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 if
e25
We observe t h a t y ( t ) , 7 depend on t h e p o i n t x , and t h i s j u s t i f i e s using t h e n o t a t i o n J,(e) f o r t h e left-hand s i d e of ( 2 . 4 ) . A t t h i s l e v e l x i s a simple parameter, but it i s of fundamental importance t o The i n t r o d u c e it e x p l i c i t l y f o r r e a s o n s which w i l l become apparent l a t e r . f u n c t i o n a l J ( 8 ) decomposes i n t o two p a r t s , an integral c o s t , corresponding t o what i s p a i d whilg t h e p r o c e s s is not stopped, and a final c o s t which i s i t s e l f expr e s s i b l e i n two p a r t s , namely +(y(O))(expJ,'c d t ) i f e < T, ( i . e . of we decide t o s t o p t h e p r o c e s s before it
0 t T , ( i . e . i f t h e process if e x i t s from t h e domain 6 ).
e x i t s from t h e domain
a
),or
h(y(T))expSo5c d t
is stopped a f t e r t h e i n s t a n t a t which it f i r s t
Since it i s p e r m i s s i b l e t o t a k e a s unknown AT = m i n ( 8 , r ) i n s t e a d Of 0. , we. may i n t e r p r e t 0 as being a c o n s t r a i n t on t h e s t o p p i n g t i m e , i . e . t h e p r o c e s s w i l l be stopped, a t t h e l a t e s t , when it f i r s t e x i t s from 0 ; where t h i s has not a l r e a d y been done, a balance w i l l be e s t a b l i s h e d between t h e i n t e g r a l c o s t N a t u r a l l y , C, may be e q u a l t o R n , i n which c a s e T = + a. and t h e f i n a l c o s t . F i n a l l y , t h e e x p o n e n t i a l term may be i n t e r p r e t e d a s an actualisation of the c o s t s ; f o r example i f c ( x ) = - 8 , where B i s a p o s i t i v e c o n s t a n t , t h e n B i s a r a t e of i n t e r e s t . ( * ) I f x ,f 0
, then
T
= 0.
(SEC. 2 )
FORMAL DESCRIPTION
3
We put u ( x ) = Inf
(2.5)
e
~ ~ ( e ) .
The f i r s t fundamental problem i s concerned w i t h t h e a n a l y t i c c h a r a c t e r i s a t i o n of t h e f u n c t i o n u ( i s it p o s s i b l e t o f i n d a set of relations of which u i s a s o l u t i o n , and i f p o s s i b l e t h e unique s o l u t i o n ? ) . Having o b t a i n e d t h i s a n a l y t i c c h a r a c t e r i s a t i o n , t h e second fundamental problem i s t h e n t o use t h i s t o deduce t h e e x i s t e n c e of an optimal stopping t i m e , i . e . a stopping t i m e 8 such t h a t
N a t u r a l l y , i n a d d i t i o n t o e s t a b l i s h i n g t h e e x i s t e n c e , it i s important t o be a b l e t o g i v e q u a l i t a t i v e i n f o r m a t i o n on 8 and a l s o t o provide some means of c a l culating the solution. The a n a l y t i c c h a r a c t e r i s a t i o n of t h e f u n c t i o n u proceeds v i a t h e d e f i n i t i o n of an adequate functional space, which poses t h e problem o f t h e r e g u l a r i t y of u a s a f u n c t i o n of x . rn So f a r , we have been d e s c r i b i n g a stationary stopping time problem; i . e . t h e f u n c t i o n s appearing i n ( 2 . 1 ) and ( 2 . 4 ) do not depend on t i m e , t h e i n i t i a l i n s t a n t i s 0 , t h e horizon i s i n f i n i t e ( i . e . 6 i s n o t bounded above, a p r i o r i , by a number T ) . The We s h a l l now g i v e t h e "nonstationary analogue" of t h e above problem. We a l s o t a k e a f u n c t i o n f u n c t i o n s g , u, f , J1, h , c i n t h i s c a s e depend on t i m e . The i n i t i a l i n s t a n t i s t < T ; t h e e v o l u t i o n of t h e (x) and a horizon T < m , i s described b y system over [ t , T ]
I f 9 i s a s t o p p i n g t i m e such t h a t
8
E
C t , T l , we put
The s t o p p i n g time T i s t h e f i r s t i n s t a n t o f e x i t from 8 , a f t e r t . ( * ) The The terms f , I), h have a meaning analogous t o t h a t given f o r ( 2 . 4 ) .
T
supplementary term u ( y ( T ) ) ~ ~ , ~ ~ ~ ce dxs p i$s ~t h e f i n a l c o s t when we decide n o t t o s t o p t h e process b e f o r e T and i f t h e process has n o t l e f t 0 b e f o r e T . The analogue of ( 2 . 5 ) becomes
( * ) If x
40
, then
T =
t.
4
(CHAP. 1)
INTRODUCTION TO STOPPING PROBLEMS
The problems which arise are essentially the same as in the stationary case.
3.
ANALYTIC CHARACTERISATION BY DYNAMIC PROGRAMMING
Dynamic programming provides a (formal but highly intuitive) technique for obtaining the relations which the function u has to satisfy. For further generality, we shall consider the nonstationary case. We put
P=Bx]O,T[,Zra8x]O,T[,r=as. A certain number of relations are obvious.
u(x,T) = c(x), u(x,t)
(3.1)
Moreover, from the definition of u ,
u(x,t) and if x E
a
8
r
=
(3.2)
I
First,
h(x,t),
x,t f Z .
we have
s Jxt(t)
and t
<
T we have J (t) = +(x,t), xt
u(x,t) e +(x,t)
,x
so that
E 8, t < T.
Let us now consider that the stopping times 9 have to satisfy the constraint
T > e l t + 6 , 6 > 0 where 6 will tend to 0 (of cowse, this assumes t < T). We also assume x E 0 ;then ‘I > t and for 6 sufficiently small (random), we have t + 6 2 T (10. At the instant t + 6, the state of the system has become (approximately)
x + bg(x,t) + u(x,t)(w(a+t)
- w(t)).
From the definition of the function u, we will have to ’pay’, at the instant t + 6, a cost greater than or equal to
u(x+6g(x,t)
+ U(X,t)(W(b+t)
- w(t)),t+6)
9
this cost having to be actualised at the instant t , and therefore multiplied by
eXp ftt+6cdB where exp
i”,’”
cds -(1+6c(x,t)).
We must also pay the integral cost
between t and t + 6, the process not having been stopped on this interval. Thus, by forcing the process not to stop on the time interval (t, t + 6 ) , we may expect to have to pay, at most, a cost (approximately) equal to
x=
bf(x,t)
+
E(l+6c(x,t))u(x+6g(x,t)
+
- w(t)),t+b).
u(x,t)(w(a+t)
However, from the definition of the function u, u(x,t)
2
X.
( * ) The formal manipulations which follow actually suppose 6 to be non-random
. ..
(SEC. 3 )
ANALYTIC CHARACTERISATION
5
If t h e f u n c t i o n u i s o n c e - d i f f e r e n t i a b l e with r e s p e c t t o t and twiced i f f e r e n t i a b l e with r e s p e c t t o x , we can w r i t e an expansion of X i n terms of 6 , a s follows
From t h e p r o p e r t i e s of t h e Wiener p r o c e s s , t h i s l a t t e r expectation i s equal t o ( t r denoting t h e t r a c e ) :
(3.4)
1
8%
2
ax
7 a@ 6 .
-tr
I t i s t h e r e f o r e of f i r s t o r d e r i n 6 and not o f second o r d e r , as might have been expected by analogy with Taylor expansions i n a d e t e r m i n i s t i c case. Moreover
(3.5)
E-
. u(w(t+6) - w ( t ) )
bU PX
= 0
.
We now reconsider t h e i n e q u a l i t y u 5 X , t a k i n g account of ( 3 . 3 ) , ( 3 . 4 ) , ( 3 . 5 ) . m e term u (of o r d e r 0 ) vanishes. We can t h e r e f o r e d i v i d e by 6 , and by making 6 tend t o 0 we o b t a i n :
We a r e t h u s l e d t o i n t r o d u c e a family, indexed by o p e r a t o r s with r e s p e c t t o x, namely
t , of d i f f e r e n t i a l
(3.7) where t h e m a t r i x a = a
i j
a=-
i s defined by
a# 2
.
'
With t h i s n o t a t i o n , ( 3 . 6 ) may b e r e w r i t t e n i n t h e form
(3.8)
- a2 ,t
+
A(t)uI f
.
F i n a l l y , it i s worth noting t h a t a t t h e i n s t a n t t we have two p o s s i b i l i t i e s : e i t h e r we s t o p t h e process immediately, o r we allow t h e process t o evolve f r e e l y
6
INTRODUCTION TO STOPPING PROBLEMS
(CHAP. 1)
over an infinitesimal time interval. Now, one or other of these decisions has to be taken. Consequently, one or other of the inequalities (3.2) or (3.8) is actually an equality. However, depending on the position of the point (x,t), this may be either ( 3 . 2 ) These considerations may be expressed by writing
o r (3.8).
(3.9) The set of relations (3,1), (3.2), (3.8) and (3.9) allow, as we shall see, one and only one function u to be characterised in a suitable functional space. The choice of the functional structure is naturally one essential element of the mathematical problem. This choice depends on the properties of the data g, U , f, c, JI, h. Later on, we shall give some idea of the various characterisations which can be deduced from (3.1), (3.2), (3.8), ( 3 . 9 ) . It turns out that these various characterisations are meaningful in different functional spaces. More precisely, some of them require spaces of more regular functions than do others. The analysis of these various possibilities will form one of our fundamental objectives in this volume. It is in fact very important to be able to solve the optimal stopping-time problem with only minimal assumptions on the data, not so much for the mathematical interest of the problem, but rather because the solution of the stopping-time problem is one stage in the solution of the Now, in this latter problem, impulse control problem (discussed in Volume 2). the value of JI will be imposed upon us by the problem, and will satisfy very weak regularity properties.
From the function u , characterised as indicated above, it is possible to obtain an optimal stopping-time via the formula:
(3.1 0) An intuitive justification of (3.10) is as follows.
C =
(3.11)
{Xtt
I
x
E 0
t E [OPT]
I
u(x,t)
Let
<
JI(x,t)].
Suppose that we are at the instant t, the state of the system being x, and let x,t E C. Now it would be absurd to stop the process at the instant t, since the cost to be paid would then be $(x,t); we know that we can do better than this, since the lower bound of the costs, i.e. u, is strictly less than JI. It is therefore necessary to continue. By extending this line of reasoning, we see that it is not necessary to stop before 8 . Nowaat this instant u = $, and we can realise u by stopping immediately. Thus 8 is optimal. For this reason, the set C is termed the continuation Set, its complement in Q being the stopping set.
4.
EXAMPLES OF OPTIMAL STOPPING-TIME PROBLEMS
A very large number of examples of optimal stopping time problems has been investigated in the literature. We may cite in particular CHERNOFF [l]. DYNKIN C1,2,31, SHIRYAEV [l], SAMUELSON-MACKEAN ClI, STRATONOVITCH C1I VAN MOERBECKE [l] and the corresponding references. These examples are in general concerned with the fields of Statistics or Economics. We shall now discuss a number of these, taken from SHIRYAEV and SAMLTELSON-MACKEAN (loc. cit.).
4.1
Bayesian Formulation of a Test Problem
Let (n,
Q
, Pa) be
a family of probability spaces which depend on a parameter
(SEC. 4)
[o,l].
a E
(4.1)
E W L E S OF PROBLEMS
We assume t h a t
p
= up + (I-a)Po. 1
a
We a l s o assume t h a t we a r e given on s t o c h a s t i c p r o c e s s w ( t ; w ) such thai;
(n, a, P a ) a random v a r i a b l e X ( w ) and a
t a k e s only t h e v a l u e s 0 o r 1 and
h
(4.2)
7
P~(A=o) 3 1-a
,
,
P ~ ( A = I )= a
(4.3)
w(t)
(4.4)
h and w ( t ) a r e independent.
i s a s t a n d a r d i s e d s c a l a r Wiener p r o c e s s
Now l e t r and u be two numbers such t h a t
(4.5)
r & O
,
<(t,o)
= rt h ( w )
o > O .
We p u t
(4.6)
+
.
u w(t,w)
The v a r i a b l e A i s n o t observable. On t h e o t h e r hand, we assume t h a t it is p o s s i b l e t o observe t h e e v o l u t i o n of t h e p r o c e s s
s(t).
However, we N a t u r a l l y , t h e information c o l l e c t e d on A i n c r e a s e s w i t h time. I n o r d e r t o avoid paying assume t h a t we pay a c o s t c p e r unit o b s e r v a t i o n time. an i n f i n i t e c o s t , we decide t o s t o p t h e experiment a t a c e r t a i n i n s t a n t ( t o b e A t t h i s i n s t a n t 9 , we make a decision: d = 0 o r 1. chosen). I f d # A , t h e decision i s i n c o r r e c t . If d = A , no c o s t i s i n c u r r e d , i n c u r a c o s t a i f d = 0 and X = 1, o r a c o s t b i f d = 1 and A = 0 . We may t h u s summarise t h e c o s t of t h e d e c i s i o n
b(d-A)+ + a(a-h)’
.
We t h u s have t h e two d e c i s i o n v a r i a b l e s
(4.7)
L ( e , d = Eu[ce
and
d
We t h e n
a s follows:
d , and a c o s t
+ b(d-h)+ + a(a-h)-]
.
The s t r u c t u r e of t h e i n f o r m a t i o n i m p l i e s (4.8)
(4.9)
i s a stopping time w . r . t .
zt = a ( c ( s ) , s
t)
,
e
d i s 3 measurable, d = 0 or 1.
The c o n d i t i o n ( 4 . 9 ) s i g n i f i e s t h a t t h e d e c i s i o n d must be based on t h e information a v a i l a b l e up t o t h e i n s t a n t 0 , t h e i n s t a n t a t which t h e experiment i s stopped. The t e s t problem t h u s c o n s i s t s o f minimising L ( 0 , d ) w i t h r e s p e c t t o t h e We s h a l l now show v a r i a b l e s 6,d which s a t i s f y t h e r e s t r i c t i o n s (4.8), (4.9). t h a t t h i s problem reduces t o an o p t i m a l s t o p p i n g time problem of t h e t y p e considered e a r l i e r . F i r s t , we d i s p o s e o f t h e d e c i s i o n v a r i a b l e
d.
I n f a c t , we have
8
(cm.
INTRODUCTION TO STOPPING PROBLEMS
Ea b(d-h)+ = Eu b d XA=G = Ea {b d Pa(h=O
(4.10)
I 3e ) )
L
Ea
1)
6(l-y(8))
i f we put
eThe r e l a t i o n s (4.10) and (4.12) r e s u l t from t h e f a c t t h a t Hence we f i n a l l y have
3 measurable.
d is
and
Consequently, we a r e l e d t o t h e problem involving t h e s i n g l e v a r i a b l e 8 :
(4.13)
Minimise
e satisfying (4.8)
Ea[Ce+
g(y(8))
3
where (4.14)
g(x)
E
m i n { b ( l - x ) , ax).
The problem (4.13) i s then a stopping-time problem of t h e type being cons i d e r e d , by v i r t u e of t h e following r e s u l t from t h e theory of s t a t i s t i c a l estimation: t h e r e e x i s t s a s t a n d a r d i s e d Wiener process, denoted by V ( t ) , such t h a t we have
(4.1 5)
dy
= r y(1-y)
Y(O) = a
dz
.
To conclude, we should j u s t i f y t h e terminology Bayesiun f o m l a t i o n . I f we consider t h e model (4.6) with A a non-probabilistic unknown parameter, which may a p r i o r i be equal t o 0 o r 1, t h e Bayesian approach c o n s i s t s o f s t a r t i n g from a ( s u b j e c t i v e ) a p r i o r i probability of X, and then deducing an The choice a p O s 8 W i O P i probability y ( t ) on t h e b a s i s o f t h e observation E ( t ) . d = 0 o r 1 i s thus a gamble on t h e value o f A . 4.2
Modelling o f a breakdown phenomenon
We consider a machine which can b e e i t h e r i n t h e "renewed" s t a t e o r i n t h e "breakdown" s t a t e . Depending on which s t a t e p r e v a i l s , t h e functioning of t h e machine i s described by a d i f f e r e n t model. Breakdown occurs a t t h e end of a c e r t a i n random time. Since i n both cases t h e f u n c t i o n i n g i s perturbed by n o i s e ,
4)
(SEC.
9
EXAMPLES OF PROBLEMS
We a r e now conit i s not p o s s i b l e t o know e x a c t l y t h e i n s t a n t of breakdown. cerned w i t h s t o p p i n g t h e o p e r a t i o n of t h e machine a t a c e r t a i n time ( f o r example I f we s t o p t h e t o c a r r y o u t maintenance), b e a r i n g i n mind t h e following c o s t s . I f we s t o p a f t e r machine b e f o r e t h e breakdown, we pay a p e n a l t y equal t o u n i t y . t h e breakdown, we pay a p e n a l t y equal t o c p e r u n i t time which e l a p s e s between The i n t e r p r e t a t i o n of t h e i n s t a n t o f breakdown and t h e i n s t a n t of s t o p p i n g . t h e s e c o s t s i s obvious. On t h e one hand, t h e r e i s no advantage i n s t o p p i n g t h e machine i n i t s normal o p e r a t i n g s t a t e ; on t h e o t h e r hand a machine c o n t i n u i n g t o f u n c t i o n i n i t s breakdown s t a t e w i l l produce f a u l t y a r t i c l e s which have t o be r e j e c t e d , so t h a t t h e c o s t i n c u r r e d i s t h e n p r o p o r t i o n a l t o t h e l e n g t h of t h e p e r i o d d u r i n g which t h e machine i s allowed t o f u n c t i o n i n i t s d e f e c t i v e s t a t e . We s h a l l now d e s c r i b e t h e model i n more d e t a i l . Let
( a , d , Pw) denote
I
~ ( w ) ,a
a family o f p r o b a b i l i t y s p a c e s .
R.V.
E
R+
We t a k e
d i s t r i b u t e d i n accordance w i t h t h e law
(4.1 6)
(4.1 7)
a s t a n d a r d i s e d Wiener p r o c e s s
w(t)
independent of
A.
We t h e n observe t h e p r o c e s s E ( t ) d e f i n e d by
t
2
I n o t h e r words, f o r t < A , A. < ( t )= r ( t - A ) + u w ( t ) .
E(t) =
a w ( t ) and f o r a l l
Let
(4.19)
at
= o ( s ( s ) , s_c t )
,
and l e t 0 be a stopping time w i t h r e s p e c t t o
t
3
(decision variable)
We p u t
Such a problem reduces t o t h e g e n e r a l form of We now have t o minimise L(B). stopping-time problems, by v i r t u e of a number of r e s u l t s from t h e t h e o r y Of s t a t i s t i c a l estimation. I n f a c t , i f we put
t h e n it can be shown ( * ) t h a t t h e r e e x i s t s a s t a n d a r d i s e d Wiener p r o c e s s f ( t ) such t h a t we have
+ f
(*)
See SHIRYAEV ( 1 o c . c i t . ) .
y(1-y)dz
INTRODUCTION TO STOPPING PROBLEMS
10
(CHAP. 1)
Furthermore, employing arguments based on the theory of martingales we may reexpress the functional (4.20) in the form
We have thus succeeded in reducing the problem to the general form. 4.3
A warrant-pricing problem
A warrant gives its owner the right to purchase which we shall take to be equal to unity to assist The owner of the warrant may exercise his right to a latest date T (possibly + m ) which is called the
a given share at a given price in clarifying the concepts. purchase at any instant before lifetime of the warrant.
