n s 1.11 and 2 . 1 t h a t t h e c o e f f i c i e n t s a i j , a i , a. a r e bounded i n 6 x R'l w i t h t h e same p r o p e r t i e s
6 = R?
as b e f o r e . We once more assume t h a t H ( x , t , u , p ) s a t i s f i e s (1.5), with t h i s time
(1.99)
h E L2(C,T;Hp)
,
p
>
0
(1.6), (1.7)
fixed.
( I n particular we car, t h e r e f o r e t a k e h = 1.) We then obtain precisely a l l t h e analogues of t h e previous statements by replacing V and H by V and H !J
1.9.
u
.
I n f i n i t e horizon.
We now adopt t h e c o n d i t i o n s assumed i n S e c t i o n 2.14, from t h e assumptions
Chapter 3.
We s t a r t
S
(SEC. 1)
509
CONTINUOUS VARIABLE AND STOPPING TIME CONTROL
Then - by t h e same methods t o the V . I . (1.14). m
Remark 1.5.
-
we o b t a i n t h e extension of Theorem 2.16, Chapter 3
I f 8 i s unbounded, we can i n t h e above, r e p l a c e H and V by
m
Hu, V,,. 1.10.
The case "Qunbounded" with unbounded c o e f f i c i e n t s .
We now adopt t h e conditions assumed f o r Section 2.17, Chapter 3 , and a l s o t h e n o t a t i o n of t h i s s e c t i o n . H ( x , t , u , p ) s a t i s f i e s (1.5),
We assume t h a t t h e function
.
h E L2(0,T;L:)
( 1 .I031
(1.61, (1.7) with
Remark 1 . 6 .
.
H?-l,T*
We can a l s o ( c f . Remark 2.26, Chapter 3) work i n t h e spaces i n t h i s case (1.103) w i l l be replaced by
( 1 .I 04)
The V . I .
.
2
h E L (0,T;H
IJ.9
i7
m
considered can now be w r i t t e n
( 1 . I 05)
t o g e t h e r with t h e analogue of (2.228),
( 1 .I 06)
9, s
us
(2.231),Chapter 3 , and
.
Q2
Remark 1 . 7 . We recover t h e s i t u a t i o n of Section 2.17, Chapter 3 i f m H ( x , t , u , p ) = 0 and i f J, = - m , J,, = J,. 1 We then have t h e analogue of Theorem 2.20, Chapter 3 , and here t h e assumption (2.241) becomes
(1 .I071 1.11.
"i E ci , at
2
L (c,T;F), i = 1,2
.
M a x i m u m weak s o l u t i o n .
Our e s s e n t i a l o b j e c t i v e i n t h i s s e c t i o n i s t o prove
Assume t h a t the t h e c o e f f i c i e n t s a i j , ai, a s a t i s f y ( 4 . 2 ) and THEOREM 1 . 6 . that H satisfies (4.5), (4.7). Assume that $, = - a, J,, = J,, w i t h (1 .I 08)
x4d
9
and t hat (1 .I 09)
f E L~(o,T;H)
(1.110)
E E H .
,
There then e x i s t s a maximum weak solution of (1.16), ( 1 . 1 7 ) .
8
F i r s t we c a r r y out a preliminary r e d u c t i o n , a s i n s t e p 1) o f t h e proof of
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
510
( C W .
4)
Lemma 1.1.: it follows from ( 1 . 7 ) t h a t
- H(x,t,v,b) I S
IH(x,t,vup,b+Dq)
C1(IvI+\mI)
;
we can t h e n reduce t o t h e c o n d i t i o n where (1.111 )
a(t;v,v+)
- c1
5.
( I v I + I D v I ) v + dc 2 a1
Ilv+ 112
V v 6 V
.
I
We s h a l l begin by proving THEOREM 1.7. We adopt t h e conditions asswned f o r Theorem 1.6. Let u be t h e solution of t h e penalised problem (1.125) (where $I, = $I and where there i s no 1 -$Il)-). Then, i f E 5 we have t e r n - -(u E
(1.1 12)
Proof.
obtain
E
U S U , E E
.
The n o t a t i o n and proofs a r e analogous t o t h o s e of Theorem 2.7 : we
But
from which t h e r e s u l t can be deduced.
8
Proof of Theorem 1.6. The p r i n c i p l e of t h e proof i s , i n every r e s p e c t , Let w be an a r b i t r a r y weak analogous t o t h a t of Theorem 2.6, Chapter 3. With t h e n o t a t i o n s o l u t i o n , and l e t u be a s o l u t i o n of t h e p e n a l i s e d problem. of Lemma 2 . 1 , Chaptgr 3 , we have
We deduce from t h i s , a s i n Lemma 2 . 2 , Chapter 3 , t h a t
( * ) It is of no b e n e f i t here t o have c a r r i e d out t h e preliminary reduction t o (1.111).
(SEC. 1)
CONTINUOUS VARIABLE AND STOPPING TIME CONTROL
(1 .I 22)
(Au
- Av,u-v)
+(H(x,u,Du)-H(x,u,D~), u-v) 2 al Ilu-vll
2
m,v E
We t h e n o b t a i n r e s u l t s which a r e analogous t o t h o s e of Section convex s e t being defined by {vI
511
v E
v ,
( v l v E V,
vs
$1
+, s v s
v
.
2.1,the
or Q~
a . e . i n 01.
Likewise we can extend t h e r e s u l t s of Section 1 without any p a r t i c u l a r difficulty
(CHAP. 4 )
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
512
2.
t o t h e case "@unbounded" with bounded c o e f f i c i e n t s ; t o t h e case "6 unbounded", with c o e f f i c i e n t s a i ( x ) unbounded.
m
REVIEW MATERIAL ON THE HAMILTON-JACOB1 EQUATION.
SYNOPSIS. I n o r d e r t o i n t e r p r e t t h e v a r i a t i o n a l i n e q u a l i t i e s with nonlinear o p e r a t o r , i n v e s t i g a t e d i n Section 1, we s h a l l need some r e s u l t s concerning t h e equation corresponding t o t h e same o p e r a t o r ; f o r t h e same reasons as i n t h e case where t h e o p e r a t o r i s l i n e a r , t h e r e s u l t s from Chapter 2 c o n s t i t u t e an e s s e n t i a l preliminary The r e s u l t s which we s h a l l need a r e c l e a r l y l i n k e d with t h e t o Chapter 3. p r o b a b i l i s t i c i n t e r p r e t a t i o n of t h e s o l u t i o n . With a view t o g e n e r a l i t y , we s h a l l adopt t h e context of d i f f e r e n t i a l games, i n t h e same way a s i n Section 1. We s h a l l i n d i c a t e t h e r e s u l t s which are s p e c i f i c t o t h e c o n t r o l case and which do not g e n e r a l i s e t o t h e games s i t u a t i o n .