The price of the share at the Stock Exchange fluctuates in a random manner. The problem facing the owner of the warrant is to know at which moment it is most profitable for him to exercise his right. This optimisation is, furthermore, a preliminary which enables a f a i r price to be fixed for the warrant. SAMUELSON and MACKEAN (loc. ci+,.) consider the following model for the evolution of the share price < at the Stock Exchange:
(4.24) where w(s) is a standardised Wiener process, u > 0, k 2 0. Let 0 E [t,T’] denote a stopping time with respect to the family indexed by s f Ct,T3 3: = o-algebra generated by w(h), t 5 A 5 S. The owner decides to exercise his right at the instant 8; his gain is then
(4.25) where 6 is the interest rate on capital. The problem consists of finding 8 so as to minimise Jxt(8). If we put
(4.26)
yxt(s) = x
+
u(w(s)-w(t))
+ k(s-t) , 8 2 t
we may re-express (4.25) in the form
(4.27)
Jxt(0) = E(exp yxt(0)-1)+
exP-B(0-t).
We thus have an optimal stopping-time problem.
5.
OPTIMAL STOPPING-TIME PROBLEM AND FREE BOUNDARY PROBLEMS
We shall now return to the set of relations (3.1), (3.2), ( 3 . 8 ) , (3.9) and to the set C. We may consider u and c as unknowns such that we have
OPTIMAL STOPPING AND FREE BOUNDARY PROBLEMS
--8a Ut + A(t)u - f (5.1)
us$
in
- bU
A(t)u
+
P
0
11
i n C,
C , u = + o u t s i d e C,
-f
0 o u t s i d e C,
u ( x , T ) = c(x) 3 u ( x , t ) E h(x,t) on X.
I f t h e s e t C were known, it i s c l e a r t h a t o b t a i n i n g u would reduce t o s o l v i n g a " c l a s s i c a l " boundary v a l u e problem ( a t any r a t e i f C has s u i t a b l e t o p o l o g i c a l p r o p e r t i e s ) , namely
- bU E +A(t)u - f = 0 i n C , u + on ac , U ( X , T ) = C(x).
(5.2)
I
Problem ( 5 . 1 ) belongs t o t h e c l a s s o f s o - c a l l e d " f r e e boundary" problems, i n which,in g e n e r a l t e r m s , t h e s o l u t i o n t o a boundary-value problem has t o be found i n a domain which i t s e l f has t o be determined, w h i l s t at t h e same time s a t i s f y i n g The c o n d i t i o n s very o f t e n i n c l u d e matching condc o n d i t i o n s o u t s i d e t h e domain. i t i o n s on t h e boundary of t h e domain, c o n s i s t i n g of t h e matching of t h e f u n c t i o n s and t h e d e r i v a t i v e s , s o as t o s a t i s f y a c e r t a i n g l o b a l r e g u l a r i t y . Such free-boundary problems crop up very f r e q u e n t l y i n Physics and Mechanics ( c f . DWAUTLIONS C11). The i n t e r e s t i n g phenomenon which a r i s e s h e r e i s t h a t problems which a r e a t f i r s t s i g h t very d i f f e r e n t , such a s stopping-time problems and c e r t a i n problems i n Physics and Mechanics, a c t u a l l y l e a d t o t h e same a n a l y t i c f o r m u l a t i o n s . We can t h e r e f o r e t a k e advantage of t h e s e d i f f e r e n t i n t e r p r e t a t i o n s s o a s t o o b t a i n We s h a l l i n one c o n t e x t p r o p e r t i e s suggested by p h y s i c a l i n t u i t i o n i n a n o t h e r . meet c e r t a i n a p p l i c a t i o n s o f t h i s i d e a i n t h e f o l l o w i n g c h a p t e r s .
To i l l u s t r a t e t h i s p o i n t , we s h a l l now simply s t a t e t h e correspondence which e x i s t s between a stopping-time problem i n one dimension and t h e s i m p l e s t S t e f a n problem ( t h e problem o f i c e m e l t i n g i n w a t e r ) . The problem i s as follows ( s e e F i g u r e 1). We c o n s i d e r an i n f i n i t e t u b e , modelled as a s t r a i g h t l i n e . A t the instant 0, t h i s c o n t a i n s i n t h e p a r t x 5 0 water a t a temperature 8 ( x ) 2 0 ( 8 (0)= 0 ) and In t h e cougse of time: t h e water i n t h e p a r t x t 0, i c e a t t h e t e m p e r a t u r e 0 . warms t h e i c e and t h e r e f o r e t h e i n t e r f a c e between t h e water and t h e i c e at t h e Moreover, s ( t ) i n s t a n t t , denoted by s ( t ) , w i l l l i e a t a p o s i t i o n t 0 . i n c r e a s e s w i t h t for obvious p h y s i c a l r e a s o n s . The problem c o n s i s t s of determining t h e curve s ( t ) , which i s t h u s a f r e e The equations governing t h e physics o f t h e phenomenon a r e shown i n boundary. I n p a r t i c u l a r , along t h e f r e e boundary S t h e temperature g r a d i e n t Figure 1. i s equal and o p p o s i t e t o t h e speed of displacement o f s. We n o t e , moreover, that
aaxe
5
0, so that
s
2
0.
12
INTRODUCTION TO STOPPING PROBLEMS
( C H A P . 1)
Fig. 1 We shall now associate the above problem with an optimal stopping-time problem. We assume that 9,(x)
(defined for x
0 ) is such that
5
We then arbitrarily take $(x,t) (regular) and we put
- 2. 3 2
f ( x , t )E 1
(5.5)
u(x) = +(x,T)
+
cp(x) if '15
We then define a function u(x,t) (5.6)
.d(x,t)
- $(x,t)
=
-5.
T-t
P
o , Q(X,T)
if x 2 0.
by the following formulas:
B(x,s)ds + cp(x) for x g 0
,
0 s
tsT,
(SEC. 5 )
OPTIMAL STOPPING AND FREE BOUNDARY PROBLEMS
13
(where h ( x ) i s t h e i n v e r s e f u n c t i o n of s ( t ) ) ,
(5.8)
u(x,t) = +(x,t) f o r x 2 0
,
t 2 T-h(x).
We s h a l l now attempt t o determine t h e r e l a t i o n s s a t i s f i e d by First, for x
5
u.
0 and 0 5 t 5 T , we have from ( 5 . 6 )
Now, d i f f e r e n t i a t i n g ( 5 . 6 ) twice with r e s p e c t t o
x , we o b t a i n
so t h a t with ( 5 . 9 ) we have
we have t h u s shown t h a t
From ( 5 . 7 ) we have
We now consider t h e region x t 0 .
:: ::
(5.1 1) But f o r t
---I
S
.
B(x,T-t)
T-h, we have h
5
T-t
d
T and hence from Figure 1,
Furthermore, s t i l l using ( 5 . 7 ) , we have
INTRODUCTION TO STOPPING PROBLEMS
14
since
6(x,h(x)) = 0 .
D i f f e r e n t i a t i n g again with r e s p e c t t o
(CHAP. 1)
x , we o b t a i n
Now, from t h e condition on t h e f r e e boundary ( s e e Figure 1) we have
Consequently, ( 5 . 1 3 ) becomes:
Taking account of ( 5 . 1 1 ) , ( 5 . 1 4 ) we o b t a i n by v i r t u e of ( 5 . 5 )
Since 0
5
0 and JI 2 0 , it i s c l e a r from ( 5 . 6 ) , ( 5 . 7 ) t h a t we have U S @
(5.16) For
f o r x S 0 , 0s t i T
x> 0 x 2 0
,
,
t 2 T-h(x)
0s t _ c T-h(x).
,
we have
u
P
+
(SEC. 6)
GENERAL DISCUSSION
We have thus shown (subject to proving that the calculations make sense) that the function u defined by ( 5 . 6 1 , (5.7), (5.8) satisfies
Now, the relations (5.18) form the analytic characterisation of an optimal stopping-time problem, so that we have indeed established a correspondence between the Stefan problem and a certain optimal stopping-time problem.
6.
GENERALISATIONS
The situation considered so far (optimal stopping-time problem for a stochastic differential equation of the type (2.1) or (2.7))may be generalised in a certain number of directions. We may simultaneously consider a 'continuous' control and a stopping time as decision variables: in other words the equation governing the evolution of the system becomes
where v = v(s) is a 'continuous' control and the criterion is written
We are interested in the function
(6.3) *
A
as well as, of course, in obtaining and characterising an optimal pair (v, 0 ) . If we use the 'formal' methodology of dynamic programming, we are led to the following analytic problem: the function u has to be a solution of
16
INTRODUCTION TO STOPPING PROBLEMS
(CHAP. 1)
(6.4)
where we have put
The problem ( 6 . 4 ) i n i t s most general sense ( i . e . when t h e c o n t r o l a c t s When simultaneously on both t h e d r i f t and t h e d i f f u s i o n terms) remains open. u = R", and under s u f f i c i e n t l y s t r o n g r e g u l a r i t y assumptions on t h e d a t a , a complete s o l u t i o n can be given. On t h e o t h e r hand, when t h e c o n t r o l a c t s on only t h e d r i f t terms a much more I t i s convenient t o re-express ( 6 . 4 ) i n t h e complete theory can be developed. form
(6.6) where
(6.7) The f u n c t i o n a l H i s termed t h e HamiZtonim. There i s a g r e a t d e a l of d i f f e r e n c e , a s f a r as t h e methods of s o l u t i o n a r e concerned, between problem ( 6 . 4 ) and problem ( 6 . 6 ) . I n t h e case of ( 6 . 6 ) , a dire c t study i s p o s s i b l e ( g i v i n g t h e e x i s t e n c e and This uniqueness of a s o l u t i o n u of ( 6 . 6 ) i n an adequate f u n c t i o n a l s p a c e ) . f u n c t i o n i s then i d e n t i f i e d with t h e right-hand s i d e of ( 6 . 3 ) . I n t h e case of ( 6 . 4 ) , a d i r e c t a n a l y t i c a l study i s , f o r t h e moment, an open question. However, it i s p o s s i b l e i n c e r t a i n cases ( s u b j e c t t o t h e r e s t r i c t i o n s mentioned, i n p a r t i c u l a r u = R n ) t o prove t h a t t h e f u n c t i o n u defined by ( 6 . 3 ) i s a c t u a l l y a s o l u t i o n of ( 6 . 4 ) , and even t h e unique s o l u t i o n .
It may be noted t h a t t h i s comes down t o proving t h a t t h e r e l a t i o n s ( 6 . 4 ) a r e I n c o n t r a s t , i f we solve ( 6 . 6 ) d i r e c t l y , and i f we i d e n t i f y u with t h e right-hand s i d e of ( 6 . 3 ) , t h e n we have t o show t h a t ( 6 . 6 ) c o n s t i t u t e s a s e t of s u f f i c i e n t conditions f o r o p t i m a l i t y .
necessary conditions f o r o p t i m a l i t y .
I..
An optimal p a i r ( v , e ) i s i n g e n e r a l defined as follows: we c o n s t r u c t ( i f it e x i s t s ) a Bore1 f u n c t i o n v ( x , t ) such t h a t we have
(SEC. 6 )
and we seek
GENERAL DISCUSSION
y,
which i s a s o l u t i o n ( i n an a p p r o p r i a t e s e n s e ) of t h e e q u a t i o n
+
df = g ( ? , Q ( f , s ) , s ) d s
9(t) = x Then
S(s) =
and
E
.
d?,s)dw(s)
9
G(f(s),s)
= i d {TZ sz t
1
u(f(s),s) = +(g(s),s)]
form an optimal pair. Another s i t u a t i o n t r e a t e d i n t h i s book is t h e problem of differential games with continuous c o n t r o l and s t o p p i n g t i m e s , o r w i t h s t o p p i n g t i m e s a l o n e . Here we c o n s i d e r two p l a y e r s , each being a b l e t o make a d e c i s i o n on a p a i r comprising The e q u a t i o n governing t h e e v o l u t i o n a continuous c o n t r o l and a s t o p p i n g t i m e . of t h e system i s t h e n
where vl, v2 a r e t h e two c o n t r o l s .
We d e f i n e t h e c r i t e r i o n
where el, O 2 a r e s t o p p i n g t i m e s E C t , T I ( * ) . t h e f u n c t i o n u ( x , t ) d e f i n e d by
Very f o r m a l l y , we a r e i n t e r e s t e d i n
(6.10)
The meaning of t h e right-hand s i d e of (6.10) needs t o be c l a r i f i e d . going i n t o d e t a i l s , we can s t a t e t h a t i n v e s t i g a t i o n of t h e f u n c t i o n u t h e following a n a l y t i c problem:
Without leads t o
( * ) I n (6.9) t h e c o s t a t t h e boundary of t h e domain i s t a k e n t o be zero t o simplify t h e notation.
18
INTRODUCTION TO STOPPING PROBLEMS
(CHAP. 1)
(6.11)
Numerous other generalisations can be put forward, regarding the dynamics of the system. Some of these will be treated in Volume 2: diffusions with reflection at the boundary of the domain, processes of the Poisson or semi-Markov type, degenerate processes, etc
...
With regard to the infomation on which the decision-making is based, we have hitherto formulated the problem on the assumption that it was possible to observe the evolution of the state of the system with the passage of time (i.e. the evolution of y). In this case, the form of the optimal decisions mentioned above has one very attractive aspect: all the information necessary for making the optimal decision at each instant is concentrated in the state of the system at that instant. Information regarding past history is superfluous. Putting it yet another way, the optimal decisions are Markov. This important property no longer applies in the case of partial infoonation; the problems still remain largely unresolved in this direction, except in the case of linear dynamics with linear observation (and, for games, in the case where each of the players has access to the same information). In this connection, we can prove the principle of separation (see Chapter 4); this signifies that the optimal decision (control and stopping time) can be obtsined at each instant as a function of the best estimate of the state of the system at that instant.
7.
VARIOUS CKARACTERISATIONS OF THE OPTIMAL COST FUNCTION
The characterisation of the function u by the relations (3.1), (3.2), (3.8) and (3.9) can actually be proved only in cases for which the data are sufficiently regular (which implies regularity properties for the function u and which justifies, in particular, the use of expression (3.9)). In the case where the data are not regular, other characterisations are possible. First, let us consider the stationary case. For simplicity, we take f = 0, h = 0. The characterisation (3.1), (3.2), (3.8), (3.9), when this is applicable, is written in the form
We then consider the process
(SEC. 7 )
CHARACTERISATIONS OF OPTIMAL COST FUNCTION
19
which is a Markov process. With this process there is associated a semi-group O(t) on the space of functions which are uniformly continuous on 8 and zero at the boundary, via the formula Q ( t ) h(x) =
E h(zx(t))
.
A characterisation equivalent to (7.1) (when (7.1) is applicable) is then as follows : u is the maximum function such that
(7.3)
u
and u
5 J,
5
@(t)u
The advantage of (7.3) is that it still has a meaning if J, is not a differentiable function. rn Another formulation equivalent to (7.1) is given by the theory of VUPiUtiOnUl Inequalities (V.I.), which in the present book will form the basic tool for solving analytical problems of the type ( 7 . 1 ) . Associated with the operator A there is a bilinear form on a Sobolev space, given by
(7.4)
a(u,v)
= z
j
i j-b
dU
aijK
j
where
We may therefore rewrite (7.3) in the form
If we compare (7.3) and ( 7 . 5 ) , which are valid under the same regularity conditions on the function J,, we can see that (7.5) gives information on the derivative of the function u. Furthermore, (7.5) is certainly more convenient In fact if for to use than ( 7 . 3 ) , us f u r as nwnerical analysis zs concerned. example g . = 0, then (7.5) is quite simply the Euler condition for the optimisation p+oblem
(7.6)
Min a(v,v) subject to the constraints V
E
Hi , v
5
JI.
If g. # 0, then (7.5) can no longer be interpreted as the Euler condition for an optikisatior. problem. However, the same numerical solution algorithms can still be applied, apart from a few very simple transformations. In the nonstationary case further difficulties arise since the regularity
20
INTRODUCTION TO STOPPING PROBLEMS
( C H A P . 1)
of the function u then has Bo be investigated with respect to the t w o variables x and t. It is in this case necessary to resort to the theory of parabolic variational inequalities. Depending on whether or not the function $(x,t) is regular with respect to t, very different situations arise. These are investigated in detail in the present book, as are the extensions corresponding to the cases considered in Section 6.
CHAPTER 2 STOCHASTIC DIFFERENTIAL EQUATIONS A N D 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 E Q U A T I O N S OF S E C O N D O R D E R
INTRODUCTION In this chapter :re present the essential elements of both stochastic differential equations and linear partial differential equations of second order (elliptic and parabolic). Although the relationships between these two theories are well known, the techniques for investigating the two fields have been developed relatively independently. We shall endeavour here to draw together the methods and viewpoints. We shall do this not merely by stating results, but by actually pointing out those techniques which we consider to be important and which are likely to remain in use for some time. We shall now give a very simple idea of the relationships which exist between the two theories. Suppose that we have to solve the Dirichlet problem
where
6 is a regular bounded domain in Rn and f
E
Co(o)
I
It is possible to give an ezpzicit (probabilistic) formula for u(x) at any point x E 6. Thus
(ii) where w(t) is the Wiener process in Rn and T(X) the first instant 2 0, at which the trajectory x + w(t) reaches the boundary aB , and where E in (ii) denotes the mathematical expectation. Generally speaking, it is possible to give explicit formulas for the solutions of elliptic o r parabolic linear P.D.E.'s of second order, by means of processes 'derived' from the Wiener process. There is in fact a very close analogy with the method of characteristics for hyperbolic equations of first order: we know that the characteristics are solutions of ordinary differential equations. For second-order equations, it is necessary to consider stochastic characteristics, these being solutions of stochastic differential equations. A stochastic differential equation has the form
(iii) where g,a are two given functions on Rn
X
[ O , T l and where w(t) is a Wiener process. 21
STOCHASTIC D.E.'s & P.D.E.'S OF ORDER 2
22
(CHAP. 2 )
We can w r i t e ( i i i ) i n t h e form
+
= g(y,t)
(iv)
U(Y,t)
dw
but t h i s i s i n f a c t only a formal expression a s t ( i t i s , however, continuous).
+
w ( t ) i s not differentiabze
Nonetheless, t h e analogy with ordinary d i f f e r e n t i a l equations i s very s t r o n g . Hence f o r a = 0 ( i v ) reduces t o (V
.
9 = g(y,t) dt
1
We s h a l l now i n d i c a t e why ( i i i ) i s a good model of a stochastic dynamic system, which i s , moreover, very concrete d e s p i t e t h e t e c h n i c a l d i f f i c u l t y of t h e nonIndeed, although we cannot r e f e r ( * ) t o d i f f e r e n t i a b i l i t y of w o r y . dw t h e increments dt, w(t+h) a r e well defined. y(t+h)
- w(t)
,
h 2 0
We may w r i t e ( i i i ) "approximately" as follows
- y ( t ) = g ( y ( t ) , t ) h + &(t),t)(w(t+h)
-
w(t))
(**)a
Thus i f y ( t ) r e p r e s e n t s t h e evolution of t h e s t a t e o f a dynamic system as a f u n c t i o n of time, we can say t h a t , s i n c e a t t h e present i n s t a n t t t h e system has t h e known s t a t e y ( t ) , i t s evolution between t and t + h ( h small) i s t h e sum o f a known term g ( y ( t ) , t ) h and a random term a ( y ( t ) , t ) ( w ( t + h )- w ( t ) ) . I t follows from t h e p r o p e r t i e s of a Wiener process t h a t E(w(t+h) E(w(t+h)
- w(t))
- w(t))2
= 0 = h
.
Consequently, t h e mean evolution of t h e system between t and t + h i s g ( y ( t ), t ) h and t h e mean d e v i a t i o n around t h i s mean evolution i s
2 I U ( Y ( t ) , t ) In .
The random term may be i n t e r p r e t e d a s a perturbation on t h e veZocity of t h e This p e r t u r b a t i o n i s d i s t r i b u t e d (approximately, f o r system a t t h e i n s t a n t t . h s m a l l ) i n accordance with a normal l a w with mean 0 and with variance a2(y(t),t)h. Furthermore, i f u does not depend on
tl # t 2 , then t h e p e r t u r b a t i o n s dt,)(w(t,+h)
- w ( t l 1)
y
and
and i f we consider two i n s t a n t s
dt2)(w(t2+h)
- w(t,))
a r e independent.
( * ) We s h a l l not dwell h e r e on t h e mathematical a s p e c t s ; f i n e d i n t h e d i s t r i b u t i o n a l sense.