A s i n Section 1, we f i r s t i n v e s t i g a t e t h e nonstationary case. 2.1.
Notation and assumptions.
(2.1)
aij E
aij = a j i V Cl-..C,
E R
c0(q) , E
B > 0
aijEiEj
.
z p E c:
,
A s u s u a l , w e assume t h a t t h e f u n c t i o n s a . . are extended t o R", i n such a way as 1J t o preserve t h e i r p r o p e r t i e s .
Let Vl, Y2 be two compact subsets of R p , Rq r e s p e c t i v e l y .
and g ( x , t , v ,v ) : 6 x Y 1 2 1 c ( x , t , v l , v 2 ) : a x v1 x y2 + R such t h a t
f(x,t,vl,v2)
-
I f ( x , t , v l ,v2)
(2.2)
f measurable,
(2.3)
c , g measurable and bounded.
For u
(2.4)
E
R, p
E Rn,
We assume t h a t , f o r v We put
f
2
I
We t a k e f u n c t i o n s
R and R~ r e s p e c t i v e l y , and
h ( x , t ) E Lp(Q)
2
f(X,t,Vl,V2)
+
lJ c ( x , t , v 1 , v 2 )
+P
.
e(w,v1,v2)
f i x e d (and x , t , u , p a l s o f i x e d ) , t h e f u n c t i o n L a t t a i n s
Yl, t h e f u n c t i o n v2
+
Max L a t t a i n i n g i t s minimum at v
The f u n c t i o n H i s c a l l e d t h e HmiZtonian.
(2.6)
+
we put
L(x,t,u,p;v1,v2) =
i t s maximum a t v1
x Y
We assume t h a t w e have
H i s a measurable function.
We note t h a t by v i r t u e of ( 2 . 2 ) , ( 2 . 3 ) H s a t i s f i e s t h e p r o p e r t i e s
2
y2'
(SEC. 2 )
HAMILTON-JACOB1 EQUATION
513
L a s t l y l e t ;(x) be such t h a t (2.9) We term t h e Hamilton-Jacobi equation (H.J), t h e equation
- H(x,t,u,Du)
(2.1 0 )
I
0 in
O x ]OPT[
,
(*)
Our o b j e c t i v e i s t o prove t h e e x i s t e n c e and uniqueness o f a s o l u t i o n of ( 2 . 9 ) and t o g i v e an i n t e r p r e t a t i o n of t h e r e s u l t obtained. belonging t o b2y1yP(Q) Before doing s o , we make a number o f observations.
Remark 2 . 1 . We can e v i d e n t l y r e v e r s e t h e r o l e s of min and max i n ( 2 . 5 ) . We t h e n o b t a i n a d i f f e r e n t equation f o r (2.10), except i n t h e case where t h e function L possesses a saddle p o i n t . m Remark 2.2. linear.
I f f , c , g a r e independent of vl, v2, t h e o p e r a t o r H becomes More p r e c i s e l y H(x,t,u,Du) = f ( x , t ) + u c ( x , t )
+
Du.g(X,t)
.
The e x i s t e n c e and uniqueness then follow from t h e theorem of
LADYZENSKAYA-URAL'TSEVA-SOLONNIKOV ( s e e Chapter 3, Section 4 ) . (2.11
1
A(t) =
-
X a 1J ,,
ij
a2 bX.bX. L J
We s h a l l put
.
It i s c l e a r l y permissible t o add a f i r s t - o r d e r term and a term of o r d e r 0 t o t h e right-hand s i d e of (2.11). To a b b r e v i a t e t h e n o t a t i o n , we consider t h e s e I n t h e same way, we can add t o t h e terms t o be included i n t h e o p e r a t o r H. This a l s o i s right-hand s i d e of ( 2 . 1 0 ) a given f u n c t i o n belong t o L p ( Q ) . absorbed i n t o H. 8
Remark 2.3. Verification of assumption ( 2 . 5 ) . It i s easy t o g i v e Since V1 a n d V 2 a r e compact, s u f f i c i e n t conditions f o r ( 2 . 5 ) t o be s a t i s f i e d .
it i s a c t u a l l y s u f f i c i e n t t h a t L should be 1 . s . c . x , t ,u,p f i x e d . 8
i n v 2 and
U.S.C.
i n vl,
with
We assume t h a t t h e r e e x i s t s Verification of assumption ( 2 . 6 ) . Remark 2 . 4 . a n i n c r e a s i n g sequence K c K2... c KL c... o f compact s u b s e t s included i n Q, such t h a t
Q-K1
UK2U
5 ...
i s of measure zero and t h e r e s t r i c t i o n of t h e f u n c t i o n s f , c , g t o KL X V1 X Y2 i s continuous i n x , t , uniformly with r e s p e c t t o vl, v2; t h e n ( 2 . 6 ) i s s a t i s f i e d , because t h e r e s t r i c t i o n of H t o K, 8 v e r i f Ted. (*)
~u =
(e,..., -)axn bU
dXl
x
R x Rn i s continuous, as may e a s i l y be
( s e e Section 1).
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
514
Remark 2.5.
(CHAP.
4)
Existence of a minimax measurably dependent on x , t ,u,p.
From assumption ( 2 . 5 ) , t h e r e e x i s t s f o r a l l x , t , u , p a minimax of t h e It w i l l be important i n t h e following discussion t o know under what Hamiltonian. conditions t h e r e e x i s t measurable functions V,(x,t,u,p)
9
$2(x,t,u,P) *
-
such t h a t f o r a l l x , t , u, p form a minimax of t h e Hamiltonian. We give a s u f f i c i e n t condition f o r t h i s , v & L s i $ on a general theorem f o r t h e e x i s t e n c e of measurable s e c t i o n s of a multivalued mapping. We assume t h a t f , c , g a r e continuous i n x , t , uniformly with r e s p e c t t o v From t h e preceding Remark, it is c l e a r t h a t H i s a continuous function of x ,
i: v2. u, p.
We a l s o assume t h a t f , c , g a r e continuous with r e s p e c t t o Y , v2 for x , t fixed. Now l e t It i s c l e a r t h a t L i s continuous with r e s p e c t t o a l l t h e v a r i a b l e s . D = [ ( x , t , u , p ; v l ,v2) IL = Then D i s closed ( * ) .
HI
-
Let
-
From assumption ( 2 . 5 ) , A = Q
x
R x Rn.
To s i m p l i f y t h e n o t a t i o n , we put z = ( x , t , u , p ) ;
W
Then ( s e e f o r example FLEMING-RISHEL C11, Appendix w = w ( z ) , measurable, such that
(z, w ( z ) ) E D
z
a.e.
E A
= (vl,V2).
B) , there e x i s t s a mapping
.
The a p p l i c a t i o r , of t h i s r e s u l t immediately shows t h e e x i s t e n c e of a minimax which i s measurably dependent on t h e d a t a .
Remark 2.6.
Let F ( x , t , u , p ) be a measurable f u n c t i o n such t h a t
t h e n we can f i n d Let M and m be two r e a l values, s p e c i f i e d but a r b i t r a r y ; control s e t s V1 9 V p which a r e compact, and functions f , c , g measurable and bounded such t h a t t h e r e l a t i o n F = H where H i s given by ( 2 . 5 ) , i s s a t i s f i e d f o r (*)Therefore i n p a r t i c u l a r o-compact, D = D1 U D2 U
...
1.1
5
i. e.
where D1 ,D2,.
..
M,
Ipl 5 m
.
a r e compact.
(SEC.
2)
where B
2
HAMILTON-JACOB1
EQUATION
515
i s a c o n s t a n t t o be chosen l a t e r ( * ) ,
then
F(x,t,h
=
&,t,v1,v2)
w )w 2' 2
1 + Iw21
+ w1 , c(x,t,vl,v2) =
hl
.
Thus
Consequently, a s can e a s i l y be . v e r i f i e d , we have
We have t h e i d e n t i t y
G= G
+
O
n
i=l
(*)
IA21
i=l
PiGi
+ F(x,t,h2,p)
+ 2 Pi(Gi(W2)-Gi(P)) Observing t h a t f o r
n
5
- F(x,t,u,p)
+ B2(p-w21 +
M we have, from t h e assumption,
The c o n s t a n t B2 w i l l only depend on B1 and M.
B1 lu-h21
.
(CHAP. 4 )
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
516
we can prove t h e following property (FLEMING [I])
where
Thus H b F. thus
But f o r v2 = ( p , u ) = vz (which belongs t o
Max
L ( x , t , u , p ; v , ,v;)
V 2 ) we have L = F 'dv,,
= F(x,t,u,p)
v1
hence H
5
F.
We can t h u s i n t e r p r e t a general f u n c t i o n having l i n e a r growth with r e s p e c t t o u,p i n t h e form of a Min Max, over any bounded domain. This w i l l allow t h e r e s u l t s m which follow t o be a p p l i e d t o general parabolic quasi-linear equations. 2.2
I n t e r p r e t a t i o n o f t h e s o l u t i o n of t h e H . J .
equation
When t h e c o e f f i c i e n t s a . . a r e r e g u l a r , t h e e n e r a - t y p e methods developed i n Section 1 a s s u r e t h e exist$Ace and uniqueness of a s o l u t i o n u E b 2 * ' , p o f equation In t h e case of coefficients which a r e merely continuous, t h e study w i l l (2.10). be based upon a r e g u l a r i s a t i o n technique and on t h e interpretation of t h e r e g u l a r i s e d Since t h i s i n t e r p r e t a t i o n i s t h e same whether t h e c o e f f i c i e n t s a r e solution. r e g u l a r o r n o t , we begin by p r e s e n t i n g t h i s i n t e r p r e t a t i o n . The i n t e r p r e t a t i o n w i l l a l s o provide a uniqueness r e s u l t . We assume t h a t t h e functions f , c , g s a t i s f y , i n a d d i t i o n t o ( 2 . 2 ) , ( 2 . 3 ) , t h e properties: (2.14) (2.1
5)
f , c, g
a r e uniformly continuous on
for fixed
x,t
E
Q
with r e s p e c t t o
v1
, y2 ,
$ , f , c , g a r e continuous with r e s p e c t t o vl, v 2
We now introduce a stochastic differential game.
A s u s u a l , we d e f i n e o ( x , t ) such t h a t
2
9a 2
a
(where a = a i j ) .
Let (Q,St , P .
I n p a r t i c u l a r it follows t h a t (when a n o t h e r subsequence has been e x t r a c t e d )
H(x,t,ur,Dur)
+
H(x,t,u,Du)
a.e.
We can t h e n r e a d i l y proceed t o t h e l i m i t i n ( 2 . 3 2 ) so t h a t it follows t h a t u m
i s a s o l u t i o n of ( 2 . 1 0 ) , (2.30)
.
A l t e r n a t i v e proof of Theorem 2.2 in t h e control case. We assume h e r e t h a t f , c , g do
(*) (**)
not depend on vl.
We t h u s have
We have a l r e a d y assumed t h a t c and g a r e bounded ( c f . ( 2 . 3 ) )
I n ( 2 . 7 ) we can t a k e h = c o n s t a n t , s i n c e f i s bounded.
522
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
(CHAP.
4)
H ( x , t , u , p ) = Min L ( X , t , u , p , v ) ( * )
1 We consider a sequence of f u n c t i o n s uo, u uo i s a s o l u t i o n of duo -+ A(t)uo Bt
I
0
,
,... ,un ,... defined
a s follows:
.
uolz = 0 , uo(x,T) = c ( x )
, uo
Eb2'1'P(Q)
.
Having defined u" E b 2 ' l Y pwe , denote by v n ( x , t ) a measurable f u n c t i o n with v a l u e s i n y , such t h a t n n n H(x,t,un,Dun) = L ( x , t , u ,Du ,v ) . We then d e f i n e un+l a s t h e unique s o l u t i o n of
--d t
+ A(t)un+l
un+l
un+1
BU"+l
(2.37)
Iz =
0
t
= Dun+'.g(r,t,vn)+ c(x,t,vn)un+' + f ( x , t , v n ) (x,T) =
E(x)
,
.