(**IWe consider dimension
1 t o simplify t h e n o t a t i o n .
can i n f a c t be de-
(SEC. 1)
PROBABILITY AND STOCHASTIC PROCESSES
23
NonetheMuch o f t h e m a t e r i a l i n t h e p r e s e n t c h a p t e r i s c l a s s i c a l i n n a t u r e . l e s s we do g i v e a number of new r e s u l t s : t h e i n v e s t i g a t i o n of monotone o r multivalued 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 , P.D.E.'s w i t h a r b i t r a r y growth of t h e c o e f f i c i e n t s of f i r s t o r d e r ; c e r t a i n p r o o f s of known r e s u l t s a r e a l s o o r i g i n a l ( a s f a r as we a r e a w a r e ) . We s h a l l now g i v e a very b r i e f o u t l i n e of t h e c o n t e n t s of t h i s c h a p t e r . S e c t i o n 1 p r e s e n t s a b r i e f account of p r o b a b i l i t y , i n p a r t i c u l a r with r e g a r d t o s t o c h a s t i c processes. S e c t i o n 2 g i v e s t h e t h e o r y of s t o c h a s t i c i n t e g r a l s and of s t o c h a s t i c d i f f e r e n t i a l c a l c u l u s . S e c t i o n 3 i s devoted t o 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 ( s t r o n g f o r m u l a t i o n ) and S e c t i o n 4 t o 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 (weak f o r m u l a t i o n ) . S e c t i o n 5 d i s c u s s e s e l l i p t i c P . D . E . ' s of Section 7 second o r d e r , and S e c t i o n 6 p a r a b o l i c P.D.E.'s of second o r d e r . gives t h e 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 o f c e r t a i n P.D.E.'s. F i n a l l y , S e c t i o n 8 i s devoted t o t h e Markov p r o p e r t i e s o f t h e 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 equation.
1.
REVIEW OF THE CALCULATION OF PROBABILITIES AND THE THEORY OF STOCHASTIC PROCESSES
1.1 P r o b a b i l i t y space.
Random v a r i a b l e s
We d e f i n e on R a o-algebra a , i . e . a s e t of p a r t i t i o n s of Let il be a s e t . R which i n c l u d e s t h e empty p a r t i t i o n and which i s s t a b l e on proceeding t o t h e denumerable complement, union and i n t e r s e c t i o n . The elements of R a r e c a l l e d elementary events and t h e elements o f a events. A p r o b a b i l i t y law on i s an a b s t r a c t measure on a , 2 0 and o f t o t a l mass equal t o u n i t y , i . e . a mapping from a + [ O , l l , such t h a t P(R) = 1, P ( 1 An) = P(An) for any sequence of events which a r e d i s j o i n t i n p a i r s . The
(R,a)
5
number P(A) i s termed t h e p r o b a b i l i t y of event A. I f P(A) = 1, we say t h a t A i s almost c e r t a i n ( 0 i s t h e c e r t a i n e v e n t ) . The t r i p l e t ( n , a , P ) is termed t h e probability space. Let A (1.1)
(1.2)
we put
denote a sequence o f e v e n t s ;
---
l i m sup A
n
=
n
n
l i m inf A = U n
n
n
U A a m
n
Am
a
.
The event l i m sup A ( r e s p . l i m i n f An) comprises t h e s e t of t h e W contained i n an i n f i n i t y (respectiv:ly, except i n a f i n i t e number) o f A We have
l i m inf A, C l i m sup A,
.
.
If t h e s e two s e t s a r e i d e n t i c a l , we p u t : lim An = l i r n i n f A, = l i m sup ,A
.
I n p a r t i c u l a r , i f {A 1 i s an i n c r e a s i n g ( r e s p . d e c r e a s i n g ) monotone sequence, then l i m A e x i s t s ( l i m n A n = l i m An or l i m An = l i m + An depending on whether we a r e i n m e i n c r e a s i n g o r d e c r e a s i n g c a s e ) .
+
The following r e s u l t s a r e very u s e f u l i n t h e c a l c u l a t i o n of p r o b a b i l i t i e s
(1.3)
P(limninf ),A
s
l i m n i d P(A )
n
s
limnsup P(A )
P(lim sup A )
n
n
.
24
(CHAP. 2 )
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
In particular i f l i m A
(1.4)
limnP(An)
(1.5)
If
e x i s t s , we have
= P(limnAn)
h P(An)
<
rn
,
.
then
P ( 1 i m sup A ) n
n
I
0
.
The r e s u l t (1.5) i s known a s t h e Borel-Cantelli Lema. A subset
a .
W
c
a,
such t h a t Q i s a l s o a a-algebra, i s termed a
sub-a-algebra
of
we use t h e terminology Bore1 o-algebra of E , Let E be a t o p o l o g i c a l space; denoted by , S ( E ) , t o s i g n i f y t h e a-algebra generated by t h e open domains o f E, i . e . t h e s m a l l e s t o-algebra containing t h e open domains o f E . I n g e n e r a l , we w i l l have occasion t o work with E = R n ; t h e a-algebra A(Rn) i s then generated by t h e open cubes. A measurable mapping from 62 -+ E i s termed a random variable w i t h values in E. Hence f i s a R . V . (random v a r i a b l e ) i f we have
f-'(B)
E0
,
VB
E B(E)
.
Let f . , i E I , be a family o f mappings from 62 -+ E. We denote by d f i , 1 E I) t h e s m a l i e s t o-algebra of p a r t i t i o n s of 0 , f o r which all t h e mappings f . a r e measu r a b l e , We c a l l 3(fi, i E I) t h e o-algebra generated by t h e f u n c t i o n s l f i .
-
Let f k be a sequence of Rn-valued R.V.'s such t h a t fk(W)
f(w)
v
i
then f ( w ) i s a R . V .
Expectation;
1.2
Let
f
c o n d i t i o n a l expectation
be an Rn-valued R . V . which i s integrable re1 t i v e t o t h e me s u r e P.
We
Put (1
-6)
Ef = jQf(u)dP(w) = mathematical expectation of
f.
We denote by LP(Q,Q,P;Rn), 1 $ p < -, t h e space of (eq$valence&lasses) of random v a r i a b l e s with i n t e g r a b l e p t h power. The space L (P,d,P;R ) denotes t h e space of t h e (equiva ence c l a s s e s ) of R . V . ' s whose e s s e n t i a l upper bound i s f i n i t e ( i f n = 1 we w r i t e L ( Q , Q , P ) ) .
5
If fk i s a sequence of R.V.'s with values i n R n , we say t h a t (1.7)
(*)
f k -. f a . s .
U o s t surely.
(*I
if{w:fk-.
f j
I
(SEC. 1)
PROBABILITY AND STOCHASTIC PROCESSES
25
N a t u r a l l y , we a l s o have a l l t h e c o n c e p t s of convergence i n t h e s p a c e s
LP,
l s p s m
( i n t h e s t r o n g , weak, s t a r s e n s e ) a l t h o u g h p r o b a b i l i s t s do n o t employ any s p e c i a l terminology i n t h i s c a s e ( e x c e p t p e r h a p s f o r s t r o n g convergence i n L2, which is sometimes termed convergence i n t h e q u a d r a t i c mean). Let X b e a
by
2 R.V. E L (Q,Q,P;Rn) and l e t B b e a sub-a-algebra of
a.
We d e n o t e
('L Q,B,P;R")
t h e c l o s e d sub-space i n
(1.9)
EPX=
L2 o f t h e W-measL a b l e R.V.'s.
x
p r o j e c t i o n of
We p u t
L~(Q,B,P;R~) ,
on
The R . V . E L2(Q,B,P;Rn) t h u s d e f i n e d i n u n i q u e manner) i s c a l l e d t h e c o n d i t i o n a l e x p e c t a t i o n of X w i t h r e s p e c t t o 4 . We may e x t e n d i t s d e f i n i t i o n t o R . V . ' s which a r e i n L1. + Suppose+that X i s r e a l - v a l u e d
X- = Min(X , n ) b e l o n g s t o L 2 .
t h e n X+ and X- a l s o b e l o n g t o L1 and and i n L1; Thus we c a n u n i q u e l y d e f i n e
From t h e d e f i n i t i o n of p r o j e c t i o n , we have
Hence, t a k i n g q = 1, we s e e t h a t
(1.11)
EY; = EX;s EX'S
EIXl
<
m
. + .
Furthermore, s i n c e t h e sequence X; i s i n c r e a s i n g , t h e sequence Yn 1 s l i k e w i s e i n c r e a s i n g . Indeed from (1.11)we have E(Yi+l
- Y;)r)>
We can t h e n t a k e
so t h a t
which i m p l i e s
if
Y+
0 t/q> 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
<
la ,
m
, then 1
.
have a more precise estimate than (1.53), namely
(1.54)
t tWe now assume that 3 is right continuous, and that X(t) is a right continuous 3 submartingale such that there exists an integrable R.V. Y, satisfying (1.55)
E(Ylgt) 2 X(t) Vt
.
We put X(-) = Y. Let S and T be two stopping times such that S 2 T. R.V's X(S) and X(T) are integrable and satisfy
(*)
Then the
It is always possible to complete by considering the o-algebra generated by d and the set of negligible partitions. This will be done systematically in what follows, without particular mention of the fact being made.
STOCHASTIC D.E.’s & P.D.E.’s OF ORDER 2
32
(CHAP. 2)
The result (1.56) is called Doob’s optional sampling theorem; it generalises the submartingale inequalities to stopping times. In practice this will generally have to be applied over a finite interval C0,al and hence S S T S a. We shall always be able to take Y = X(a) = X(t) for t t a and (1.54) will still hold. STOCHASTIC INTEGRALS
2.
2.1
The Wiener process
Let (B,C7,P) be a probability space. A real-valued process w(t), t 5 0, is a Wiener process (or Brownian motion process) if it satisfies the following properties: (2.1)
V tl,..,t,,
(2.2)
w(tl)
Ew(t)w(s)
...w(t ) I
is a Gaussian vector with mean 0.
min(t,s) V t , s Z 0
.
We shall assume the existence of such a process (see, for example, MACKEAN [ll). We note the following properties (2.3)
,
w(0) I 0
(2.4) w ( t 2 )
Indeed
-
w(tl),
w(t4)
Ew(tI2 = t
- w ( t 3)
-
are independent if t
1
5
t2
S
tg
5
t,,
.
- -
-
EC(w(t2) w(tl))(w(t4) W(t3))l’ t 2 t, t2 + tl = 0 and thus w ( t 2 ) - w ( t l ) , w ( t ~ + ) - w ( t 3 ) are uncorrelated Gaussian R.V.’s, and are therefore independent.
Let pt be the o-algebra generated by w(s), 0 (2.5) p
E[w(t)
IF’]
t
w(s)
vt
2 s
S s S
.
Indeed, w(t) - w(s) is independent of w(A), for martingale and we have
(2.6)
E[(w(t)
t. We have
h
E [O,s]
.
Hence w(t) is a
- w ( s ) ) 2 Ips 1 = t - s (t 2 s) .
In fact, the property (2.6) characterises continuous martingales which are zero at 0 and which are Wiener processes. More precisely we have the following result, due to P. Levy ( * ) .
Let w(t) be a continuous re al process satisfying w(0) = 0,and THEOREM 2.1. l e t 3 t b e an increasing f a m i l y of sub-0-algebras of Lz such that w(t) i s a 3 martingale and such t hat (2.7) (*)
E[(w(t)
- w(s))’I#]
= t
-
8
Vt 2
A proof is given in the book by DOOB [11.
;
(SEC. 2 )
33
STOCHASTIC INTEGRALS
then w ( t ) is a Wiener process. The above c h a r a c t e r i s a t i o n i s very simple t o check i n a p r a c t i c a l a p p l i c a t i o n , s i n c e it assumes nothing about t h e d i s t r i b u t i o n s . The o r i g i n a l proof ( s e e DOOB C11) i s somewhat t e c h n i c a l . We s h a l l g i v e ( s e e Section 2.5) a simpler proof following MTNITA-WATANABE [ll, but t h i s i s based on 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 c a l c u l u s which w i l l be developed l a t e r . I n t h e following we s h a l l u s e t h e n-dimensional Wiener process. By d e f i n i t i o n t h i s i s a s e t of Ewi(t)wj(s)
(2.8)
n
s c a l a r Wiener processes w . ( t ) such t h a t
= bij min (+.,a),
i, j
I 1.
.n
.
We say t h a t w ( t ) i s a strmdardised n-dimensional Wiener process.
t
More g e n e r a l l y , we s h a l l t a k e an i n c r e a s i n g family 3 of o-algebras, and w ( t ) w i l l be an ?-valued continuous st martingale s a t i s f y i n g w(0) = 0 , and
where I i s t h e i d e n t i t y matrix. By t h e above c r i t e r i o n , w ( t ) i s a standardised ni s not known, we can always t a k e dimensional Wiener process. I f t h e family for t h e o-algebra generated by w ( 8 ) , 0 S 8 S t .
st
zt
2.2
I n t r o d u c t i o n of s t o c h a s t i c i n t e g r a l s
t
t
,
t E [OPT] and w ( t ) a continuous 3 martingale We t a k e an i n c r e a s i n g family a with v a l u e s i n Rn, zero at 0 and s a t i s f y i n g ( 2 . 9 ) ( * ) . We introduce t h e following spaces of (equivalence c l a s s e s o f ) processes: (2.1 0)
Lz(O,T) = (q(t;U) a . s . I z I q ( t ) lPdt
(2.1 1 )
<(O,T)
i s measurable, adapted, scalar-valued, and
<
= ( q E L:(O,T)
a]
,p2
s
1
/ E :llp(t)lPdt
-
< + -1
We s i m i l a r l y d e f i n e L$(O,T;Rn) and #(0,T;Rn) f o r processes with values i n Rn. The space M $ ( o , T ; R ~ ) i s a Banach &space of Lp((O,T) x Q,dt Q dP;Rn). We s h a l l be concerned more p a r t i c u l a r l y with t h e c a s e p = 2 and with t h e H i l b e r t space
$O,T;&
,
which w i l l be w r i t t e n a s 0 t o simplify t h e n o t a t i o n . Suppose we have a number N which w i l l u l t i m a t e l y t e n d t o +
-,
and l e t k =
T 7.
(*)
For t h e moment we s h a l l not make use of t h e f a c t t h a t w ( t ) i s a Wiener process, s i n c e t h e proof of Theorem 2 . 1 which we g i v e depends on developments which w i l l be described l a t e r .
(**)
It would be p r e f e r a b l e t o w r i t e
flt(O,T)
3
, but
t h i s would be t o o cumbersome.
3L
(CHAP. 2)
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
With any function (2.12)
'p
E
=
'Pk(t)
The sequence 'pk E 0 . that
, we
0
I 1,;
associate 'pk defined by
~ j ( ds)ds,nks ~ - ~ t) < ~ <
t
k
.
.
2
'p in L (0,T;L2(Q,g,P;Rn)) Furthermore, it is clear Thus 'pk -. 'p in 9. We have thus proved the following property:
'p
the set of piecewise-constant functions
(2.13)
E
0 is a dense subspace in 0 .
Suppose we now have a piecewise-constant function a decomposition of C0,TI:
to = 0 with (2.14)
Since For 'p
'p
E
n l 1
("+Ilk
< tl <
t2
..* < t * =
'p 6
there thus exists
Q ;
T
v ( t ) = 'pn in [t,, tn+,[, n = 0 , t E 0 , 'p" must be 3 n measurable.
... N-1 .
@ a piecewise-constant function, we put (see (2.14))
(2.1 5)
where 6.n denotes the inner product of two vectors E,n in R". We have (2.1 6)
t since
E3
Dl w ( t m t l )
=
W(tm)
, from the
- w(t,))
EI'pn.(w(tn+l)
-
t
= E([E3 n ( w ( t n + l )
n 2
= (tncl-tn)El?
I
properties of martingales. Furthermore,
12
- w(t,))*I
w(tn))(W(tn+l)
, from
'pn.
9")
(2.Y), hence (2.16). 2
The mapping I is thus linear from a dense subspace of 0 to values in L (n,Q,P) and continuous as regards the 0 norm. This extends by continuity to 0 in a form
'p
-
I(?)
'.
We have for
'p
E 0
(SEC. 2 )
STOCHASTIC INTEGRALS
35
The R.V. I(cp) is termed the stochastic integral (of '7 ) , the following proposition:
We have thus proved
-.
The stochastic integral ( 2 . 1 7 ) defines a continuous linear PROPOSITION 2 . 1 mapping from Q L ~ ( Q , Q ,PI, such t h a t EI('p) = 0
(2.18)
V
'p f Q
2
(2.1 9)
EI(d2 =
(2.20)
EI((p) I(+) = E l i dt).C(t)dt
11p ' 119
V cp,+
f
Q
.
2 The property (2.20) is an immediate consequence of the fact that L (Q,O,P) and are Hilbert spaces, and of (2.19): Remark 2 . 1
If cp(t) is a matrix,
Jr
'1
is the vector with components
,
'pi(t).dw(t)
where 9 . is the ith row of
cp(t)dw(t)
0
'p.
It can easily be shown that
(2.21 Remark 2.2 to Lz[O,T;Rn]
-
We can also define the stochastic integral for processes cp belonging In fact, suppose we have
The sequence Put
if
The sequence 5 by definition is
5
s
N
, x..(z)
converges on
I
o
if z > N .
pN , and therefore a.s. on S l ,
to a R.V. which
36
STOCHASTIC D.E.'s & P.E.D.'s OF ORDER 2
(CHAP. 2)
From this we deduce, in particular, the estimate
In fact we have
1:
$'(t).ddt)
qN(t).dw(t)
S'
o l q ( t ) I 2d t s N
if
and hence (2.22).
There follows from (2.22) the following property (2.23)
lolcp,(t) T
if
1: J-:
2
cp(t) I d t * 0
dT
Tn(t).dw(t)
Remark 2.3 (2.24)
in probability, then
cp(t).dw(t)
in probability.
rn
If [ a , @ ] c [O,T] we put cp(t).dw(t)
=J:
X r a , @ ) ( t ) cp(t).dw(t)
where characteristic function of [ a , @ )
X[.,p)
Let 5 be a bounded
3'-rneasurable
5 x[,,@)(t)cp(t) E Q
R.V.;
,
.
the function cp
if
c
P
,
and thus (2.25) (2.26)
E j
E
cp(t).dw(t) =
c2( j : "(t).dW(t)I2
0
= El:
c2 x[,,@)(t)
= E C2f:lp(t) Relation (2.26) also holds with 5 instead of
-
c2
I2dt
Icp(t)I2dt
.
(use
5= (Y-O2 and the linearity). From the definition of conditional expectations, (2.25) and (2.26) mean that (2.27)
EL/:
cp(t).dw(t)
laa]
=
0
(SEC. 2 )
2.3
Let
STOCHASTIC INTEGRALS
37
Properties of the stochastic integrals as a function of the upper bound 'p
f @
I
I(t) =
(2.29)
.
$[O,T,Rn]
s"
We can define for all t
E
C0,Tl:
'p(s).dw(s)
We are thereby defining a stochastic process. We assume that we have chosen a modification of I(t) which is separable (see (1.44)). Furthermore, I(t) is a E(I(t)
martingale, since for t
la")
= I(s)+
= I(S)
E(
,
j:
'p(A) .dw(A) from (2.27).
Z s
we have:
la")
We can therefore apply the estimates (1.53) and (1.54) for separable martingales (we apply (1.54)with a = 2 , which is permissible since
E I(t)2 = .Ej:l'p(s)
.
I2ds
We thus have
Furthermore, if is piecewise constant, then I(t) is a continuous process (in We shall now show that by virtue of ( 2 . 3 0 ) , view of the continuity ( * ) of w(t)). and for arbitrary q, in 0, we can find a modification of the process I(t) which is continuous. Consider a sequence of piecewise-constant functions may have to extract a subsequence, we may assume that 2 1
Il'p,
-
'pll
5-
'p
*
2"
Therefore we also have
b"+,-1I,'
2
4 2"
s-
Furthermore, from (2.30) applied to
(*)
'p
n+l
-%
we have
We have not needed to use this property until now.
-
'p
in 0.
Although we
38
(CHAP. 2)
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
4
Since the series 2 4 n converges, it follows from the Borel-Cantelli Theorem 2"
that a.s. we can find N(w) such that n
t
N(w) implies
Thus a.s. the series
converges, uniformly in t, to a function which is continuous in t. a.s.
5"
'p,.dw(s)
However for all :],t
In other words
-. a function continuous in t.
cpn.dw(s)
-
I ( t ) a.s. (although we may have to extract a
new subsequence). We have thus established the result. 2 This property extends to the case in which 'p E LW(O,T;Rn).
In fact, we have
Since the right-hand side of (2.32) i s continuous, the left-hand side is likewise. Furthermore
It then follows that
and consequently we can improve (2.23) as follows:
2.4
Ito's formula
Let a, b (2.35)
be two processes such that a E Li(O,T;Rn)
.
; b E L:(O,T;&(Rrn;R"))(*)
Consider a family S(t) of adapted continuous processes with values in Rn, which is dependent upon the initial value S(0) (R.V. with values in R", ZO-rneasurable) and which is defined by
( * ) In other words, b(t) is an n
x
m matrix and b. (t)E L$O,T)
.1J
Vi,j
.
STOCHASTIC INTEGRALS
(SEC. 2 )
THEOREM 2.2
Asswne t h a t ( 2 .3 5 ) holds.
Rn x CO,Tl, belonging t o
L e t Y ( x , t ) Se a functional on
x i0,Tl);
C2”(Rn
39
ue then have:
(2.37)
Proof. I n o r d e r t o keep t h e n o t a t i o n t o a minimum, we s h a l l c o n f i n e o u r s e l v e s t o proving t h e r e s u l t i n t h e s c a l a r c a s e . We t h u s have t o show t h a t (2.38)
Y(C(t),t)
= Y(g(O),O)
+I:+ (Y;
Suppose t h a t we have proved ( 2 . 3 8 ) f o r with
and with E N ( t ) corresponding t o a N , bN.
a N ( t ) -. a(t)
a.s.
-
1
Yia +?Y;2 a,
b
b bounded.
W e can a p p l y ( 2 . 3 8 )
We have
in
L’(o,T)
in
L2(0,T)
N - . m
b&t)
a.s.
b(t)
and hence from ( 2 . 3 4 )
sup
Wst
]CN(t)
- <(t)l
I:
0
(in probability).
To w i t h i n an e x t r a c t e d subsequence we may t h u s assume t h a t we have
(2.39)
a. s .
a -. a N b N -. b SUP
GS41T
lCN(t)
in
L’(0,T)
in
L2(0,T)
- Sit) I
-
0
.
Using t h e p r o p e r t i e s of Y and t h e f a c t t h a t ,€ i s continuous, w e e a s i l y deduce by proceeding t o t h e l i m i t t h a t ( 2 . 3 8 ) holds w i t h a , b We can t h u s assume t h a t
.
(“I
w ( t ) has v a l u e s i n Rm, where m can be f n , f o r g e n e r a l i t y .
40
(CHAP. 2)
STOCHASTIC D.E.’s & P.D.E.’s OF ORDER 2
we have
We shall now prove (2.38) in the case t = T. The proof is the same for all t. This will prove (2.38) for t fixed, a.s. w, in other words the process on the righthand side of (2.38) is a modification of the process on the left-hand side. Since both these processes are continuous, it then clearly follows that a.s. all the trajectories are equal. Consider a subdivision of C0,Tl into o , k ,
Y(C(T),T)
-I?-1 Y(E(O),O)
...., Nk
N-1
5
= T.
Jo [Y(C((n+l)k),(n+l)k)
= k-0 Z [Y(E((n+l)k),(n+l)k)
We have
- Y(C(nk),k)]
- Y(E((n+l)k),nk)]
Applying TaylorIs formula, we obtain
Furthermore, the second summation on the right-hand side of (2.41) is equal to
N-1
C
kk0
{kl)’
g(E(nk),nk)
a(t)dt +
N-1
C
k 0
g(C(nk),nk)lP)’ b(t)dw(t)
We note that
However, a.s.
1-1’ s(C(t),t)a(t)dt 0
br
a.s.
= I + If
,
(SEC. 2 )
41
STOCHASTIC INTEGRALS
We have t h u s shown t h a t
where
c ( t ) represents
( E ( t ) , t ) t o shorten t h e notation.
However, using (2.40) we have
since
N- 1 a. s .
n--0
&% ( C ( n k )
J‘z’)k
b2(t)dt)-jo
T
dt ( e ( t ) ) b 2 ( t ) d t
we can s e e t h a t t o o b t a i n ( 2 . 4 6 ) it i s s u f f i c i e n t t o show t h a t
STOCHASTIC D.E.’s
42
&
P.D.E.’s OF ORDER 2
(CHAP. 2 )
Let
We have
(2.48)
€1 I
PiIZI >
PI
SUP
4%T
lS(t)I
> 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
x ) .
44
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
Additionally,
from the properties of martingales.
X2 X X
6m,f
--
6 E.X2 ( & Xm)2 n n m
(CHAP. 2)
.
Hence by using (2.52) we have = Ck
-
g m&
c%2
E XG
N(N-I)
c2~2
5-
2
4 It then follows from Fatou's lemma that m(T) 5 2a CT is arbitrary, (2.54) clearly holds with C1 = 3Cz.
Remark 2.4
+
The formula (2.37) is called I t o ' s formula. Y € C2"(Rn
, and
3C2T2
since
cy
If
,
x ]O,T[)
the formula is applicable between two arbitrary instants tl of ( 0 , t)).
30,TC (instead
5 t2, E
Remark 2.5 To simplify the notation, we shall express (2.36) and (2.37) by saying that c(t) possesses the stochastic differential dS(t)
(2.59)
I
a(t)dt
+
b(t)dw(t)
and then Y(c(t),t) possesses the stochastic differential
= (Y;
dY(g(t),t)
(2.60)
1
+ Y;.a
+ ? tr Y;2.bb*)dt
+ Y;.b
dw(t)
.
\
We see that, in comparison with the rules of ordinary differential calculus, we now have an extra item. To express this, it is sometimes said that dw(t) is of the order of vdt (see P. LEVY [l], E. WONG [l]). 2.5 Applications of Ito's formula As a first application we shall give the following proof:
Boof of Theorem 2.1 We apply (2.37) with Y(x,t) = exp i U. it is also valid with Y complex). We have dY(w(t))
1
= ?YIuI
2
dt
+Y
(The linearity of (2.60) implies that i u.dw(t)
.
Integrating between s and t and taking the conditional expectation with respect to '3' , we obtain (2.61)
E[Y(w(t))
- Y(w(8))
I $1
=
- $lu12 E[f:
Y(w(h))dhl$]
,
(SEC. 2)
45
STOCHASTIC INTEGRALS
Let r- be a bounded and $-measurable
R.V.
E y ( w ( t ) ) q = E 's(w(s))tl
It follows from ( 2 . 6 1 ) t h a t
- 4u ' j t ,
.
~(i(w(h))tldh
If we put f ( t ) = E Y(w(t))tl we have f o r t t s f(t) = f(s)
-$lU/'j:
f(A)dA
,
and hence
We have t h u s proved t h a t
2 which proves t h a t w ( t ) - w ( s ) i s independent of $and Gaussian law with mean 0 and with v a r i a n c e t - s .
t h a t w ( t ) - w ( s ) follows a
Before g i v i n g a second example of a p p l i c a t i o n o f I t o ' s formula, we s h a l l f i r s t consider a s t o c h a s t i c i n t e g r a l with a stopping time a s an upper bound. Let T be t
3 and l e t '3 be t h e o-algebra of t h e events
a stopping time with r e s p e c t t o previous t o T .
We put
X,W
(2.62)
=
1
if
t < z
I 0
if
t2.7.
We observe t h a t X T ( t ) E 0. adapted
.
We can consider
I(T)
More g e n e r a l l y , l e t
=
rl E
' 3
, then
j,:
'?(t).dw(t).
be two stopping times w i t h
T 1, T~
0 1 7, 1 we have
If i n a d d i t i o n
'F
2-
a.s.
;
t
-+
q
q ( t )is
46
OF ORDER 2
STOCHASTIC D.E.'s & P.D.E.'s
Let
q
be a bounded and 3 -measurable R.V.
we deduce from (2.63) after multiplication by
(CHAP.
2)
Taking account of the fact that
q
that
(2.64) (2.65)
and hence that
(2.66)
Jz
The relation (2.66) also follows from Doob's optional sampling theorem ( * ) (see
(1.54)) since
cp(s).dw(s) is a martingale; the same applies for relation (2.67) since the process
is also a martingale, as may readily be verified.
Remark 2.6. - Ito's formula can also be expressed using stopping times with t respect to 3 , since ( 2 . 3 7 ) holds a.s. for all t. We shall now give a second example of application of Ito's formula.
(*)
t
At least if 3 is right-continuous.
(SEC. 2)
STOCHASTIC INTEGRALS
THEOREM 2.3 - L e t
T~
p E MPCO,T;Rnl, p
w
t is a stopping time with respect to 3
.
5
2; then
We apply Ito’s formula to
We apply (2.70) with t = which is permissible. We take the mathematical expectation and we note that the expectation of the stochastic integral is zero. Hence
Now <(tAT ) is a martingale.
N
Thus from (1.53) we have
47
48
STOCHASTIC D.E.’s & P.D.E.’s OF ORDER 2
(CHAP. 2)
‘(2.72)
Taking i n p a r t i c u l a r t = T, and n o t i n g t h a t ~~2 T and
we deduce from ( 2 . 7 1 ) and ( 2 . 7 2 ) t h a t
from which we e a s i l y deduce t h a t
Making N +
Remark 2.7
and hence
+
-
and n o t i n g t h a t TN + T a . s . , we t h e n deduce ( 2 . 6 9 ) .
9
We deduce from ( 2 . 6 9 ) t h a t
I(y) i s
a l i n e a r , continuous o p e r a t o r from
To conclude t h i s s e c t i o n we g i v e t h e following theorem:
t Let T be a stopping time with respect to 3 and let w ( t ) be an THEOREM 2 . 4 n-dimensional standardised Wiener process which is a 3%martingale; then the process (2.74)
n(t)
I
w(t
+
2)
-
W(.)
is an n-dimensional standardised Wiener process and a
t
martingale with
qt = 3t+t ,
Proof. We s h a l l l i m i t o u r s e l v e s t o t h e s c a l a r c a s e .
From t h e g e n e r a l p r o p e r t i e s of
(SEC. 3 )
49
STRONG FORM
STOCUSTIC D . E . ' s :
t h e processes ( s e e S e c t i o n 1.4), we s e e t h a t G ( t ) i s f o r e s u f f i c i e n t a s a next s t e p t o apply Theorem 2 . 1 .
0t
measurable. We have
It i s t h e r e -
which i s e x a c t l y t h e same as E[G(t+h)
-
E[(i(t+h)
- G(t)l
3.
a'
.
]
= 0
- ;(t))*I ct]
so t h a t t h e r e s u l t t h e n f o l l o w s .
Go
t
fJ
=
h
We n o t e i n p a r t i c u l a r t h a t w ( t ) i s independent of
STOCHASTIC DIFFERENTIAL EQUATIONS: STRONG FORMULATION
3.1
D e f i n i t i o n of t h e problem
Let g ( x , t ) , o ( x , t ) be f u n c t i o n s of x € Rn, t E [o,T], measurable, with v a l u e s i n Rn and ~ ( R " ; R ~ ) Let w ( t ) be a s t a n d a r d i s e d Wiener p r o c e s s w i t h v a l u e s i n Rm. We seek a s o l u t i o n y ( t ) of t h e e q u a t i o n ( w r i t t e n , f o r t h e moment, i n formal f a s h i o n )
.
dy = g ( y , t ) d t + c J ( y , t ) d w ( t ) . The analogy w i t h 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 s i s obvious.
If o =
0,
then
y
= g(y,t).
satisfies
We now need t o make t h i s more p r e c i s e , i n p a r t i c u l a r with r e g a r d t o t h e i n i t i a l v a l u e s , which may be random ( t h i s being important i n a p p l i c a t i o n s ) .
t
We t a k e an i n c r e a s i n g family of o-algebras attfor which w ( t ) i s t & 3 martinga l e . Let T be a stopping time w i t h r e s p e c t t o 5 and l e t 5 be a J measurable R.V. We put 1 if t < r
x,
(t) =
0
if
t > 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 - <
+ / o ( x , t ) ) 2 c <(l +
< +=
.
1x12)
K/x
- x' I
(*)
Then there e x i s t s a process y(t) .which i s a solution of ( 3 , 1 ) , ( 3 . 2 ) , ( 3 . 3 ) such that
There i s uniqueness i n the following sense. (3.1), ( 3 . 2 ) , ( 3 . 3 ) and ( 3 . 7 ) , then (3.8)
a.s.
Proof.
1
y ( t )= y2(t)
for
Uniqueness. We nave for t
t E [O,TI E
KO, TI
and taking the mathematical expectation we have
If
.
y',
y'
are two solutions of
(SEC. 3 )
STOCHASTIC D.E.'s
Noting that the process (1 follows that a.s. (1
:
STRONG FORM
- X,(t)) /y1 (t) - y2(t) 1 - x,(t))ly'(t) - y2(t)l
is continuous, it then
=
o v
t
which is the same as (3.8) Existence. (3.1 2)
We put YO(t) = 5.
t
and
- x7(s))g(yn-1(s),a,ls
(3.l3)yn(t) = 5 +i:(1
-
+ i : x,(s)) ( l a(yn'l(s),s).dw(s)
which is permissible since (1
- x,(s))a
(yn-l(s),s)
E L,(~,T;R~) 2
.
From estimates analogous to those in the uniqueness proof, we obtain (3.14)
- y"(t)
EIy"+l(t)
where
L
I
2(T+1)8
12s L I Z Elyn(s)
.
By iterating (3.14), we obtain
But E1y1(s)
- F;I2sL T I?(1+E)5.12).
Consequently, we have
(3.1 6)
Elyn+'(t)
Furthermore, we have
and hence (from (2.31))
- yn(t)I21
C
F
.
- y"-'(s)
1 ' 4 s
STOCHASTIC D . E . ' s
52
&
(CHAP. 2 )
P.D.E.'s OF ORDER 2
I t then follows t h a t
+zo
Consequently, s i n c e t h e s e r i e s on t h e right-hand s i d e of (3.17) converges, a . s . the series 5 ( y n + l ( t ) y n ( t ) ) converges uniformly i n t which means t h a t
(3.18)
-
yn(t) - y ( t )
uniformly i n t .
Since y n ( t ) is a continuous p r o c e s s , it then follows t h a t y ( t ) i s a l s o a conFrom ( 3 . 1 3 ) , y ( t ) = 5 f o r t E [ O , TI and, by r e c u r r e n c e , ( 3 . 1 ) tinuous process. i s seen t o hold. By noting t h a t
E l y n ( t ) I 2 s 3{E1cl2+
f
+
TJ:(1
we o b t a i n
+
E l y n ( t ) 1 2 s C(l
E/yn-'(s)I2)ds
E / E (2)
+
Cl:
+
f[:(l
E\Yn-l(s)l2ds
and by recurrence ( t a k i n g C > 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@
<
(x,t) I
(1
+
1 x 1 ~ );
then t h e conclusion of Theorem 3.3 s t i l l holds.
Proof. For uniqueness, only t h e condition (3.34) i s needed. For demonstrating existence, t h e proof of Theorem 3.3 shows t h a t t h i s amounts t o proving
We use I t o ' s formula, which gives
and by v i r t u e of (3.45)
STOCHASTIC D.E.'s : STRONG FORM
(SEC.3)
59
We thus again have the property (3.46), so that the result follows. 3.4 Use of monotonicity methods In general, if the coefficients of the stochastic differential equation are only continuous, as opposed to being Lipschitz continuous, we do not know whether there exists a solution (in the strong sense) of the equation. The concept of a solution in the lJeak sense was introduced with the precise objective of avoiding this difficulty. We shall see, however, that the property of monotonicity (see (3.49) below) in conjunction with simple continuity also allows existence and uniqueness to be established in the strong sense. Additionally, we shall be able to treat arbitrary growth at infinity. We take a mapping h : Rn x [ o ,Tl
(3.47)
+
R" satisfying
h(x,t) is continuous with respect to x to t,
and measurable with respect
where
The property (3.49) is termed the monotonicity property. We also take functions g,o such that
(3.50)
g,o satisfying (3.4), (3.5).
For simplicity, we shall take the initial instant to be
(3.51)
T
= 0, and we take
5, a R.V. with values in Rn, $-measurable such that E ( C I4 < + m.
THEOREM 3.5 Under the a s s m p i o n s (3.47), t o ( 3 . 5 1 ) , there e x i s t s an Rnvalued process y(t), s a t i s f y i n g t
,
(3.52)
y(t) is adapted to 3
(3.53)
y(t) is continuous on [ o , T l ,
(3.54) a.s. Y(t)
+I:
h(y (s),s)ds = 5
+I:
g(y(s),s)ds
+
(3.55)
2
Furthermore, if yl(t) = y (t) s a t i s f y ( 3 . 5 2 ) ,
. . ., ( 3 . 5 5 )
then
60
(CHAP. 2)
STOCHASTIC D.E.’s & P.D.E.’s OF ORDER 2
(3.56)
a.5.
1 y (t) = y 2 ( t )
vt c
[o,~]
.
Proof * Uniqueness. Let yl, y2 be two solutions. d(yl so
- y2)
= [-h(yl)
+
that
We have
(U(Y
+ h(y2) + g ( y l ) 1
1-
U(Y
2
1).
- g(y2)] d t
dw
Taking the mathematical expectation and using (3.49) and (3.50), we obtain
and hence (3.56) (see the proof of uniqueness in Theorem 3.1)
Ezistence. We shall treat the right-hand side of (3.54) as a Lipschitz-continuous perturbation(as is suggested bg its expression). We start by solving the following problem. Let f(t)be an R -valued process, which possesses the stochastic differential
(3.57)
d f ( t ) r a ( t ) d t . + b(t).dw(t)
9
f(0)
= 0
where
We shall construct a process y(t) satisfying (3.52), (3.53) and
(3.59)
y(t)
5
+j:
I 5
+ f(t)
h(y(s),s)ds =
a(s)ds
.
+I:
b(s).dw(s)
(SEC .3 )
STOCHASTIC D.E.'s
and we seek a s o l u t i o n
z
(3.60)
z(t)
+I:
: STRONG
61
FORM
of h(z(s)
+
f(s),s)ds
t
.
E
Equation (3.60) i s , f o r w f i x e d , an ordinary d i f f e r e n t i a l equation i n t h e We s h a l l show t h a t it possesses one and only one solsense of Carathgodory. ution. Uniqueness follows from a p p l i c a t i o n of t h e monotonicity property. In t h e proof of e x i s t e n c e , Carath6odory's theorem cannot be a p p l i e d without prec a u t i o n s , s i n c e it i s not c e r t a i n t h a t an adapted process w i l l thereby be obtained.
a)
Solution o f (3.60) by d i s c r e t i s a t i o n
, O,k, ...,Nk
We consider a subdivision of [o,T] We put which w i l l tend t o + m ) .
(3.61)
f n = f(nk)
(3.62)
hn(x)
P
Jz-l
k
)k h b , t ) d t
= T (where N i s an i n t e g e r
.
We next consider t h e equation ( i n R n )
(3.63)
$ +hn(x)
= q E R"
.
This possesses one and only one s o l u t i o n . P(x)
=
I n f a c t t h e mapping
$ + hn(x) - 1
i s continuous and
*
h(o,t)dt). x
(n-1 )k
B
o
for
1x1 o p
,p
- q.x
sufficiently large.
From a v a r i a n t of Brouwer's f i x e d p o i n t theorem ( s e e f o r example J . L . Lions Uniqueness t h e n follows as a [ 2 1 ) t h e r e e x i s t s E , 151 S p such t h a t P ( 5 ) = 0 . r e s u l t of t h e monotonicity. Moreover, t h e mapping q + x
We have
i s continuous.
I n f a c t , l e t rl
J
-+
rl and
62
STOCHASTIC D.E.'s
P.D.E.'s OF ORDER 2
&
(CHAP. 2 )
or
I X J. 15
from which it follows t h a t
constant.
t h e c o n t i n u i t y of hn we s e e t h a t x!