Since t h e equation (2.37) i s l i n e a r , t h e r e e x i s t s one and only one s o l u t i o n of (2.37) inb2""(Q).
Since g , c , f a r e bounded, we have
(2.38) Furthermore, we have
- @+ A(t)u" Bt
= h n . g ( v n - l ) + c(v"-')un+ 2 Dun.g(vn)+ c(v")u"+
f(v"-l)
f(v")
,
which, bearing i n mind ( 2 . 3 7 ) , implies
- aa( ut " + ' (Un+'-
un)+ A(t)(un+l- u") s D(u"+l-
U")(T)
I
0
.
u"). g(vn) + c(v")(uncl-
u")
,
From t h e maximum p r i n c i p l e it follows t h a t Un+l - un 2 0 . Since f,; a r e bounded, and t a k i n g account a l s o of ( 2 . 3 8 ) , it t h e n follows t h a t
n
u
(2.39)
u
-
and
un - u
weakly i n
b2"'p(Q).
I t can a l s o be deduced, through compactness, t h a t
uE
(2.40) For v
(*)
E
u
strongly i n
V , we have
We w r i t e v i n place of v2.
b1 ,O,P ( Q ) .
(SEC. 2 )
(2.41)
HAMILTON-JACOB1 EQUATION
- Dun.g(v) - c(v)u" - f ( v )
a U"
- a ~ A(t)u" +
< -
a U" + --
=
-&(U"-
at
523
- k n . g ( v n ) - c(vn)un - f ( v n ) un+') + A(t)(u"- u"") - D(un- uncl).
A(t)un
g(v")
- c(v")(u"-
u"").
The right-hand s i d e of (2.41) converges weakly t o 0 i n LP(Q). Proceeding t o t h e weak l i m i t , we t h u s have
-
(2.42)
dU
+ A(t)u
- Du.g(v) -
C(V)U
- f(v)
0
and f u r t h e r dU
--+at
(2.43)
A(t)uzS L ( x , t , u , k , v ) a . e .
But we have moreover
(2.44)
all -+ A(t)u at
=
- H(x,t,u,Du)
- ia(tu - u n + l )
2
V v
E V
,
BU -a t + A(t)u - Du.g(v") - C(V")u
+ A(t) (u-un+l )
- D(u-u"+l
) .g(v")
-
C(V")(U-U"+~
- f(vn) )
and t h e right-hand s i d e of ( 2 . 4 4 ) again t e n d s t o 0 weakly i n L P ( Q ) . W e t h u s have
-% + A(t)u at
- H(x,t,u,Du) 2 0
which t o g e t h e r with (2.43) c l e a r l y shows t h a t Jacobi equation.
u
i s a s o l u t i o n of t h e Hamilton-
Remark 2.11. Theorems 2.1 and 2.2 can be extended s l i g h t l y , merely by assuming N a t u r a l l y f i s no t h a t ( 2 . 1 4 ) , ( 2 . 1 5 ) hold good with ( x , t ) E Q i n s t e a d of Q. longer bounded, and we have t o u t i l i s e assumption (2.2) and t h e upper bounds of t y p e Lp a s i n Lemma 3.1, Chapter 3. This i s u s e f u l f o r i n t e r p r e t i n g t h e H.J. equation i n t h e case where s a t i s f y (2.9).
does not
I f t h e a . . a r e r e g u l a r , t h e methods of Section 1 allow us t o prove t h e e x i s t ence of the'dolution of t h e H . J . equation, f o r example i f
E V
and
f H
A(");
.
I n o r d e r t o i n t e r p r e t t h e s o l u t i o n , we may f o r i n s t a n c e introduce t h e function 6 , which i s a s o l u t i o n of
--+aP at
A(t)B
81, = 0 and we can put
z = u
Y
- 6.
Then z i s a s o l u t i o n of
i
O
,
P(x,T) = t ( x )
,
(CHAP. 4 )
STOPPING TIMES AND STOCKASTIC OPTIMAL CONTROL
524
But H(x,t,z
+
p,Dz
+
+
Bp) = MinMax.[f
cp
+ g.Dp + cz + g.Dz]
v2 v1 so t h a t we have r e t u r n e d t o t h e c a s e where i s r e g u l a r , changing f i n t o f + C B + g.DB, which i s continuous on Q x V1 X Y 2 but not with r e s p e c t t o T . of
We can a l s o proceed by r e g u l a r i s a t i o n , as i n Theorem 4.5. u can again be deduced from t h i s .
*
The i n t e r p r e t a t i o n
Remark 2.12. We assume t h a t , over and above t h e assumptions of Theorem 2.1, we have t h e property
w w
(2.45)
L(x,t,u,p;vl,v2)
= M~;E Mia L(x,t,u,p;vl,v2) v1
v2 v1
;
v2
it i s t h e n p o s s i b l e t o supplement t h e r e s u l t s ( 2 . 2 1 ) and ( 2 . 2 9 ) . In f a c t , t h e c o n t r o l s v ( s ) , v ( s ) of t h e p l a y e r s 1 and 2 a r e processes adapted t o 3‘ 1 2 t with values i n y, ,y2 r e s p e c t i v e l y . We t h e n d e f i n e f v v ’ g v v ’ c v v by 1 2 1 2 1 2
I
(2.46)
then
0 i s such t h a t time f o r p l a y e r 2. Application o f I t o ' s formula t o $he p e n a l i s e d problem g i v e s (putting Hc(x,t) = H(x,t,uE,DuE))
We t h e n l e t
E
-+
0 , i n (4.23).
u
H~
-
4
Due t o t h e f a c t t h a t
u
in
CO(Q)
H
in
L'(Q)
,
-
kE DU
in
L'(Q)
we can proceed t o t h e l i m i t i n ( 4 . 2 3 ) , n o t i n g t h a t c , g a r e bounded and b e a r i n g i n I t t h e n follows t h a t mind Lemma 3.1, Chapter 3.
536
STOPPING TIMES AND STOCHASTIC OPTIML CONTROL
(CKAP.
4)
Taking i n t o account t h a t
we o b t a i n
We can then l e t h t e n d t o 0 i n ( 4 . 2 5 ) ,
bearing i n mind t h a t f e
From t h e c o n t i n u i t y of u, we o b t a i n
xt
+ E ‘lV2
u(y(TASkAS2))exp
F i n a l l y we l e t 6 t e n d t o 0.