-t
J
By e x t r a c t i n g a subsequence and using
x.
Since t h e l i m i t i s unique f o r any sub-
sequence, we have x. + x . J We next consider t h e recurrence r e l a t i o n s n -
y
k y
p
n-1
+ h ( y ) = F
,
fn-l
n
>
l
,
(3.64) znryn-fn
,
z
0
E y
0
= g .
We put
fk(t) = fn
in
[nk,(n+l)k[
,
I
By recurrence, we s e e t h a t y n , z" a r e a r e adapted processes.
(3.66)
e,(t)
-
Skmeasurable.
We s h a l l now show t h a t , a t f i x e d w
z(t)
Hence y ( t ) and z , ( t ) k , we have
tl t
where z ( t ) i s a s o l u t i o n of ( 3 . 6 0 ) . From t h i s w i l l then c l e a r l y follow t h e exi s t e n c e of a s o l u t i o n z ( t ) which i s an adapted process. Let us t h e r e f o r e prove We deduce from (3.64) t h a t (3.66).
n
z - 2 k
n-I
*
hn(zn
+ fn)
R
or
en
I
'\'
-
n k
n-I
)+ hn(zn + f n )
0
. en
I
0
;
STOCHASTIC D.E.'s
(SEC. 3 )
63
: STRONG FORM
however
(3.67)
1znj2
-
lz
n-1 2
I +
mn(fn). zns 0
.
From ( 3 . 4 8 ) , it can e a s i l y be seen t h a t I h n ( f n ) I s C ( w ) (independent of n ) . deduce from ( 3 . 6 7 ) t h a t
IZk(t)
12<
+
C2T
+j:
lZk(')
We
I2dB
and from G r o n w d l ' s i n e q u a l i t y we see t h a t
(3.68) For f i x e d t
k
I
independent of t ( b u t dependent on w )
lz ( t ) 5
.
we use t h e n o t a t i o n nt = i n t e g e r p a r t of t f k .
By adding t h e r e l a t i o n s (3.64) between 1 and nt, we o b t a i n
and t h u s
From (3.68), we have l y k ( s ) I S p ' , and hence, from (3.48) h ( y k ( s ) , s ) l i e s i n a 2
bounded domain of L (0,T;R"). by t h e index k, such t h a t
zk(t) - z ( t ) h(yk(t),t)
+
We can t h u s e x t r a c t a subsequence, again denoted
weauy star i n cp(t)
L"(~,T;R~) 2
weakly i n L (o,T;Rn)
It then follows from ( 3 . 7 0 ) ( n o t i n g t h a t z k ( t ) = z
(3.72)
z(t)
+I: cp(s)ds
= C
"t
.
) that
64
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
and (3.66) then follows a l s o . ( 3 . 6 0 ) , we have
(3.73)
cp(s)
It now remains t o show t h a t , d e f i n i n g y ( t ) by
= h(y(s),s) = h(z(s)
We s h a l l f i r s t show t h a t
Now we have
Now f r o m (3.72)
i(t)
I
- q(t)
and hence
s o t h a t we deduce from (3.75) t h a t
But we a l s o have
(CHAP. 2)
+ f(s),s) a.e.
65
STOCHASTIC D.E.'s : STRONG FORM
(SEC. 3 )
and t h e r e f o r e noting t h a t h ( y ( s ) , s )
E
L 2 , we o b t a i n
s o t h a t we indeed have ( 3 . 7 4 ) .
h
I n order t o then prove (3.73) from ( 3 . 7 4 ) , we use t h e monotonicity property of as i n MINTY [11, "21. Let p ( t )
E
2
L (0,T;R").
We next make A gives
+
0.
We have
By v i r t u e of (3.48) we can use Lebesgue's Theorem, which
and hence ( 3 . 7 3 ) . b)
A p r i o r i estimate f o r t h e s o l u t i o n of ( 3 . 5 9 )
We s h a l l mw prove t h e following e s t i m a t e f o r t h e s o l u t i o n of (3.59):
(3.76)
Eldt) 1 ' .
+
C (
-.
We again use t h e d i s c r e t i s a t i o n procedure.
(3.77)
E lynI2
<
We note t h a t
This may be v e r i f i e d by r e c u r r e n c e , from ( 3 . 6 4 ) .
(*)
We r e c a l l t h a t lbI2 = bb*
.
f'
B olb(s)
I t then follows t h a t
1'88)
(*I.
66
STOCHASTIC D.E.’s
&
P.D.E.’s OF ORDER 2
(CHAP. 2)
It then follows, using (3.77) and (3.64), that we also have n n 2 Elh(y)I
(3.78)
<
0 1 .
Furthermore, from the relation
we deduce
Likewise tyn- yn-l) .fn + khn(yn) .f”
!3.80)
E
fn.(fn-
.
fn-l)
However we note that (3.81)
Now (3.82)
&Y””
.(fn-fn-’)
m
Eyn-’.
- yn-1 .(fn-
s
nk a(t)dt (n-1)k
’[ ’*
Eyn’l
Similarly (3.83)
(n-l)k
.(fn- fn-l
I
a(t)dt
n-I k ~ f .a
fn-1).
+ Q”-’ .E[jnr
b(t)dw(t)
’k]
(n-l)k
= kEyn-’.an
.
.
We thus obtain from ( 3 . 8 0 ) , taking account of (3.81), (3.82), (3.83):
&Yn .fn
(3.84)
+
- Eyn-l.fn-1
kEhn(yn).fn
I
- wn-.an l+ Elfn12 - Elfn-’ l 2 - k Efn-’ .a
and by addition
- k j=1i
Ehj(yj) .fj = E/fnI2 Eyj-’ .a’ + k j=1 Combining (3.79) and (3.85) we obtain
(3.85)
Eyn.fn
(3.86) ElynI2
- E/fnI2- ElF;I2 + 2 J;kEhJ(yJ).yj-
Taking into consideration that hj(yj) .y’ 1. h’(o) .yJ we deduce from (3.86) that
’
- kE
fj-l. a’. j=l
2 k Z, E(yj-’- f’-’).a’S
0
.
(SEC. 3 )
67
STOCHASTIC D.E.'s : STRONG FORM
which may be rewritten in the form
By using Gronwall's inequality and Fatou's lemma, we then deduce ( 3 . 7 6 ) .
c)
Further a priori estimation We shall now prove the following estimate which is more precise than ( 3 . 7 6 ) :
We introduce the processes
and bN, which has a parallel definition. same way as f is defined from (a,b). corresponding to fN and let ZN
= YN
- fN
Let f be defined from (a,,bN) in the N Let yN be the solution of ( 3 . 5 9 )
-so
First we have
t eN(s).h(fN(s),s)ds
(3.90) Moreover
-o
Hence there exists a subsequence N' such that
sup QS%T
IfN,(t)
- f(t)I
We thus have sup
IfN,(t) 15 C(w)
It follows from ( 3 . 9 0 ) that
,
N'
a.s.
-. +
.
6a
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
(CHAP. 2 )
To within a new extracted subsequence, we may thus assume that
-
aNl(t) -z(t) h(yNl(t),t)
weakly star in L~(o,T;R~) weakly in L2(0,T;Rn)
q(t)
e N , ( t ) -. e ( t ) = t
-
Using an argument based on the monotonicity, as before, we proceed to the limit in (3.57) written with fN, from which we deduce that a.s. e
N'
(t)
-
,
V t
e(t)
z(t)
a solution of (3.60).
Ito's formula applied to yNl gives
Squaring, we obtain
Taking account of the fact that (3.92)
E ( Y " ( ~ ) ) ~ SC
.
&",
bNl
are bounded, we deduce
Passing to the mathematical expectation and using the Cauchy-Schwarz inequality, we deduce from (3.91) that
From Gronwall's inequality and making N' d)
-f
a,
(3.89) then follows readily.
Solution of (3.541
4
Consider a mapping S : Mw[O.T;Rnl z(.)
= S(Y(.))
where z ( . ) is a solution of
+
4
Mw[O,T;Rnl defined by
(SEC. 3 )
STOCHASTIC D.E.'s
:
STRONG FORM
Frog t h e p o i n t s a ) , b), c ) , t h e mapping that S
i s a contraction for Zi(.)
= S(Yi(.))
k $
S i s w e l l defined. s u f f i c i e n t l y l a r g e . Now i f
i = 192
We s h a l l now show
9
we have d(z1-B2)+ ( h ( z l )-h(e2))dt = (g(y,)%(Y2))dt
+
(a(Yl)-
U(Y,))
-
dw
from which we deduce
or
Squaring, and t a k i n g t h e mathematical expectation (which i s permissible s i n c e yi,
zi E M;[O,T;f])
we o b t a i n ,
E / z l ( t ) - z 2 ( t ) 1 4 s C (EJElel-z21 4d s
+ EI~/Y,-Y,I~~s)
so t h a t , f i n a l l y , we have
By i t e r a t i n g we o b t a i n
(3.95) there exists a k We have, s i n c e
k k s s h t h a t S 0 i s a c o n t r a c t i o n . Let y be a f i x e d p o i n t of S 0. S 0 y = y , and t a k i n g account of (3.95)
There t h u s e x i s t s a r e p r e s e n t a t i v e of y and making p + m , we o b t a i n Sy = y. (namely Sy) with continuous t r a j e c t o r i e s . This process i s c l e a r l y a s o l u t i o n of ( 3 . 5 2 ) , ( 3 . 5 3 ) , (3.54). To prove ( 3 . 5 5 ) , we use
?1l Y ( t ) I 2 s $1E.I2
+jz
g(y).yds +jzY.u(y)dw + j : t r
uo*(y)ds
-jz
h(o,s).y(s)ds
Then squaring and t a k i n g t h e expectation (which i s permissible s i n c e y m then r e a d i l y deduce ( 3 . 5 5 ) .
E
4
Mw),
we
70
STOCHASTIC D . E . ' s
1 sign x
-1
I
if
x > O
if
x
<
P.D.E.'s OF ORDER 2
&
0
a r b i t r a r y value between - 1 ,
1 if
x = 0
(CHAP. 2 )
.
This equation was i n v e s t i g a t e d i n MACKEAN [ll, using a monotone approximation method. N a t u r a l l y , none of t h e r e s u l t s obtained e a r l i e r i s a p p l i c a b l e h e r e . Nonet h e l e s s , we s h a l l show t h a t t h i s equation f a l l s i n t o a general c l a s s , as long a s we use multivalued o p e r a t o r s .
h : Rn
We t a k e a multivalued mapping assume t h a t
(3.96)
v
(3.97)
sup YEhb)
x
,
h(x)
,
f b
-P(Rn) = a
h(o) 3 0
s e t of p a r t i t i o n s of Rn.
We
,
lYi s ~ ( 1 + 1 ~ 1 ~ )
(3.98)
The property (3.98) i s again ( formally )
(h(xl)
c a l l e d t h e monotonicity property.
- h(X2)).Cx1J2)
We w r i t e
20
On t h e b a s i s of (3.99) we say t h a t h i s maximal monotone., This i s j u s t i f i e d by t h e following argument. We d e f i n e on t h e s e t of multivalued o p e r a t o r s t h e order r e l a t i o n
hi c h
(3.1 00)
2
. )
vx ,
h l b ) chp(x)
and we note t h a t t h e s e t of monotone multivalued operators ( i . e . t h o s e s a t i s f y i n g ( 3 . 9 8 ) ) i s i n d u c t i v e f o r t h i s order r e l a t i o n . A monotone multivalued o p e r a t o r s a t i s f y i n g (3.99) i s t h e n maximal f o r t h i s order r e l a t i o n , i n t h e s e t of monotone multivalued o p e r a t o r s .
ExcnnpZe:
I f n = 1 and i f
i s an i n c r e a s i n g mapping from
f
R
-t
R , then
x -. h(x) = [f(x'),f(x+)] i s maximal monotone.
I n p a r t i c u l a r , s i g n x i s maximal monotone.
w
A fundamental c h a r a c t e r i s a t i o n of a maximal monotone o p e r a t o r i s as follows ( s e e BREZIS C21):
(3.101)
V h
>
+ U ( x ) 3 y has a unique s o l u t i o n x, v y fixed, = J X y . The mapping JX i s a contractionfrcan Ef.+ Rn.
0, t h e equation x
denoted by ( I + Xh)-'y
(SEC. 3 )
STOCHASTIC D . E . ' s
We put
: STRONG FORM
S-Jh
(3.1 02)
A
h A E
'
It can be shown ( s e e BREZIS
[el) that
hA i s maximal monotone and L i p s c h i t z continuous with r a t i o 1 / A .
(3.103)
Finally, since 0
E
h ( o ) , we have
We note t h a t we have
In f a c t
= \(XI
+ \(XI
.(X-JAX)
= A IhA(x)
.JA(X)
l2
+ \(x) .JA(x)
.
Now
\(XI
-
,
f h(JA(x))
and from (3.98) (with y2 = 0 , x2 = 0) y1
h(xl)
which t h e n gives (3.105).
,
yl.x, 2 0
We t a k e functions g , u such t h a t
and
5 , a R . V . with values i n Rn such t h a t 5 is 30 measurable and
(3.1 07)
I <
Eli 4
.
0
We s h a l l now prove t h e following theorem:
...,
THEOREM 3.6 Under the assumptions (3.96h, (3.99) and (3.106), (3.107), there e x i s t s a process y ( t ) with values i n R s a t i s f y i n g
t
y ( t ) i s adapted t o 3 and continuous,
(3.1 08) (7.109)
ass.
Y(t)
+I:
ds)ds
P
5
+J: g ( y ( s ) , s ) d s +[:
u(y(s),s).dw(s)
where q ( t ) is an adapted measurable process such t h a t q(t;w) f h(y(t;w)) (3.1 10)
V t f [O,T]
,
a.e.t
,
a.s. w
E / Y ( t ) 1 4 S C(E(CI4
+
, 1)
.
STOCHASTIC D . E . ’ s
72
& P.D.E.’s OF ORDER 2
(CHAP. 2)
The soZution of (3.108), (3.109), (3.110) is unique in t h e sense of ( 3 . 8 ) . UniqUeneSS. Let y1 , y 2 , n 1 , n2 be two p o s s i b l e s o l u t i o n s . 1 2 1 2 d(y -Y [-(n -rl + g(Y1) g ( y 2 ) ] d t + ( d y 1 ) - d y 2 ) ) .dw Using t h e monotonicity p r o p e r t y , we deduce from I t o ’ s formula t h a t
Proof.
.
-
We have
and t h e proof i s then concluded as for Theorem 3.5.
Existence. We s t a r t by solving t h e following problem. process which possesses 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 (3.111)
df(t)
I
a(t)dt
+ b(t).dw(t)
where (3.1 12)
la(t)I4dt
<+
-,
, f(0)
EJ:
Let f ( t ) be a n R“-valued
= 0
lb(t)I4dt
< +-
;
we s h a l l c o n s t r u c t a process y ( t ) which s a t i s f i e s (3.108) a s w e l l a s
(3.1 13)
dt)
+I:
h(y(s))ds
3
t;
+
f(t)
More p r e c i s e l y , t h e r e exists n ( t , w ) , a process which s a t i s f i e s t h e p r o p e r t i e s of (3.109) and (3.1 14)
Y(t)
+jz
n(s)ds
=g +
f(t)
.
Putting z(t) = Y(t)
- f(t)
we seek t h e s o l u t i o n of t h e multivalued equation
(3.1 15)
+ h(z(t)+ z(0)
f(t)) 3
0
5
P
Let us now c l a r i f y t h e meaning of (3.115). Let H E L2(Q,O,P;R’I) , Through an abuse of n o t a t i o n , we s h a l l s t i l l denote by h t h e monotone multivalued opera t o r from H + H defined by
- -E
H
.
We s h a l l show t h a t t h e r e e x i s t s a unique z ( t ) : 10, T I
+
Y’ E h ( t )
. )
Y”(w) E h(;(w))
a.s.
v I,
y
H such t h a t
STOCHASTIC D.E.'s : STRONG FORM
(SEC. 3 )
dz dt E
(3.1 16)
- dz
a.e. If we now put
c
CO(O,T;H)
E h(z(t)+ f(t))
-=, dz
dt) =
,
t is 3 measurable, a.e. t
, Z(t)
V t
,z
L=(O,T;H)
Y(t) = Z(t)
73
, z(o)
t is 3 measurable,
= 5
.
+ f(t)
then the pair (y, rt) meets our requirements. a)
Solution of (3.116)
Let zx be a solution in C 1(0,T; H) of
-+
(3.117)
dz h dt
h
z,(o)
= 5
which exists since x
-t
(Z
h
+ f )
I
0
h
hh (x + f) is Lipschitz continuous from H
-f
H.
Then yh possesses the stochastic differential (3.11 8 )
dyh(t)
+ \(yh)dt
= df(t)
.
From Ito's formula it then follows, taking account of (3.105), that
Squaring and taking the mathematical expectation ( * ) , we obtain (3.119)
Ely,(t)l4S
C(E15I4
+ Ejrl*(t)14dt + El:lb(t)l4dt)
However, since Jh is a contraction and since J IJh(Yh)
I5
h
(0)
.
= 0, we obtain
lYhl
so that EIJh(Yh(t)) 14S E / y h ( t )
l4
.
But then, by virtue of ( 3 . 9 7 ) , it follows that (*)
It is necessary first to consider a bounded approximation of a and b and then to proceed to the limit.
74
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
I
E I ~ ~ ( Y ~ I( ~C(lGIJh(Yh(t)) ) )
[I 11
I n o t h e r words, i f have (3.120)
i4) s c1
(CHAP. 2 )
from
(3.119).
denotes t h e norm i n H , it follows from (3.117)t h a t we
de
I1h II 5 c1 dt
Furthermore, we have
However,
By v i r t u e of t h e monotonicity of h , we have
Consequently
and
Consequently, zA i s a Cauchy sequence i n C o ( O , f o r e e x t r a c t a subsequence z such t h a t
T ; H) when A + 0.
We can t h e r e -
An
e
(3.1 21)
hn
-.
e
in
C(O,T;H)
-
dZX dz 3 weakly i n L~(o,T;H). dt dt The l i m i t z thereSy obtained s a t i s f i e s t h e r e g u l a r i t y p r o p e r t i e s (3.110) dzx ( n o t i n g i n p a r t i c u l a r t h a t Y h,eA(t,w) and -(*;id) a r e measurable, and moreover clt t h a t V t , zA i s measurable, a . e . is measurable). dt +
at
3
at
kEC. 3)
STOCHASTIC D.E.'s
:
STRONG FORM
75
we also have
Then let E L2(0,T;H)
'p
and 1 E L2(0,T;H)
such that a.e. Y(t) E h('p(t)+ f ( t ) ) We have T
Jo
dzL ((r + Y(t.1,
+ f(t)
q(t)
- JX(zL+ f ) ) ) d t 2 0
and proceeding to the limit we obtain
1;
(3.1 23)
- e(t)))dt Z 0 .
((%+ Y ( t ) , + ( t )
If we now define a multivalued operator h"(cp)
h'
on L2(H) by
E h(cp(t)+ f ( t ) )
a.e. then is maximal monotone (it is clearly monotone and R(I$) (3.122) implies dz -m E h"(z)
h"
= L2(H))
and
and hence (3,116)- By applying Fatou's l a m a we deduce from (3.119) the estimate
(3.1 24)
E / Y ( t ) 14S C(E1EI2
+I:
EIa(t)l4dt
+I:
.
Elb(t)I4dt)
Note that the uniqueness of z , the solution of (3.116),or of y, the solution of (3.113), follows immediately from the monotonicity property. b)
Solution of (3.109)
4
We now consider the Banach space M [O,T; Rnl. As in Theorem 3.5, we shall W define a mapping
S : M:[O,T;Rn] as follows: for where z
(3.125)
y(.)
-. M:[O,T;Rn]
E M,4[o,T;Rn]
we put
is the (unique) solution of z(t)
+J:
h(z(s))ds 3 5
+I:
g(y(s),s)ds
S(Y(.)) =
+I:
5(.)
dy(s)s)dw(s)
The remainder of the proof is identical to that of Theorem 3.5. To conclude, we shall give a nunber of supplementary results.
rn
.
76
STOCHASTIC D.E.’s & P.D.E.‘s OF ORDER 2
(CHAP. 2 )
THEOREM 3.7. I f 3 i n t h e statement of Theorems 3.5 and 3.6, t h e asswnptions ( 3 . 5 0 ) o r (3.106) are repZaced by ( 3 . 3 5 ) , then t h e eoncZusions remain v a l i d . We s h a l l only v e r i f y it This i s analogous t o t h a t of Theorem 3.3. Proof. as i n t h e proof of We consider g f o r t h e replacement involving Theorem 3.6. N’ ‘N Theorem 3.2. Let 5 be defined by
1
N
‘No
5 if 161 s N 0 otherwise
and l e t y be a s o l u t i o n of N
+I:
YN(~) and
h(YN(s))ds 3 t;N
eN =
(*I
i d It
z
+Sz
gN(YN(s),a)ds
Ol(yN(t),t)
Pie,
<
TI
=
PI
SUP
E S T
IYN(t)l
z
I
uN(YN(S),s)dw(s)
P %f .
We s h a l l f i r s t prove t h a t
(3.126)
+
N)
-
o
.
Now we have
Squaring once again and t a k i n g t h e expectation, we o b t a i n
From t h i s we deduce ( 3 . 1 2 6 ) , a s f o r Theorem 3.3. In t h e c a s e of Theorem 3 . 5 , we make use of t h e f a c t t h a t h ( x ) . x 2 h ( o ) . x . The remainder of t h e proof i s analogous t o t h a t o f Theorem 3.3. COROLLARY 3 . 1
have
(3.1 27)
-
Under t h e asswnptions of Theorem 3.7 and if in addition we
EIEI&<+-
,
m z z
,
then t h e solution y defined i n Theorem 3.6 s a t i s f i e s We have r e t a i n e d t h e assumption E1Cl4 < + m , which s i m p l i f i e s t h e s i t u a t i o n (*) r e l a t i v e t o t h a t i n Theorem 3 . 3 ; it i s no longer necessary t o introduce t h e function
(SEC. 3 )
STOCHASTIC D.E.'s : STRONG FORM
For the sozution
(3.129)
J:
y
77
defined i n Theorem 3.5 it i s aZso necessary t o assume t h e t
Ih(o,s)/ads
< +-
.
We then have (3.130)
E/y(t)lh
<
C(1
+
E\Cla
+jr
/h(o,s)Ihds)
.
Let us now prove t h i s p r o p e r t y , f o r example i n t h e c a s e o f Theorem 3.5. t h u s assume t h a t (3.127) and (3.129) h o l d . Consider yN, a s o l u t i o n of
We
From I t o ' s formula, we have
S i n c e gN, uN a r e bounded, it r e s u l t s from (3.131) t h a t Ejy,(t)12P < + '=, f o r
a l l p. We can t h e r e f o r e t a k e t h e e x p e c t a t i o n of b o t h s i d e s of ( 3 . 1 3 2 ) . Taking account of t h e f a c t t h a t h ( x , s ) . x L h ( o , s ) . x and of t h e growth assumptions on g , o , we o b t a i n
A p p l i c a t i o n o f Gronwall's lemma t h e n l e a d s i m e d i a t e l y to ( 3 . 1 3 0 ) .
m
78
STOCHASTIC D . E . ' s
4.
& P.D.E.'s
STOCHASTIC DIFFERENTIAL EQUATIONS:
(CHAP. 2 )
OF ORDER 2
WEAK FORMULATION
Synopsis Suppose we a r e given g ( x , t ) : Rn x C0,Tl + Rn and o ( x , t ) : Rn X C0,TI +I?(*;Rn). We s h a l l now d e f i n e a concept of 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 (4.1)
dY = g b , t ) d t
+
dY,t)dW(t)
which i s more g e n e r a l than t h a t introduced i n S e c t i o n 3. We s h a l l no longer t a k e t h e s e t (Q,a,P,zt,W(t)) w i l l appear a s an unknown, j u s t as does y ( . ) .
t o be f i x e d a p r i o r i , and t h i s
We s h a l l say t h a t we have solved ( 4 . 1 ) i n t h e weak s e n s e , i f t h e r e e x i s t s a of sub-a-algebras of U and p r o b a b i l i t y space (Q,a,P) , an i n c r e a s i n g family
at
two processes y ( t ) , w ( t ) with values i n Rn and Rm, such t h a t y ( t ) , w ( t ) a r e continuous and adapted t o 3t ,
(4.2)
(4.3)
on
w ( t ) i s a s t a n d a r d i s e d Wiener process and a
st
martingale,
The advantage of t h i s concept i s t h a t it allows us t o weaken t h e assumptions g and a which a r e necessary f o r e s t a b l i s h i n g e x i s t e n c e ( * )
Thus SKOROKHOD [11 has given an e x i s t e n c e theorem f o r s o l u t i o n s of (4.1) i n t h e weak sense, under t h e following assumptions (4.5)
g(x,t),
u(x,t)
a r e continuous on
f x [O,T]
,
(4.6) Also STROOCK and VARADHAN under t h e assumptions
(4.1)
C11 have proved t h e e x i s t e n c e
(4.7)
a ( x , t ) continuous on Rn
(4.8)
g ( x , t ) measurable and bounded.
x
CO,T],
o f a weak s o l u t i o n Of
bounded,
For t h e proof of t h e s e r e s u l t s i n t h e i r f u l l g e n e r a l i t y , we r e f e r t h e reader t o t h e works c i t e d ; f o r our p r i n c i p a l o b j e c t i v e h e r e , namely s t o c h a s t i c c o n t r o l , it does not present much of a problem i n a p p l i c a t i o n s i f we assume a c e r t a i n r e g u l a r i t y on u ( f o r example, a property of L i p s c h i t z c o n t i n u i t y with r e s p e c t t o x). On t h e o t h e r hand, it i s important t o remove t h e r e g u l a r i t y assumptions on g ( s i n c e t h i s i s t h e term s u b j e c t t o t h e c o n t r o l ( * * ) ) . This i s p o s s i b l e , q u i t e simply, by using t h e Girsanov transformation technique, a s we s h a l l see l a t e r .
Further( * ) We do not i n general have uniqueness without s p e c i a l assumptions. more, s e v e r a l concepts of uniqueness a r e p o s s i b l e ( s e e YAMADA-WATANABE [11). (**)
See chapter 3.
(SEC.
4)
STOCHASTIC D.E.'s
: WEAK
FORM
79
I n p a r t i c u l a r , ( s e e Section 4 . 2 ) , we s h a l l prove t h e following theorem:
THEOREM 4 . 1
We assume rn = n , u, u -1 continuous and bounded on Rn
(4.9)
(4.10)
Iu(x,t)
- a ( x l , t ) 15 CIXJI
[O,T],
x
I v X,XI
E R~
,
g ( x , t ) measurable and bounded.
(4.1 1 )
Then f o r (P,ff) givvn, there e x i s t s a probability measure P on (Q,d, a s well a s an increasing f a m i l y 3t and two Rn- valued processes y ( t ) , w ( t ) , such t h a t ( 4 . 2 1 , ( 4 . 3 ) , ( 4 . 4 ) are s a t i s f i e d .
4.1
Fundamental lemma
t
Let (n,c?,P) be a p r o b a b i l i t y space, 3 an i n c r e a s i n g family of sub-aa l g e b r a s of 0 ,w ( t ) a standardised n-dimensional Wiener process which i s a 3 martingale; we s h a l l now prove t h e following lemma: LEMMA 4 . 1 Let a ( t ) be a process folloLling assumptions i s s a t i s f i e d :
(4.1 2)
E exp(i
(4.1 3)
3
p , >~
+ 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)
,
<
E exp p l a ( t ) I 2 s
- '12 ' 0
Ia(t)
l2
cv
dt] =
We f i r s t suppose t h a t (4.12) i s s a t i s f i e d .
t c [o,T~
1
,
.
We approximate a by a
piecewise c o n s t a n t .
k'
We put
nk
a
(n-1) k
a(t)dt
,ny
1
,
and
= an f o r t E [nk, (n+l)k[
ak(t)
.
We s h a l l now show t h a t ( 4 . 1 4 ) i s t r u e f o r a without any p a r t i c u l a r assumpk tions. We need t o show t h a t we have (4.1
5)
Let us denote by Hk t h e q u a n t i t y i n square .brackets i n ( 4 . 1 5 ) .
E$ =
N-1 do an .(w((n+l)k) - w(nk)) -7d0 k/anI2 1
N-I
We have
.
STOCHASTIC D.E.’s & P.D.E.‘s OF ORDER 2
80
(CHAP. 2 )
Now
Since aN-l i s
3 ( N - 1 ) k measurable and w(Nk) - w((N-l)k) i s independent of
3(N-1)k, t h e R.V. aN-’.(w(Nk)-w(
(N-l)k)) possesses with r e s p e c t t o 3 c o n d i t i o n a l p r o b a b i l i t y l a w which i s Gaussian with mean 0 and with v a r i a n c e
I
k aN-l
I *.
Thus
Erexp aN-l .~~(Nk)-(w(N-l)k))l~(~-’’~]= exp k
12
N-1
and
We next t a k e t h e c o n d i t i o n a l expectation with r e s p e c t t o 3(N-2)kp and by recurrence we thereby deduce ( 4 . 6 ) . We know, moreover, t h a t
(4.1 6)
a.s.
and hence
[i
(4.17)
ak
ak.dw
+
in
a
2
a.dw
L‘((o,T;R~)
.
We may t h e r e f o r e e x t r a c t a subsequence, a l s o denoted by
ak.dw
(4.18)
-s
a.dw
k, such t h a t
c1
a.s.
Consequently
exp
(4.19) where
H
9d e x p H
a.s.
r e p r e s e n t s t h e q u a n t i t y i n square b r a c k e t s i n ( 4 . 1 4 ) .
We s h a l l now e s t i m a t e
a
(SEC. 4)
81
STOCHASTIC D.E.'s : WEAK FORM
We u s e H b l d e r ' s i n e q u a l i t y and we n o t e t h a t EJIP = 1, so t h a t
1
1
E[exp 9 ] ' + ' S (E exp ~ ( I + Z ) ~q -( 9 ) JE1ak12dt)
'/
.
Now
2
$ql+E)
if we t a k e q
- q (l+E) 2
=
(l+cq)
, andcswht'nate
2:
2 q
Consequently, we o b t a i n ( s i n c e
E[exp
%]'+'<
(E exp
I T1 + 51
- -91
+ c(l+q)
J:
1+w(l+q+eq) q-1 Io
+
6
.
lakI2dt s jT,ia12dt) 1
($+ 6){:
i . e . , E[exp %]lieSC.
/aI2dt)
It t h e n f o l l o w s t h a t
QA[exp
%] -.
0
Eexp%-
uniformly i n k , when P ( A )
-+
0 , and consequently
EexpH=l,
so t h a t we have ( 4 . 1 4 ) i f ( 4 . 1 2 ) i s s a t i s f i e d . We can a l s o deduce t h a t f o r 0
(4.20)
E{exp[j::
tl
5
t 2 Z T , we have
5
-
a(t).dw(t)
lact)
1 2at]/
tl 3
1=
1
ass.
I n f a c t we f i r s t v e r i f y ( 4 . 2 0 ) f o r ci , i n t h e manner adapted above. proceed t o t h e l i m i t , on t h e b a s i s o f t h e same upper bounds. We now assume ( 4 . 1 3 ) t o be t r u e .
1:
exp[h
la(t) 12dt]
I
A1
j tt" l A(t"-t')
exp A(t"-t')
'"I'-
E{exp[Jey a(t).dw(t)
with A(s.
-si) < 1-1.
1+1
We have
S2"
<
-
2
Oa
.
t * la(t) 12dt]13t']
Ct,,t21 a s f o l l o w s :
We next c o n s i d e r a d i s c r e t i s a t i o n o f
<
1:
la(t) I2dt]
Ia(t) I2dt
<+
/a(*) I2dt]
Consequently, from ( 4 . 2 0 ) we have
t l = s1
>
< 1-1, it follows from (4.13) t h a t we have
I f t h e r e f o r e t"-t' s a t i s f i e s A ( t " - t ' )
E exp[h/:y
We t h e n
From J e n s e n ' s i n e q u a l i t y , we have f o r A
exp[--
5
.
8
m
I
t
2
= 1 a.s.
STOCHASTIC D . E . ' s
82
&
(CKAP. 2)
P.D.E.'s OF ORDER 2
We then approach (4.14) step by step.
Rmark 4.1. In the course of the proof of Lemma 4.1, we showed that, without making any particular assumptions of the type (4.12) or (4.13), we have
Remark 4.2. the process
It follows from (4.20) that under the assumptions (4.12) or (4.13)
t is a 3 martingale.
Remark 4.3.
If a(t) has complex values, i.e. if a(t) = 6 ( t ) + iy(t)
then (4.14) must be replaced by
(4.23)
$5;
E exprj; B(t).dw(t)
--2 Jf cT
IB(t) I2dt
+
+ is:
y(t).dw(t)
Iy(t) I2dt
- ij;
1
.
C V t f [G,T]
.
B(t).y(t)dt]
I
The property (4.23) is true if
3 q > 1,
(4.24)
L
>
0
such that
where (4.25)
3 p, C
>
0 such that E exp p(\y(t)
i2
+
\P[t)l2)
<
The proof is along lines analogous to those of Lemma 4.1. 4.2
Girsanov's Theorem
We shall now prove the following theorem: THEOREM 4.2.
We t a k e a famiZy
(Q,O,P,Jt,w(t))
and a process a(t) s a t i s f y i n g
(SEC. 4)
STOCHASTIC D.E.'s : WEAK FORM
one or other of the conditions (4.12), (4.13). probability law
a3
We associate with (W7)the
then for this probability measure the process ;(t)
(4.27)
I
w(t)
-/:
a(s)ds
t
is, on CO,TI, a standardised Wiener process and a 3 martingale.
woof.
Let JI
E
L2 (o,T;R") (non random).
We put (cf. (2.17))
(4.28)
since hcp + pY is deterministic. From Remark 4.3, and taking account of the fact that a satisfies the conditions of Lemma 4.1, and that cp,Y are deterministic, we see that the quantity in the curly brackets is equal to unity, so that we have (4.30). Furthermore, we shall now also show that we have
(4.31)
g[exp
ihI((p)]$]
f o r all
cp
= exp
E &O,T;Rn)
--$St I T ( ~ ) ds ]~
such that cp(s) = 0 for a
<
t
.
If in fact we write out explicitly the left-hand side of (4.31), taking account of the fact that (p 0 for s t, then we obtain for 5 3t measurable
(*)
Which is clearly a probability law on 51 by virtue of Lemma 4.1.
STOCHASTIC D.E.’s
84
If we now denote by X formula, we have
&
(CHAP. 2 )
P.D.E.‘s OF ORDER 2
the conditional expectation appearing in the above
But then, from ( 4 . 3 2 ) and using Lemma 4.1, we clearly have ( 4 . 3 1 ) .
,...,
Let ei e be a basis for Rn, and let Xt ( s ) = 1 if s > t , and 0 if s > t. The process W(t) defined by ( 4 . 2 7 ) is identically
which proves, of course, that w(t) is a standardised n-dimensional Wiener process. Furthermore for t 2 s
Since X , ( e )
- Xs(e)
I
0 if
e
< s , it follows from ( 4 . 3 1 ) that we have
which proves that w (t) - w ( s ) is independent of k This completes the proof ofkTheorem 4.2.
9 and wk(t) is a 3t
martingale.
Rmurk 4.1+. The transformation ( 4 . 2 6 ) , (4.27) will be called the Girsanov transformation. The result of Theorem 4.2 is due to GIRSANOV C11. Girsanov’s transformation allows a solution (in the weak sense) to be defined for certain stochastic differential equations. First, we note the following corollary:
(SEC.
4)
a?
STOCHASTIC D.E.‘s : WEAK FORM
We take a process
The conditions are those of Theorem 4.2. COROLLARY 4.1 y ( t ) which possesses the stochus6ic d i f f e r e n t i a l
(4.34)
dy(t) = a(t)dt
where
a ( t ) E LLIO,T;Rn]
+
b(t)dw(t)
.
b ( t ) E L:[O,T;d(Rn;Rn)]
and
A f t e r Girsanov’s transformation, t h e process y ( t ) possess t h e st o c h a st i c differential dy(t) = (a(t)+b(t)a(t))dt
(4.35) Proof.
From ( 4 . 3 4 ) we have
(4.36) P a . s .
y ( t ) = y(o)
+h
a(s)ds
+[:
1
a(s)ds
+jz
We have t o prove t h a t
(4.37) Let b
a.s.
k
y ( t ) = y(o)
+
st
.
+ b(t)d;(t)
b(s)dw(s) V t
b(s)a(s)ds
E [O,T].
+jz
b(s)d;(s),
be a piecewise-constant approximation of b ( a s i n ( 2 . 1 2 ) ) . y k ( t ) = Y(O)
+
a(S)ds
+j:
However, it can e a s i l y be shorn t h a t
j:
1:
bk(s)dw(s)
Furthermore, f o r a subsequence
j: j:
bk(s)dw(s)
-.
bk(s)dw”(s)
-.
1’0
1:
jz
bk(s)dw”(s)
W € lO,T]. We put
.
bk(s)dw(s)
+i:
bk(s)a(s)ds
.
,
b(s)dw(s)
a.s.
P
b(s)dw”(s)
a.s.
P” ,
b k ( s ) a ( s ) d s -.
Since P and P a r e a b s o l u t e l y continuous with r e s p e c t t o each o t h e r , t h e l i m i t s a r e equal a.s. P o r P, and hence ( 4 . 3 7 ) . Application of Corollary 4 . 1 allows us t o prove Theorem 4.1.
Proof of Theorem 4 . 1 We s t a r t from (Q,C7,P,,;gt and w,(t)), where w (t) i s a standardised n-dimens i o n a l Wiener process and a 3 t ~ a r t m g a l e , We Aefine ( i n unique f a s h i o n ) t h e process y ( t ) , a s o l u t i o n of
(4.38)
dy(t)
=
dy(t),t)dw,(t)
which i s l e g i t i m a t e i n view of t h e assumptions on u.
(4.39)
We next put
a(t) = o”(y(t),t)g(y(t),t)
which s a t i s f i e s a l l t h e assumptions o f Theorem b . 2 .
Hence Corollary 4 . 1 i s
a6
(CHAP. 2)
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
applicable.
Defining
(4.40)
w(t)
I
-
w,(t)
Jz
a(s)ds
and P by
(4.41) t we see that P, 3 Remark 4.5.
, y, w
meet our requirements.
We can replace assumption (4.11)by g(x,t), measurable such that
In particular if u = I and if
(4.43)
Ig(x,t)
(4.44)
E
IS
K(1+. Ix 1)
~ X Pkoly(O)l2
<
m
, ,
assumption (4.42) is satisfied (note that y(t) = y(o) + wl(t)).
In fact this
last result is still true even if u # I but satisfies ( 4 . 9 ) , (4.10). This However, follows from the upper bounds on the 'probability density' of y(t). this is anticipating the results of Section 8. Remark 4.6. The concept of a weak solution clearly brings out the somewhat In general we take s2 = C(O,T;Rn), 0 a 'accessory'character of (Q,a,P,st,w(t)). The Bore1 u -algebra, a'=u -algebra generated by w ( s ) , 0 i s 5 t,if w E n. measure )I on C(O,T;Rn) associated with the process y(t;w)()I = P if s2 = C(O,T;Rn)) is in fact the most important concept; this is defined in unique fashion under the STROOCK-VARADHAN assumptions, ( 4 . 7 ) , (4.8) (see loc. cit.), and if moreover 2 a 1
5.