Since S$ 4 S , we o b t a i n
(s) i s bounded. 1 2
(SEC.
4)
537
HAMILTON-JACOB1 V . I . ' s
But s i n c e
UI
-
0 and u(x,T) = s
2-
1
Z t
,
so t h a t , t a k i n g i n account ( 6 . 3 5 ) , we can deduce t h a t
lcpj(sl)-cpj(s2)I S Ih(sl)-h(s2) The s e t
Vj
[ +
Is1- s21
C(w ,)
*
i s t h u s ( f o r a l l f i x e d w ) equicontinuous and bounded on C t , T I .
From A s c o l i ' s Theorem, t h e r e e x i s t s a subsequence cp which converges uniformly
'k
t o q , and proceeding t o t h e l i m i t i n ( 6 . 3 4 ) , we f i n d t h a t cp i s a s o l u t i o n of The (6.32). Now t h e sequence 'p i s a sequence of processes adapted t o 3;. s&e
j
i s t h e r e f o r e t r u e f o r cp( s )
.
rn
4)
(SEC. 6 )
555
PRINCIPLE OF SEPARATION
R T a r k 6.4. I f we consider t h e l a s t i n t e g r a l equation i n (6.311, we n o t i c e t h a t y a s previously constructed i s a s o l u t i o n of
+
?(s) = x + I:(F-XH)f(h)dh
(6.36)
[:
B(h,v(JC(h),h))dh
dK which i s a l s o an i n t e g r a l equation i n y , i f we consider z as having been given. I f we examine (6.36) d i r e c t l y by t h e methods o f Lemma 6.1, we can prove t h e exi s t e n c e of a s o l u t i o n which i s adapted t o z ( s ) (i.e. to Zf). But s i n c e we cannot confirm t h e uniqueness of t h e s o l u t i o n o f (6.368, it i s not p o s s i b l e t o conclude I f t h e i n t e g r a l equation from t h i s t h a t t h e process i ( s ) i s adapted t o Zt. (6.36) possesses a uni ue s o l u t i o n , t h e n we may thereby conclude t h a t t h e Kalman 3 f i l t e r i s adapted t o 2 (and t h e r e f o r e 5‘ = Z i ) .
t
t
It i s c l e a r t h a t we s h a l l have uniqueness i f we assume
-
/ B ( ~ , v , ) B(A,VJ
Is
c l v , - v21
and i f t h e function v ( x , s ) s a t i s f i e s
lv(x,s)
- v(x1,s) 1 I
CIX-x’
I
This i s e s s e n t i a l l y t h e feedwhere t h e constants depend on t h e i n t e r v a l [O,TI. back c l a s s considered by WONHAM [ll, i n t h e case of a s i n g l e c o n t r o l (without stopping t i m e ) . 6.3.
- - -
Variational inequality
We consider t h e functions f , $, p a r t i c u l a r t h a t we have (6.37)
?(x,v,t)
=
defined i n Chapter 3 (4.147).
f(x+C,v,t)(exp
-1
We r e c a l l i r
(p”(t)E,E))dE
t h i s formula being v a l i d i f we have
(6.38)
P ( t ) i n v e r t i b l e , P - ’ ( t ) bounded.
difficulty that f , ( 6 . 2 0 ) , ( 6.21) ) .
We have analogous formulas for $ ( x , t ) and { ( x ) . I
$,;
We can v e r i f y without
_
s a t i s f y p r o p e r t i e s analogous t o t h o s e of f , $ , z ( c f . (6.191,
Next we introduce t h e Hamiltonian (6.39)
& x , t , p ) = Min
{f(x,v,t). + p.B(t,v)]
V E V
t h e minimum being a t t a i n e d s i n c e we a r e minimising a continuous f u n c t i o n on a compact s e t . We s h a l l assume t h a t (6.40) (6.40)
L ( x , t , v , p=) =? ( X ? (, X ) p.B(t,v) + p.B(t,v) e minimum t h et hminimum of o f L(x,t,v,p) v ,,tv) , t + a unique p o pi noti.n t . i s i ast taatitnaei dn eadt aat unique
556
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
(CHAP.
4)
Let V ( x , t , p ) be t h e unique minimum defined i n (6.40). It i s easy t o v e r i f y The Hamiltonian H s a t i s f i e s t h a t V i s a continuous function of i t s arguments. the properties
We then w r i t e ( c f . Chapter
3, (4.148))
where
(6.43)
.
a ( s ) = P(s)EE*(s)R-'(s)H(s)P(s)
We assume t h a t
(6.44) Then
H-l(s) e x i s t s and i s bounded.
(6.38) and (6.44) imply
(6.45)
k(s)zar
,
a > o
.
We then consider t h e V . I .
(6.46)
By applying arguments used s e v e r a l times a l r e a d y , ( c f . proof of Theorem 5 . 1 ) t h e r e e x i s t s one and only one s o l u t i o n of (6.46). Furthermore f o r p. > n+2, Du i s a continuous function of x , t , t h u s
v(x,t) = ?(x,t,Du) i s continuous i n x and measurable w . r . t . t . As we have seen, we can a s s o c i a t e with it an admissible c o n t r o l denoted by 8 ( s ) . We have
(6.47)
e(s)
= O($(s),s)
.
Also, l e t
We then have THEOREM 6.1. Under t h e assumptions and notation of Section 4.12, Chapter 3, together with (6.1), (6.19), (6.20), (6.21), (6.38), (6.40), (6.44),then ( j ( s ) , 5 ) defined by (6.47), (6.48)s a t i s f i e s
(SEC. 6 )
Proof.
PRINCIPLE OF SEPMATION
This i s analogous t o t h a t o f Chapter 3 , Theorem 4.12.
Remark 6.5.