,
a
>
0
(independent of x, t)
LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS OF SECOND ORDER Synopsis
Important relationships exist between the theory of stochastic differential equations and the theory of linear partial differential equations. These relationships will be used continually in the present work, and they will be described in Section 7. We start by giving the essential elements from the theory of linear partial differential equations (P.D.E. Is), which will be sufficient for our aims. Tne present section is concerned with elliptic P.D.E.'s of second order. 5.1 5.1.1
Preliminary results
Sob0 lev spaces
Let. 0 be an open subset of Rn. We denote by a(8) the set of indefinitely differentiable functions with compact support in 8 . We shall denote the
(SEC. 5 )
ELLIPTIC P.D.E.'s
x = (xl,
d e r i v a t i v e s of a function ?(x), i s a system of n i n t e g e r s 2 0 :
87
OF ORDER 2
...,x
) , i n t h e following manner: i f
j
we put
We r e c a l l t h a t i f
-
i s a sequence from
n
t h e supports of 'p
- tl j ,
n
J
D 'pn
+
do),we
say t h a t 'p
n
-. 0
i n a(0) i f
l i e i n a f i x e d compact subset of 0
0 uniformly i n 0 ,
We denote by &(a) t h e space of d i s t r i b u t i o n s on 0, i . e . t h e duaZ space of 8 ( 6 ) o f t h e l i n e a r forms 'p -+ ( f , ' p ) which a r e continuous on .&& . ( * ) If f i s l o c a l l y summable on 0, then 'p
- l0
f@%
=
(f,d
d e f i n e s a d i s t r i b u t i o n on 0, a l s o denoted by f , which i s permissible i f we i d e n t i f y f u n c t i o n s and c l a s s e s of f u n c t i o n s t o be equal except perhaps on a s e t of measure zero.
. . , j n ) , we denote by DJf t h e d i s t r i b u t i o n
If f i s a d i s t r i b u t i o n and j = (jl,. defined by
(DJf,'p) We then denote by DJf
6
Lp(O.),
v
IE
(-?)I'l(f,DJ'p)
ptp(0) the
where
Ijl
=
j,+..+jn
.
space of f u n c t i o n s f E L P ( d such t h a t
/ j l s m.
We equip ?yp(0)
with t h e (Banach) norm
(5.1)
The spaces
pYp are
We w r i t e
c a l l e d Sobolev spaces.
f(0) = wm#2(0)
(5.2)
which f o r t h e norm ( 5 . 1 ) with p = 2 , is a H i l b e r t spacer
do)is n o t
dense f o r m > 0. We denote by Worn'' (0)t h e c l o s u r e of &) We n o t e t h a t if 8 =Rn then but
$"(Rn)
= pnp(Rn)
in
&a)
pf].' (0) ( f o r
i s dense i n p * p ( 6 ) , p = 2, Worn''
=
.
We can d e f i n e WsYp(Rn), f o r an a r b i t r a r y r e a l PEETRE C11, LIONS-MAGENES c11, MAGENES [11)
.
s , by i n t e r p o l a t i o n ( s e e LIONS-
The following i n j e c t i o n p r o p e r t i e s of Sobolev spaces w i l l be very u s e f u l :
(*)
$1.
Throughout t h i s volume we s h a l l confine our a t t e n t i o n t o t h e r e a l case.
a8
(CHAP. 2 )
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
(5.3)
u.
If
<
P P p ( R n ) C?-"
then
P
q(p)(Rn)
(continuous i n j e c t i o n )
where
I n p a r t i c u l a r i f p = m , we have
(5.4) m < n
if
c Lq(m)(Rn)
wm'p(Rn) P
1 1 m m= ;; .
with
'
(continuous i n j e c t i o n ) ,
Furthermore:
(5.5)
m = F , P * P ( R ~ c) L;",,(R~)v
if
q
finite
i s a Frechet space f o r t h e family of norms
where L:Oc(Rn)
(
,
jrxks l f ( X ) l q ~ ) " q
;
N n 1,2,...
n
,
a 0 < a < 1 then u E P v p ( R n ) P is a . e . equal t o a continuous f u n c t i o n and we have
m
if
(5.6)
/4x)
O - +
- U(Y) I s c
llull
?PP
Ix-Yla
where C denotes a c o n s t a n t ;
(5.7)
+ 1, t h e n u E P y p ( R n ) i s a . e . equal t o a continuous f u n c t i o n an4 P f o r any compact s e t K and any a < 1, t h e r e e x i s t s a constant C ( K , a ) such t h a t if m =
lU(X)-dY)
I5
,
E K
.
C ( K , ~ ) ~ ~ U ~ ~ ~ X , Y~ ~ X - Y ~ ~
We s h a l l also use t h e following SoboZev inequaZity:
(5.8)
If
u
E
tYmYp(Rn) and
u
i f t h e numbers m , p , q , j , r , a 1
'
1
r
n
P
- = J + a (-
Lq(R"),
E
then
u
E
W J y r ( R n ) and
satisfy the relations 1 S p , q S
- 2n ) +
1
(14)-t
I n p a r t i c u l a r i f q = p , a = 1,m-j
9
Am S
a s 1
,
m-j-n/p
-
:
0s
<
ic i n t e g e r
m t 0.
= p , we recover ( 5 . 3 ) .
The p r o p e r t i e s ( 5 . 3 ) , (5.8) extend t o t h e spaces W""(O), provided t h a t d i s r= i s an m-times continuously d i f f e r e n t i a b l e manifold o f dimension n - 1, 8 b e i n g on only one s i d e of r
r e g u ~ a r( * ) , i . e .
do
.
( * ) This i s not t h e only p o s s i b i l i t y , b u t we s h a l l confine our a t t e n t i o n t o t h i s case.
(SEC. 5 )
ELLIPTIC P.D.E.'s
a9
OF ORDER 2
I f now 8 i s a bounded open s e t , of which t h e boundary then
(5'9)
t h e i n j e c t i o n from W1'p(o)
r
i-
-. Lp(o) i s compact, p L 1.
2 1 and j , m a r e i n t e g e r s 0 S j <
=
is C
b8
1
,
More g e n e r a l l y i f
m , then t h e i n j e c t i o n from
i s compact f o r a l l p L 1 such t h a t
-pl > -jn 5.1.2.
+ -l - - m r
n
.
Trace Theorems
We s t a r t by introducing t h e spaces H s ( R n ) f o r s n9n -ni n t e g e r . We use t h e f a c t t h a t Fourier transformation i s an isometry from L ( R ) i n t o i t s e l f . If u
L ~ ( R " ) we w r i t e :
then
We extend t h e Fourier transformation by c o n t i n u i t y t o t h e SCHWARTZ space 8' of tempered d i s t r i b u t i o n s . We f i r s t d e f i n e t h e s e t
a
= 1U1x" U'D
E L?R~)
,v
a , p f (where xa = xla1..x2)
equipped with t h e family of seminorms
f o r which 8 i s a Frechet space.
It may be v e r i f i e d t h a t f o r a l l
u
E
8 we have
t % I = U V U E 8. and hence 3 f &((a;8). Likewise 5 6 & ( 8 ; 8 ) and Thus 3 i s an isomorphism from 8 on t o i t s e l f , with i n v e r s e 3. We note t h a t we have
s,. (ah
dx = /Rn u(3v)dx
v
u,v
E 8
.
The space I' i s t h e dual space of 8 , equipped with t h e dual s t r o n g topology. Hence i f u E 8' we have We d e f i n e 3,s E 6 8 ' ; I ) b y t r a n s p o s i t i o n .
< a , ?= > < L I , ~V >C
~E
8 .
It may be v e r i f i e d t h a t t h e formulas ( 5 . 1 0 ) a r e s t i l l v a l i d on
8.
We then
STOCHASTIC D.E.'s
90
define Hs(Rn)
(CHAP. 2 )
OF ORDER 2
& P.D.E.'s
by
8 ( R n n ) = {uIu E 8' , ( l + l E I 2 )
(5.11)
(where
2
151 = E l + . . . < , ) .
arbitrary, r e a l .
I n (5.11),s i s d u a l of H S ( R ~ ) .
2
5/2
2
3~ E L2(Rn)]
is identically the
We note t h a t H-'(Rn)
I f s = m, i n t e g e r , t h e d e f i n i t i o n ( 5 . 1 1 ) i s c o n s i s t e n t with t h e u s u a l defini t i o n of
H". a r e c a l l e d fractionaZ Sobolev spaces.
The spaces Hs(Rn)
(*),
We now consider an open subset 6 of Rn s a t i s f y i n g
r= - l),
(5.12) The boundary dimension ( n
(5.13)
of 8 i s an i n d e f i n i t e l y d i f f e r e n t i a b l e manifold of
r ,
8 being l o c a l l y on j u s t one s i d e of
6 bounded
6 =
(except i n t h e cases
Rn o r 8 = h a l f s p a c e ) .
We denote by dr t h e s u r f a c e measure on
induced by d x .
We t a k e a family 6.,j = 1..v of bounded open s u b s e t s of Rn, covering I', such t h a t f o r a l l j , t h e r d e x i s t s an i n d e f i n i t e l y d i f f e r e n t i a b l e mapping -*
c p p
= Y cp. J
from 6.+. Q = { y l y = ( y l , y n ) , Iy'I < 1, -1 < yn < l}, such t h a t J
t h e mapping Y
-*
=
cp;'(Y)
=
- IY~Y
a l s o being i n d e f i n i t e l y d i f f e r e n t i a b l e from Q -+ 6.
cp,
sends
'
J '
furthermore
E Q, Y, > 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).
<
u
E
&(8)
.
1
such t h a t u + u i n H ( 8 ) , and a sequence vn Now (un-vn)-
E
E
There e x i s t s a sequence
a(& such t h a t vn
.*
v in
1 Ho(0), and hence l i k e w i s e (u-v)-.
THEOREM 5.2. Under the asswnptions ( 5 . 2 9 ) , i 5 . 4 1 ) , ( 5 . 4 4 ) , ( 5 . 4 5 ) , ( 5 . 4 6 ) , there exists one and only one solution u E V n L (0)of (5.43). pro0f
Existence. Let us assume f o r a moment t h a t t h e problem has been solved f o r f E L m ( 0 ) , f t 0 , and then show t h a t it i s solved i n t h e g e n e r a l c a s e where f not n e c e s s a r i l y t 0; i n f a c t i f we igtroduce m = i n f f, t h e f u n c t i o n f - m i s b 0 and hence t h e r e e x i s t s u1 E V n L such t h a t (5.47)
and s i n c e -m b 0 , t h e r e e x i s t s (5.48)
vv
a ( u , , v ) = (f-m,v)
u2
E
v
V n Lm with
vv
a(u2,v) = (-m,v)
E
is
E
v
I n f a c t t h e same then u = u - u2 i s a s o l u t i o n . We t h u s suppose 0 5 f 5 C . 1 remark shows t h a t t h e g e n e r a l i t y w i l l not be l i m i t e d i f we suppose t h a t (5.49)
<
0
fo<
f(X)%
c
.
We put uo = 0 and we d e f i n e a sequence un by (5.501,
a(un,v)
+
h(un,v) = h(un-' ,v)
+
(f,v)
v
v
.
.
Taking account of
v E
We s h a l l now show t h a t we have (5.51)
uo
I
O'U
1
s u2
... s u"...
5;
*
1-
I n (5.50) we t a k e v = ( u ) We show f i r s t t h a t u1 2 0. t h e f a c t t h a t a(w,w-) = a(w- = a(w-1 (*)I
- a ( ( u ' 1-1 - h I (u' )-12 so t h a t (ul)- = 0 , i . e . u1 2 0. t h a t un L un-'.
(*)
= ( f , (u')-)
n-1 We suppose t h a t u
By d i f f e r e n c i n g (5.50), and (5.50),-,
where a ( w ) = a ( w , w ) , by d e f i n i t i o n
Un-2 ; we s h a l l now show we o b t a i n
96
STOCHASTIC D.E.'s
(CHAP. 2 )
OF ORDER 2
& P.D.E.'s
We then t a k e v = (un - u n - l ) - and by applying arguments s i m i l a r t o t h o s e used e a r l i e r t h e r e s u l t t h e n follows. 5 = c 6 1' 1 We f i r s t consider t h e case V = H (0); we can t h e n t a k e v = (un - c l ) + i n
We s h a l l now show by recurrence t h a t un
(5.50)n.
We o b t a i n
+ c
a(un- c l , ( u n - c,)+)
+ h(unor
Un-l,(un-
+ But
. a
h(c,- u"-',(un-
- f 5 0 and by
C1)+)
+ h1(un-
a((un- c,)+)
1
J 0 a 0(x)(un-
I
C,)+)
I
recurrence c
-
1
+
+so (i
(f,(u"-
c1)+12
c l ) + cix
0
.
C,)+)
ao- f)(un- cl)+
un-l t 0 . so t h a t ( u n
-
dx
e l ) + = 0.
I n t h e c a s e of g e n e r a l V, s a t i s f y i n g t h e assumptions o f t h e theorem, t h e property (5.51) s t i l l holds s i n c e , as we s h a l l s e e , a t each i t e r a t i o n t h e s o l u t i o n
.
i n t h e c a s e of V i s l e s s than o r equal t o t h e s o l u t i o n i n t h e c a s e of H1
Gn t h e s o l u t i o n i n $. We s h a l l now
t h e moment l e t us denote by n o t a t i o n u" f o r t h e c a s e of
.
insun From (5.451, (ul - 8')a(y1
-
(t',
V and f H 1 ( 0 )
E
For
t h e c a s e of V, r e t a i n i n g t h e show t h a t
.
We t h u s have
- ~l(u'- a1 )-I 2 = c 1 -1 u t u . The same argument a p p l i e s g e n e r a l l y . -
(u1 a')-)
so t h a t (ul -
By v i r t u e of (5.51)'and from t h e f a c t t h a t 8. i s bounded we have lunl 5 c o n s t a n t , and consequently w e can deduce from (5.50) t h e e s t i m a t e
(5.52)
1 1 aII 5
constant.
We deduce from ( 5 . 5 1 ) t h a t a.e.
un(x)
+
u ( x ) from below
and hence also
(5.53)
0 5 u(x) 5
c. B
However, it follows from (5.52) t h a t u E V. We a l s o n o t e t h a t un + u weakly i n V. We can t h e n proceed t o t h e l i m i t i n (5.50),, g i v i n g t h e r e s u l t t h a t u i s a solution.
(SEC. 5 )
ELLIPTIC P.D.E.'s
97
OF ORDER 2
Uniqueness. f
Let us assume f o r t h e moment t h a t uniqueness i s e s t a b l i s h e d i n V n L satisfies
(5.54)
f 1 f
> 0 (a.e.), f
Eo
when
chosen a r b i t r a r i l y > 0.
I f ( 5 . 5 4 ) i s not s a t i s f i e d , we introduce a constant k > 0 such t h a t f
the s o l u t i o n w
+
k 1 f
a.e.
of a(w,v) = (k,v)
v
v
E
V , w E V n Lm.
Then
a(u+w,v) = (f+k,v) V v E V
so t h a t
u + w i s unique, which t h u s e s t a b l i s h e s t h e uniqueness.
If V contains We s h a l l show t h e uniqueness i n t h e s e t of p o s i t i v e s o l u t i o n s . 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 ; i f i n f a c t u , u2 a r e two p o s s i b l e s o l u t i o n s , at l e a s t one of t h e s e not being a . e . 1 0 , we inkroduce
5 = min I i n f . u l , i n f . u21, 6 s and we n o t e t h a t
0,
s ( u i - 5,v) = (f-aoS,v) and f d O E 2 f 2 f o and
U i S
2 0
.
If V does not contain t h e c o n s t a n t s , t h e n we can conveniently adapt t h e followWe s h a l l show i n f a c t t h a t i f ing proof.
a(ui,v)
(5.55) then u1 = u
I
(f,v)
f 2 f o > 0 ,
vv
€
V
,
i=1,.2,
ui20
2'
It t h i s i s not t h e c a s e , t h e g e n e r a l i t y w i l l not be l i m i t e d i f we suppose t h a t u1 u2 ( a p a r t from p o s s i b l y having t o interchange u1 and u 2 ) .
We then l e t ( * ) a b e t h e
largest r e a l number 1 0 such t h a t
(5.56)
aul
5
u2 a . e .
We have
(5.57)
O < a < l .
We n o t e t h a t t h e r e e x i s t s a 5 such t h a t a < p < 1
(5.58)
pf
+ hpu,
I n f a c t , from (5.56), pf
+
5 f
+ hu2 a.e.
.
(5.58) is t r u e a f o r t i o r i if hpul If
+ haul
( * ) Here we a r e following t h e i d e a of LAETSCH [11 f o r t h e proof of t h e uniqueness of t h e s o l u t i o n s of c e r t a i n c l a s s e s of Quasi-Variational I n e q u a l i t i e s , which w i l l be i n v e s t i g a t e d i n Volume 2 .
STOCHASTIC D . E . ' s
98
(CHAP. 2 )
& P.D.E.-fs OF ORDER 2
, and t h e r e f o r e i f ( 1 - @ ) f o 2h(p-a)sup. u1 ; hence a l s o i f (1-p)f 1 h(p-a)ul and it i s p o s s i b l e t o choose B such t h a t t h i s i n e q u a l i t y h o l d s , with a < B > 1. We now note t h a t
(5.59) (5.60) But the s o l u t i o n of a(u,v)
+
h ( u , v ) = (F,v.) v v E
i s increasing with r e s p e c t t o F; that I u2 ,
v,
E
v
from ( 5 . 5 8 ) , we t h e n deduce u s i n g ( 5 . 5 9 ) , (5.60)
w1
and i n t h i s c a s e , s i n c e B > a , t h i s c o n t r a d i c t s t h e choice of a as being t h e l a r g e s t r e a l number with ( 5 . 5 6 ) . I n t h e e x i s t e n c e proof f o r Theorem 5.2, we used a monotonely-increRemark 5 . 1 a s i n g approximation procedure. A decreasing approximation procedure can be cons t r u c t e d by s t a r t i n g from uo = c o n s t a n t , s u f f i c i e n t l y l a r g e (even i n t h e case i n We s h a l l f r e q u e n t l y have occasion t o use both t h e s e types of V). which 1 procedures i n what follows.
4
Local r e g u l a r i t x Suppose t h a t t h e a i j s a t i s f y ( 5 . 2 9 ) , and l e t u E H1
(a) be
a s o l u t i o n of
Then, without any further asswnptions on t h e a . ., u possesses a remarkable l e t 01 c 6, 6'bounded, Bdch t h a t v y f 6' the ball r e g u l a r i t y property: Ix - y ( 6 i s contained i n C+ and Let u be a solution o f u2 dx = 1 ; then there e x i s t two a ( i n ( 5 . 2 9 ) ) , on 6 , on the dimension n
This r e s u l t (which i s fundamental f o r c e r t a i n nonlinear problems) i s due t o another proof has DE G I O R G I [ll and NASH [ll who employed d i f f e r e n t methods; been given by J. MOSEFi [ll , 5.3
H
2
r e g u l a r i t y and i n t e r p r e t a t i o n of t h e problem t o b e solved
We s h a l l now assume t h a t
(5.61)
d i s a bounded open s e t of which t h e boundary
v (5.62)
~'(0)or H;(O)
.
r
i s of c l a s s C
2
,
(SEC. 5 )
ELLIPTIC P.D.E.'s
99
OF ORDER 2
We s h a l l now prove t h e following theorem:
Under the asswnptions of Theorem 5.1 and (5.61), (5.62), the solTHEOREM 5.3 ution u of (5.41)s a t i s f i e s
u E H2(B)
(5.63) Proof.
.
1
Since u E H (0)
(5.64)
ao(u,v)
+
,
it w i l l be s u f f i c i e n t t o prove t h a t t h e s o l u t i o n of
h(u,v) = (g,v)
with g 6 L2(B) , s a t i s f i e s ( 5 . 6 3 ) . namely ( o . , v . )( j = 1 . . N ) . 3 J
o1...
v
v
E
v
We t a k e a system o f l o c a l maps f o r
r
P
do,
Then d U U VN c o n s t i t u t e s a covering o f 5 with which we a s s o c i a t e a aN such t h a t ( i f we w r i t e 0 I0) p a r t i t i o n i n g o f u n i t y a o , a1
..
ai 6
SCB~)
N
E a . = ~ on
Since u
t
110
=
N Z
aiu
,
i=O
3. 2
it w i l l t h u s be s u f f i c i e n t t o show t h a t a u E H (0) V i i
We s h a l l d i s t i n g u i s h between two cases depending on whether i = 0 o r not. simplify t h e n o t a t i o n , we s h a l l w r i t e a i n s t e a d of a. o r ai i n each case.