P. VAN MOERBECKE.
Theorem 6.1 was obtained i n c o l l a b o r a t i o n with
557
This Page Intentionally Left Blank
BIBLIOGRAPHY
ANCONA C11 S u r les espaces de Dirichlet, principes, fonctions de Green. Journal de Mathdmatiques Pures et Appliquses, 54 (1975),pp. 75-124. BARDCS, C. C11 A regularity theorem for parabolic equations. Journal of Functional Analysis, v o l . 7, No. 2, April 1971. BENSOUSSAN, A., FRIEDMAN, A. c11 Nonlinear variational inequalities and differential games with stopping times. J . Funct. Analysis, 16 (1974), 305-352. BENSOUSSAN, A., LIONS, J.L. [11 Problsmes de temps d'arrst optimal pour les systdmes distribugs stochastiques, en hommage au Professeur LEPAY, Annali Scuola Normale Superiore di Pisa, 1977. C21 On the support of the solution of some variational inequalities of evolution. J. Math. SOC. of Japan, 28 (1976),pp. 1-27. C31 Probldmes de temps d'arrst optimal et indquations variationnelles paraboliques. Applicable Analysis, 3 (1973), pp. 267-294. C41 Insquations variationnelles non lindaires du premier et du second ordre; C.R. Acad. Sc. Paris, 276 (1973), pp. 1411-1415. BENSOUSSAN, A., LIONS, J.L., PAPANICOLAOU, G. C11 Asymptotic Methods i n Period S t r u c t u r e s ; North Holland, 1978. BISMUT, J . M . [ll Dualitd convexe, temps d'arrst optimal et contrdle stochastique. Z. Wahrscheinlichkeits theorie V. Gelo. 38 (1977),pp. 169-198. BLUMENTHAL, R.M., GETOOR, R.K. [I1 Markov processes and potential theory. Academic Press, 1968.
New York,
BONY, J.M., COURREGE, P., PRICURET, P. 111 Semi-groupes de Feller s u r m e varidt6 5 bord compacte et problemes aux limites integro-diffsrentiels du second ordre donnant lieu au principe du maximum; Annales de 1'Institut Fourier, 18 (1968), 369-521. BREZIS, H. C11 Equations et Insquations non lingaires dans les espaces vectoriels en dualitd; Annales Inst. Fourier, 18 (19681, 115-175. "21 Opdrateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland Math. Studies, 5, 1973. C31 Problsmes unilat6raux, J.M.P.A. 51 (19721, 1-168. BREZIS, H., EVANS, L.C. A variational inequality approach to the Bellman-Dirichlet equation for two elliptic operators; Arch. Rat. Mech. Anal., 71 (1979) p. 1-14. BREZIS, H., FRIEDMAN, A. C11 Estimates on the support of solutions of parabolic variational inequalities. Illinois Journal of Mathematics, 20 (1976), p. 82-87. BREZIS, H., KINDERLHERER, D. C11 The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. V. 23 (1974), 831-844. BREZIS, H., SIBCNY, M. El1 Equivalence de deux insquations variationnelles et applications. Arch. R.M.A. CHARRIER, P., HANOUZET, B., JOLY, J.L. C1l C.R. Acad. Sc. Paris, 1976.
559
560
BIBLIOGRAPHY
CHARRIER, P., TROIANIELLO, G.M. c11 Sur la solution forte d ' u n probldme unilateral d'kvolution avec obstacle dependant du temps. Istituto Matematico G. Castelnuovo, Universita degli studi di Roma, 1976-1977 and C.R. Acad. Sc. 28 (19751, pp. 621-623. C2l Un resultat d'existence et de regularits pour les solutions fortes d ' u n probldme unilateral d'evolution avec obstacle dependant du temps. C.R.A.S. 1975. CHERNOFF, H. C11 Optimal stochastic control, Sankhya, A ( 3 0 ) , (1968). CODDINGTON, E., LEVINSON, N. Ell Theory of ordinary differential equations; McGrau-HiZZ, New York, 1955. DA PRATO, G., GRISVARD, P. C11 Sommes d'operateurs lineaires et equations differentielles opkrationnelles. J.M.P.A. 54 (1975), pp. 305-387. DELEBECQUE, F., QUADRAT, J.P. C11 Application of stochastic control methods in the management of hydro power production. First International Conference on Mathematical Modelling, Aug-Sept. 1, 1977, St. Louis (Missouri). DENY, J., LIONS, J.L. C11 Les espaces du type de Beppo-Levi. Ann. Inst. Fourier, 5, (1953-54), 305-370. DOOB, J.L. L11 Stochastic processes. New York: Wiley, 1953. DWAUT, G., LIONS, J.L. C11 Les insquations en mgcanique et en physique. Dunod, Paris, 1972. DYNKIN, E.B. [ll Markov processes. Springer-Verlag, Berlin (1965). [ e l The optimum choice of the instant for stopping a Markov process; Dokl. Acad. Nauk SSSR, 150 238-240 (1963). C31 Theory of Markov processes. Pergamon Press, New York, (1960). DYNKIN, E.B., YUSHKEVICH, A.A. 111 Markov processes, theorems and problems. Plenum Press, New York, 1969. EVANS, L.C., FRIEDMAN, A. El1 Optimal stochastic switching and the Dirichlet problem for the Bellman equations. Trans. A.M.S. FLEMING, W. [ll The Cauchy problem for degenerate parabolic equations. Journal of Mathematics and Mechanics, v o l . 13, No. 6 , Nov. 1964, pp. 987-1008. C2l Optimal continuous-parameter stochastic control; SIAM Rev. 11, (1969). FLEMING, W., RISHEL, R. [l] Optimat deterministic and stochastic Control. SpringerVerlag, Berlin, 1975. FRIEDMAN, A. C11 Regularity theorems for variational inequalities in unbounded domains and applications to stopping-time problems. A.R.M.A. 52 (1973), 134-160. C21 Stochastic games and variational inequalities. A.R.M.A. 51 (19731, 321-346. 131 Parabolic variational inequalities in one space dimension and smoothness of the free boundary. J.F.A. 18 (1975), 151-176. C41 Partial differential equations of parabolic type. Prentice Hall, Englewood Cliffs, New Jersey, 1964. C5l Stochastic differentiat equations; 2 volumes, Academic Press, N.Y., 1976. FRIEDMAN, A., KINDERLHERER, D. El1 A one-phase Stefan problem; Indiana Univ. Math. J. 24 (19751, pp. 1005-1035. FRIEDMAN, A., ROBIN, M. [ll The free boundary for V.I. with non-local operators. SIAM. J. on Control, 1978. GAVEAU, B. C11 Methodes de contr63e optimal en analyse complexe. I. Resolution d'equations de Monge Ampdre. J. Funct. Analysis 25, pp. 391-411, (1977). See also C.R.A.S. 284 (1977),pp. 99 and 593, Paris. GIKMAN, I.I., SKOROKHOD, A.V. c11 Stochastic differential equations. SpringerVerlag , Berlin, 1972.
561
BIBLIOGRAPHY
DE GIORGI, E. c11 Sullz differenziabilitz e l'analicitk della estremali degli integrale multipli regolari, Mem. Accad. Sc. Torino, cf. Scie. Fis. Mat (3), 3 (1957), 25-43. LIE GIORGI, E., COLOMBINI, F., PICCININI, L.C. C11 Frontiere orientate di misura minima e questioni collegate. Publ. della Scuola Normale Superiore - Pisa 1972, 1-172. GIRSANOV, I.V. c11 On transforming a class of stochastic processes by absolutely continuous substitution of measures. Theory Probab. Appl. Vol. 5 , 1960, p . 285-301. GLOWINSKI, R., LIONS, J.L., TREMOLIERES, R. c11 Analyse nwnkrique des inkquations variationnelles. Dunod, Paris, 1976. Also English version with substantial new appendices: Numerical Analysis of Variational I n e q u a l i t i e s , North Holland, July 1981. GOLJRSAT, M., MAAREK, G . c11 Nouvelle approche des problhes de gestions de stocks. Comparaison avec les mgthodes classiques. Laboria Report No. 148, March 1976. GOURSAT, M. QUADRAT, J.P. Ell Analyse numsrique d'ingquations associ6es 2 des problhes de temps d'arrgt optimaux en contrBle stochastique. Laboria Report No. 154, January 1976. C21 Analyse num6rique d'ingquations quasi variationnelles elliptiques associ6es 2 des problbmes de contrdle impulsionnel. Laboria Report No. 186, August 1976. GRIGELIONIS, B.I., SHIRYAEV, A.N. 111 On Stefan's problem and optimal stopping rules for M a r k o v processes. Theor. Prob. Appl. 11 (1966), 541-558. GRISVARD, P. [l] Equations op6rationnelles abstraites dans les espaces de Banach et problsmes aux limites dank des ouverts cylindriques. Annali S. Norm. Sup. Pisa, XXI (1967), p. 307-347. HANOUZET, B., JOLY, J.L. El1 M6thodes d'ordre dans l'interpr6tation de certaines insquations variationnelles; to appear. 121 Formule de Green par des mgthodes d'ordre; applications aux 6quations et in6quations. HARTMAN, P., STAMPACCHIA, G. 111 On some nonlinear elliptic differential-functional equations. Acta. Math. 115 (1966), 271-310. IL'IN, A . M . , KALASHNIKOV, A.S., OLEINIK, O.A. El1 Equations lingaires paraboliques du second ordre (in Russian), Uspehi Mat. Nauk 17 3, 3-146 (1962). JOLY, J.L. El1 Thesis, Grenoble (1970). KALMAN, R.E., BUCY, R.S. [l] New results in linear filtering and prediction theory.
Journal of Basic Engineering, 1961, pp. 95-107. KATO, A. C11 Perturbation Theory for Linear Operators.
1966.
Springer Verlag, Berlin,
KRYLOV, N.V. El! Control of a solution of a stochastic integral equation. Theory of Probability and its Applications, Vol. 27, No. 1, 1972, pp. 114-131. C21 On uniqueness of the solution of Bellman's equation, Izv. Akad. Nauk SSSR, Seri Mat. Tom. 35 (1971); No. 6. KUNITA, H., WATANABE, S . C11 On square-integrable martingales, Nagoya Math. Journal 30, 1967, 209-245. KUSHNER, H. C11 Probability methods for approximations i n stochastic control ond f o r e l l i p t i c equations. Academic Press, (1977), New York. LADYZENSKAYA, O.A., URAL'TSEVA, N.N. C11 Equations e z l i p t i q u e s l i n k a i r e s e t quasilinbaires. Academic Press, 1968. LADYZENSKAYA, O.A., SOLONNIKOV, V . A . ,
l i n k a i r e s e t quasi-linbaires.
URAL'TSEVA, N.N. c11 Equations paraboliques Moscow, 1967.
562
BIBLIOGRAPHY
M T S C H , T. c11 A uniqueness theorem for elliptic quasi-variational inequalities. Journal of Functional Analysis, 12, 226-87 (1975). LEGUAY, C. 111 Third-Cycle Thesis; Paris, 1975. L W , P. C11 Processus stochastiques et mouvement bromien; Gauthier-Villars, Paris, 1965.
second edition,
LEWY, H., STAMPACCHIA, G. [11 On the smoothness of superharmonics which solve a minimum problem. Journal dIAnal. Math., 23 (1970), 227-236. LIONS, J.L. 111 Equations diffsrentielles opkrationnelles et problkmes a m limites. Springer-Verlag, 1961. C2l Quelques mOthDdes de r8solution des problkmes a m limites nonlindaires. Dunod, Gauthier-Villars, 1969. C31 Problbmes aux limites et conditions b l'infini. C.R.A.S. Paris, 237 (19531, 1617-1620. LIONS, J.L., MAGENES, E. C11 ProblDmes aux limites non homogDnes et applications; Vol. 1 and 2, Paris, Dunod, 1968, vol. 3, 1970. LIONS, J.L., PEETRE, J. C11 S u r m e classe d'espaces d'interpolation. Inst. Hautes Ets. Scientifiques, Pub. Math. 19 (1964), 5-68. LIONS, J.L., STAMPACCHIA, G. C11 Variational inequalities. Corn. P. Appl. Math. XX (1967)3 493-519. LIONS, P.L. 111 Rdsolution des problbmes gdn6rau de Bellman-Dirichlet; C.R. Acad. Sci. Paris, ssrie A, 287 (1978)p. 747-750. Detailed article, to appear; thesis, Paris 1979. C2l Contrdle de diffusions dans RN, C.R. Ac. Sc. Paris; sdrie A, t.288 (1979)p. 339-342. Detailed article, to appear; thesis, Paris 1979. C31 Equations de Hamilton-Jacobi - Bellman ddgdndr6es - C.R. Acad. Sc., t.289 (1979),p. 329-332. LIONS, P.L., MENALDI, J.L. C11 Problhes de Bellman avec le contrBle dans les coefficients de plus haut degrd, C.R. Acad. Sc. Paris, sdrie A.287 (1978) p. 409-412. MACKEAN, Jr., H.P. C11 Stochastic integrals.
Academic Press, New York 1969.
MAGENES, E. El] Spazi di interpolazione ed equazioni a derivate parziali. Atti del VII Congresso, U.M.I., Genoa, 1963, p. 1-64.
[el S u l problema di Dirichlet per le equazioni lineari ellittiche in due variabili; Ann. Mat. Pura Appl. 4, 48 (19591, p. 257-279.
MAURIN, S. El1 Mdthodes de ddcomposition appliquses aux problhes de contr6le impulsionnel; Communication Congrbs IFIP, Sept. 1975, Nice.
MENALDI, J.L. C11 On the optimal stopping time problem for degenerate diffusions. Siam. J. of Control; to appear. 121 Thesis, Paris 1980.
MIGNCT, F., PUEL, J.P. 111 Solution maximum de certaines indquations d'dvolution paraboliques et insquations quasi-variationnelles paraboliques; C.R.A.S. 280, sdrie A. (1975), p. 259 and ARMA, 1976. MINTY, G.J. [11 Monotone (nonlinear) operators in Hilbert spaces. Duke Math. J. 29 (1962), 341-346. [2] On a monotonicity method for the solution of nonlinear equations in Banach spaces. Proc. Nat. Acad/Sc. U.S.A. 50 (19631, 1038-1041. MIRANDA, C. C11 Equazione alle derivate parziali di tip0 e/litico. Berlin, 1955.
Springer Verlag,
MORREY, C.B. El1 Second-order elliptic systems of differential equations. Contributions to the theory of partial differential equations; Ann. of Math. Studies, No. 33, Princeton Univ. Press, 1954, pp. 101-159.
BIBLIOGRAPHY
563
MOSCO, U. C11 Some quasi-variational inequalities arising in stochastic impulse control theory; Lecture at the Int. Summer School on "Theory of Nonlinear Operators", Berlin 6 DR, Sept. 1975. MOSCO, U., TROIANIELLO, G.M. C11 On the smoothness of solutions of unilateral Dirichlet problems. MOSER, J. [I1 A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations; Comm. Pure Appl. Mathematics, 13 (1960),
457-468.
NASH, J. C11 Continuity of the solutions of parabolic and elliptic equations. American J. of Math., 80 (1958), 931-954. NECAS, J. Cll Les mlthodes directes duns l a thLorie des Lquations e l l i p t i q u e s . Czech. Acad. of Sci., Prague, 1967. NISIO, M. El1 Remarks on stochastic optimal control; Japan J. Math., Vol. 1, No. 1,
1975.
[21 On a nonlinear semi-group attached to stochastic optimal control; to appear. PUCCI, C. [11 Un problema variazionale per i coefficienti di equazioni differenziali di tipo ellittico; Annali Scuola Norm. Sup. Pisa, 16, (1962), p. 159-172. QUADRAT, J.P. C11 Optimal stochastic control: numerical analysis of the Bellman equation and applications. Working Conference on Modelling of Environmental Systems. Tokyo, April 26-28, 1976. c21 Existence de solution et algorithme de r6solution numgrique, de probleme de contrdle optimal de diffusion stochastique d6g6nbr6e ou non. Siam. J. of Control, March 1980. QUADRAT, J.P., VIOT, M. [11 M6thodes de simulation en programmation dynamique stochastiques. (R.A.I . R . O . , April 1973, R - 1 , p. 3-22). ROBIN, M. El1 Contrdle impulsionnel des processus de Markov;
thesis, Paris, 1977.
Appendix: SAMUELSON, P.A., MACKEAN, H. El1 Rational theory of warrant pricing. A free boundary problem for the heat equation arising from a problem in Mathematical Economics. Industrial Management Review, 6, (1965), 13-39. SCHAEFFER, D.G. El1 A stability theorem of the obstacle problem. Advances in Mathematics, 16 (1975). SCHWARTZ, L. El1 Th6orie des noyawc. Math. (1950),1, 220-230.
Proceedings of the International Congress of
SHIRYAEV, A.N. C11 Sequential s t a t i s t i c a l analysis.
AMS Pub. Providence, 1973.
SKOROKHOD, A.V. C11 Studies i n the theory of random processes. New York, 1965.
Addison Wesley,
STAMFACCHIA, G. El1 Formes bilin6aires coercitives s u r les ensembles convexes, C.R. Acad. Sc. Paris, t. 258 (1964), 4413-4416. [21 Le problsme de Dirichlet pour les 6quations elliptiques du second ordre 2 coefficients discontinus; Ann. Inst. Fourier, Grenoble, 15, I. (1965), 129-258. STRATONOVITCH, L. 111 Conditional Markov processes and their application to the theory of optimal control; Elsevier. New York, 1962. STROOCK, D.W., VARADMN, S.R.S. C11 Diffusion processes with continuous coefficients; Communications on Pure and Applied Mathematics, "01. XXII, 1969, pp. 345-400 (Part 11, pp. 479-530 (Part 2). TROIANIELLO, G.M. El1 On the regularity of solutions of unilateral variational problems. Rend. Acad. Sci. Fi. Math. Napoli, 1975. C21 On the jmoothness of solutions of time-dependent variational inequalities.
564
BIBLIOGRAPHY
VAN MOERBECKX, P. [l] On optimal-stopping and free-boundary problems. Mech., 1976.
Arch. Rat.
VENTZEL, A.D. Ell General boundary problems connected with d i f f u s i o n processes; Uspehi Mat. Nauk 1 5 :2 (92) , pp. 202-204 (1960). WONG, E. 111 Stochastic processes i n infomation and dynamical systems. H i l l , New York, 1971. WONHAM, W.M. 111 On t h e s e p a r a t i o n theorem o f s t o c h a s t i c c o n t r o l . Vol. 6 , No. 2 , 1962.
McGraw-
SIAM J. Control
YAMADA, T . , WATANABE, S. [ll On t h e uniqueness of s o l u t i o n s of s t o c h a s t i c d i f f e r e n t i a l equations. J . Math. Kyoto Univ. 11.1 (1971), p. 155-167.
YOSIDA, K. C11 Functional analysis.
Springer Verlag, B e r l i n , 1965.