.
To
It can be seen from a simple c a l c u l a t i o n t h a t au s a t i s f i e s c
n
(5.65)
where
J ga=ag-
Y 'ij
i3
Bu ; i ; ; axi
2
In t h e case where a = a , we s e e t h a t , s i n c e a , a u , g have compact support i n 6 , we can extend a , a u , g byoO o u t s i d e 6 ,and we can extgnd t h e a . i n an a r b i t r a r y manner such t h a t they $ m a i n C ' ( R n ) , bounded, t o g e t h e r with t h e i $ j d e r i v a t i v e s . We t h e n have
100
STOCHASTIC D . E . ' s
& P.D.E.'s
OF ORDER 2
Wg next 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 . on R ; we put =
d X 1 + h,x2,
(CHAP. 2 )
Let Q be a f u n c t i o n defined
...,xn )
and we note t h a t
We then e a s i l y deduce from (5.66) t h a t
and by s u b t r a c t i n g (5.66) we o b t a i n ( a f t e r d i v i d i n g by h ) : 7 au-au
h
(5.67)
By t a k i n g v = 'h au - au h
7 au-au
l l h l l
,
and noting t h a t t h e right-hand s i d e of (5.67) i s
n s c .
h H(R) We can e x t r a c t a subsequence hi such t h a t
7
But
in
au-au
hi
hi
R' (Rn) and hence
-b (au)
-
1
n
w weakly i n H ( R ) .
E H1(Rn).
dxi
It can be shown by t h e same method t h a t t h a t au E H 2 ( R n ) , i . e . t h a t aou E H 2 ( S )
.
a
xi(au)
E H1(Rn)
,V
i
, and
hence
We now consider t h e case a = a . . We w r i t e Z = d and s i n c e a , au, ga have compact support i n 2 , we can re-&press (5.65) i n t h ei form
(5.68)
(SEC.
In
5)
ELLIPTIC P.D.E.'s
OF ORDER 2
101
( 5 . 6 8 ) , we t h e n apply t h e change o f v a r i a b l e x
-1
I
'p
(y)
= y ( y ) (where
'p
= 'pi i s a diffeomorphism a s s o c i a t e d w i t h Z =
Oil
I f we denote by S ( y ) t h e t r a n s f o r m v ( P ( y ) ) of a f u n c t i o n v(x) on 6 n Z , we deduce from ( 5 . 6 8 ) a transformed equation
(5.69)
where t h e b . . ( * * ) s a t i s f y t h e same p r o p e r t i e s as t h e a i j , 1J
b.
4
E C1(Q+)
and
ij-1
n
b..(y) 1J
5 5 > C I: Si2 i j
-
i-1
,
i n particular
C > OV51~.-5,
1 +
9
1 +
and where a i s an a r b i t r a y element of H (Q ) ( o r Ho(Q ) ) . Moreover 6 , b il, have compact support i n Q u QO, s i n c e a , au, g, have compact support i n Z It t h e n follows t h a t i f we p u t
1
b,
nb
.
1
and i f we extend b , b 8, g by 0 i n t o 0,- ,'Q t h e n B , B 6 E H (R) ( y - H (z2)). Giving t h e b . . a r b i t r a r y vayues i n R - Q , provided t h a t t h e b . E C (z2)'are bounded, tog$$her w i t h t h e i r d e r i v a t i v e s , we can w r i t e , u s i n g ( $ ? 6 9 ) ,
.
(5.70)
for a l l t
6
~ ' ( n ) (or $ ( n ) ) .
Using 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 , we can t h e n show t h a t
Vi,j
provided t h a t a t l e a s t one of them
# n.
The method does not work, however,
Nonetheless, it follows from ( 5 . 7 0 ) t h a t , i n t h e sense of d i s t r i b u t i o n s , we have
(*)
For t h e d e f i n i t i o n of ,'Q
(**) $'pi 8'pj bij = I: a kl klaxk Kd
s e e s e c t i o n 5.1.2
.
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
102
- z
a Kff) -
-a
ij d y i ( b i j a Y j
+ A R d r - Z-
(CHAP. 2 )
a
ij
and hence, by v i r t u e of t h e p r o p e r t i e s of t h e b i j
b Taking account of t h e f a c t t h a t bnn t C > 0 , we can then r e a d i l y deduce t h a t
"Jn 2 + Hence d ii E H ( a ) and2in f a c t B 9 E H (Q 1. Returning t o t h e coordinatzs x , it then follows t h a t au € H (0 n 2) and t h a t it has compact support i n 2 U d and t h e r e f o r e a u E H2(0) , which concludes t h e proof of t h e theorem.
,
By making use of t h e r e g u l a r i t y r e s u l t ( 5 . 6 3 ) and Green's formula, it i s p o s s i b l e t o i n t e r p r e t problem ( 5 . 4 3 ) . We consider t h e o p e r a t o r A defined i n ( 5 . 2 1 ) and we Put (5.71)
A = B o +
i
v f &0)
I t follows from ( 5 . 4 3 ) , t a k i n g (5.72)
Au = f
.
b f a i F + a
a.e.
, that
we have
6
in
2 and hence AU E L (8) (which i s a l s o a consequence of, t h e property ( 5 . 6 3 ) ) . We V = Ho(d) , then we know t h a t , from must then d i s t i n g u i s h between two c a s e s . I f t h e c h a r a c t e r i s a t i o n ( 5 . 2 6 ) , we have
(5.73) 1
H2p{
On t h e o t h e r hand, i f V = H (0) , then by v i r t u e of t h e f a c t t h a t u C we can use t h e Green's formula ( 5 . 2 3 ) , from which f o r v an element of H' 8 , it follows t h a t
Taking account of (5.72) and ( 5 . 4 3 ) , we o b t a i n P
(5.74)
a u vdr
Jr?
I
0
.
1 1 Since t h e t r a c e mapping i s s u r j e c t i v e l f r o m H (0) i n t o H ' ( ( T ) , (5.74) holds i f we t a k e v t o be an a r b i t r a r y element of H ' ( T ) , and by d e n s i t y an a r b i t r a r y element of L 2 ( I ' ) ; t h i s implies t h a t
We have t h u s proved t h e following theorem:
(SEC. 5 )
ELLIPTIC P.D.E.’s OF ORDER 2
103
THEOREM 5.4 Under the asswnptions of Theorem 5.3, t h e solution u of ( 5 . 4 3 ) which s a t i s f i e s ( 5 . 6 3 ) is a solution of ( 5 . 7 2 ) , ( 5 . 7 3 ) , ( t h e Dirichlet problem) o r o f ( 5 . 7 2 ) , ( 5 . 7 5 ) ( t h e Newnann problem) depending on whether V = HL(0) or V = H‘ ( 0 ) .
.
Remark 5.2 Under t h e assumptions of Theorem 5 . 3 , it i s s u f f i c i e n t t o have V = HA(0) I n f a c t , we apply t h e following a (x) 2 0 i n ( 5 . 2 9 ) i n t h e c a s e cRange of unknown f u n c t i o n u = u where
w(x) = 2
with y that
iP
- exp - yoIx-xoI ,
a c o n s t a n t t o be chosen.
(5.76)
- izj -bXi(ba .
.w
1J
xo
,l b
The f u n c t i o n
1+E bXj dZ
bZ
-(aiw
bXi
z
-
We p u t X = F a ib w z 1
- E - b( a . , - )bw . i j b x . 1J dX
j
We have
+
yo
x-x
and hence
We can choose yo i n such a way t h a t X 2 5 > 0 .
, s a t i s f i e s z [ r = 0 and A ( o z ) = f ,
bw
J
STOCHASTIC D . E . ' s
104
& P.D.E.'s
I t may be noted t h a t 1 5 w S 2 and t h a t z i n which (5.29) i s s a t i s f i e d with a o ( x ) 5 6.
5.4
OF ORDER 2
(CHAP. 2 )
i s a s o l u t i o n of a D i r i c h l e t problem w
W'YP r e g u l a r i t y
The r e g u l a r i t y r e s u l t s r e l a t i v e t o W 2 y p , p $ 2 f o r t h e s o l u t i o n u of (5.43) i n particular a r e much more complicated t o e s t a b l i s h t h a n t h e preceding r e s u l t s ; we r e f e r t h e reader t o L A D Y Z E N S K A Y A - W ' T S E V A [ l I , MORREY [ l l , STAMPACCHIA c11. We s h a l l be content here merely t o s t a t e t h e r e s u l t s which a r e of relevance f o r what follows. We s t a r t by s t a t i n g a . l o c a 1 r e g u l a r i t y r e s u l t , which we s h a l l e s t a b l i s h f o r t h e We denote by ptP(0) t h e s e t of d i s t r i b u t i o n s u on 6 such t h a t c a s e p = 2.
(pu E
loc
P"(f3) ,
V
'p
E 6(8)
.
We suppose t h a t
(5.77) I n t h e following, we s h a l l work i n a compact subset of @ ( s i n c e we a r e i n t e r e s t e d i n l o c a l r e s u l t s ) . We can t h e r e f o r e as_sume, without r e s t r i c t i n g t h e genera l i t y , t h a t (5.77) holds with Rn i n s t e a d of 6 and t h a t a . , a . , a , t o g e t h e r with ij 1 o t h e i r f i r s t and second d e r i v a t i v e s , a r e bounded. Let v
E
Using Av, we s p e c i f y t h e d i s t r i b u t i o n
L2(Rn).
(5.78)
AT
We note t h a t Y E H-~(R"). THEOREM 5.5
-f
2 n < Av, Y > i s continuous f o r t h e norm H ( R ) and hence
Assumption ( 5 . 7 7 ) .
Suppose we have
.
u E
qoc(0)such t h a t
; then If G i s a bounded open subset Au f E L:,,(6) u E WfiZ(0) c G , then we have and G' an open subset such t h a t I
CO
,
t h e constant C being dependent on t h e c o e f f i c i e n t s of A on G , as well a s on G and on G I . We s h a l l only prove t h e r e s u l t i n t h e highly s i m p l i f i e d case where p = 2. t h u s have t o show t h a t
l'''lH2(G,
)
C(
If12 L (0,)
+
llull1 H (Go)
We s h a l l prove t h a t i f Go i s an open subset such t h a t
(5.80)
We
' c Go
and
zoc
G we have
(SEC. 5 )
ELLIPTIC P.D.E.'s OF ORDER 2
105
(5.81)
The result then follows-immediately by addition. First, let us prove (5.81). We consider a coverhg of G by a finite number of balls B. of sufficiently small radius 6, such that Bi c G . ' We associate with the B. a pirtitioning of unity Z 'pi e 1 in Go). We shall prove tha4 'pi('pi E .B (Bi) and satisfies
v
i
'piu E
1
H (Bi)
(5.82)
the constant C
being dependent on 'p and on the values of the coefficients Tilb i of A on Bi, as well as being dependent on 6. It will then follow that
and hence the result (5.81) will follow by addition. We shall suppress the index i, to sirnolifi the notation. A simDle calculation shows that. in the sense of distributions-,we have A(cpu) = 'pf + u(+aocp) 2 4 aiJ = 'pf + gq
-
and since u E coC(O)
,
gT
-1
z,9
n
J
E H (R ) and furthermore
(5.83) The constant C depends on
'p
and B, but not on u.
We shall then successively prove the estimates
The second estimate, taking account of ( 5 . 8 3 ) , clearly proves (5.82). A:=
where
- Z a .- b 2 ij i j bxibx. 0
J
By Fourier transformation, we have
We put
(CHAP. 2)
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
106
so that we obtain
(5.85)
Moreover, we have for Y
E
H2(Rn )
(5.86)
We note that
We can always suppose that Y has compact support in B, since this case we have the result
'p
has also.
In
where C2(6) + 0 with 6 (see f o r example LIONS-MAGENES [ll, Vol. 1, p. 135). Hence we deduce from (5.861, (5.87), that we have (5.88)
where C ( 6)
+ 0.
Finally
For 6 sufficiently s m a l l , we deduce from (5. 88), (5.89) and (5.85), the first inequality in (5.84). To obtain the second inequality in ( 5 . 8 4 ) , we use the differential quotients technique described in Theorem 5.3. We put
ELLIPTIC P.D.E.'s OF ORDER 2
107
and h s u f f i c i e n t l y small t h a t t h e support of p h cp u s t i l l belongs t o B. ying t h e f i r s t i n e q u a l i t y i n (5.84) t o p h q u i n s t e a d o f g u , we o b t a i n
and furthermore, by v i r t u e of t h e f a c t t h a t a . . , a. 1J
lIA(Ph cp.)
- PhA(Tld11E-2s
1
E
C
2
E
1 Hloc(6).
we have
C l P I 2
L
Proceeding t o t h e l i m i t as h -+ 0, we then deduce t h a t f i n a l l y we o b t a i n t h e second i n e q u a l i t y i n (5.84). We have t h u s proved t h a t u
By appl-
acpu E -q
L2
, so
that
To prove ( 5 . 8 0 ) , we proceed i n an
-
i d e n t i c a l manner, using a system of l o c a l c h a r t s covering G' and contained i n For each l o c a l c h a r t , we have t o prove t h e equivalent of ( 5 . 8 2 ) , i . e .
(5.90) We s t i l l have
But
hence
Go.
108
STOCHASTIC D.E.'s & P.D.E.'s
(CHAP. 2)
OF ORDER 2
so t h a t we have ( 5 . 9 0 ) , which completes t h e proof of t h e theorem. Under t h e assumptions of Theorem 5.5, it can be shown by applying Remark 5.3 t h e d i f f e r e n t i a l q u o t i e n t s technique t h a t i f 'P(R~) , t h e n u E w31P (R")
w;oc
2
loc
,
and hence i f p > 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
<
m]
we
.
Then, under t h e assumptions of Theorem 5.7, b u t p o s s i b l y with a d i f f e r e n t constant i n ( 5 . 9 4 ) , we have: ((5.117)
if' f E
:L
then
u E Lp
.
we We may v e r i f y t h i s by supposing p > 2 (which is t h e c a s e o f i n t e r e s t here; wish t o u s e t h i s type of information t o o b t a i n r e g u l a r i t y p r o p e r t i e s - s e e Remark 5.9
116
STOCHASTIC D.E.'s
below - hence we t a k e p " l a r g e " ) .
eR A ou + eR A
P.D.E.'s
OF ORDER 2
(CHAP. 2 )
We s t a r t from e q u a t i o n ( 5 . 9 6 ) , so t h a t
+ eR
~ U
and m u l t i p l y i n g ( * ) by +8R(ulp-2u
&
a u o
, if we
eRf put
However,
(*ITechniques
o f t h i s t y p e w i l l crop up q u i t e f r e q u e n t l y i n t h e r e s t of t h i s work.
ELLIPTIC P.D.E.‘s OF ORDER 2
117
and hence
from which we deduce ( 5 . 1 1 7 ) The following supplementary information i s obtained:
Remark 5.9.
From t h e g l o b a l r e s u l t (5.117) we deduce
wfi:(Rn) then u E W z:i
Aou E L:oo(Rn)
,
and
q u i t e n a t u r a l l y , t h e growth a t i n f i n i t y cannot a f f e c t hence t h a t u E t h e l o c a l r e g u l a r i t y of t h e s o l u t i o n s i n problems of e l l i p t i c n a t u r e . Furthermore, if
f E
wibpc
so t h a t u
E
m
C 2 i f p > n.
Remark 5.10 If f E L ” ( R n ) , we can t a k e 6 > 0 , a r b i t r a r y , i n (5.94). s u f f i c i e n t t o use t h e i t e r a t i v e procedure A un+l + a1un+’ + a un+l +
bun+'
= kun +
It i s
f
0
as i n t h e proof of Theorem 5.2.
5.5.2
Bounded coefficients
The case i n which t h e c o e f f i c i e n t s of t h e operator a r e bounded i s simpler a s it i s then p o s s i b l e t o u s e a v a r i a t i o n a l formulation i d e n t i c a l t o t h a t used with Theorem 5.1. We can a l s o consider f u n c t i o n s f which a t ’ i n f i n i t y i n c r e a s e more r a p i d l y than any polynomial. For t h i s , we can no longer t a k e polynomials a s weighting f u n c t i o n s , but we w i l l be a b l e t o use exponentials. We denote by ?ypy’’
t h e space
Of
f u n c t i o n s u(x) such t h a t t h e q u a n t i t y
(5.121) Equipped with t h e norm ( 5 . 1 2 1 ) ,
Py’= f Y 2W” . e again previous s e c t i o n .
aij
i s a Banach space.
k?rp’’
We put
consider t h e d i f f e r e n t i a l o p e r a t o r A defined i n t h e We now 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. s a t i s f y
=
ji , aij , ai,
a0
E L~(R~)
118
STOCHASTIC D . E . ' s
& P.D.E.'s
and we d e f i n e a continuous b i l i n e a r form on
i n which we have put
m
(XI
= exp
P
(CHAP. 2 )
OF ORDER 2
V by means of t h e formula i-1
- ~1x1 .
When t h e r e i s no r i s k o f ambiguity we denote t h e norms i n H
11 I(
respectively.
i-1
+ kjul2
a(u,u)
(5.1 24)
and V
i-1
by
[
I
and
It can e a s i l y be seen t h a t t h e r e e x i s t s a A > 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 <
{x
.
R]
We denote by uR t h e s o l u t i o n o f t h e
D i r i c h l e t problem i n &lR, namely
From Theorem 5.6,
%
i s d e f i n e d uniquely i n
...
0, Q and a p a r t i t i o n i n g We now c o n s i d e r a covering of E by t h e open s e t s 4, a s s o c i a t e d wigh t h i s c o v e r i n g , such t h a t t k e following propof u n i t y Q , erties are satisfieti:
...
(5.131)
- B,... 6 h o i n L~R.
... .
i n t e r s e c t anR and %o+l Bh d i m Bi < 6 (independent of R ) ,
- E Dbi) - any p o i n t x 'pi
i
P
-
E ilR
,
I...h
E
-
t h e r e e x i s t s an open covering 1 c B. and cp. 2 on P i ,
1 2 1 , 3 3cp
-1
a
z1
...
@;
J k
1 s N2
for x
6
t/
i = 1,
...,h
1.
J
PA
ER such
of
that
@i
where N2 i s a c o n s t a n t independent of
-
4, N1 being
N1
'pi
i n t h e form
sets
belongs a t most t o N,
independent of R ,
-
a r e contained
, the
set
Oi n
bC$
i
and R , can be r e p r e s e n t e d
@(xl . . . x j-1' x j+l"'Xn)
f o r a c e r t a i n j and
where t h e c o n s t a n t C* depends upon n e i t h e r t h e open s e t nor on R .
6i
120
STOCHASTIC D.E.’s & P.D.E.’s OF ORDER 2
It is possible to construct
el...oh and
‘p,
.-.Th
(CHAP. 2)
in such a way that all the
conditions (5.31) are satisfied (see A. FRIEDMAN C41). We put
OiR
= OR n Bi
We note that the function
\‘pi
.
is a solution of a Dirichlet problem in 6iR .
the right-hand side of (5.132) belonging to Lp(eiR). From this we deduce, using Theorem 5.6 and the special properties (5.131), that we have
the constant C being independent of i and of R , or
(5.133)
We multiply both sides of (5.133) by exp - ( i & ! p and we note that
C, for x cQi.
4-~ ~ 1 x I 1 )exp
(-
where Ci is the centre of 8. 1’
PPIs~I)
Q
c2 exp (-
PPIXI)
We then deduce from (5.133)
From (5.31), and by adding the above relations for all the indices i, it then follows that
(SEC. 5 )
ELLIPTIC P.D.E.‘s OF ORDER 2
121
Furthermore, from t h e f i r s t r e l a t i o n i n ( 5 . 1 3 0 ) we deduce, f o l l o w i n g m u l t i p l i c a t i o n by
l%Ip-2
%
exp
-
pplxl and i n t e g r a t i o n o v e r 0
R,
By v i r t u e of ( 5 . 1 2 6 ) t h e r e e x i s t s a
u
such t h a t f o r p
5
that
uo, ( 5 . 1 3 5 ) i m p l i e s
and we t h e r e f o r e o b t a i n
(5.136)
s o g
We now u s e t h e f o l l o w i n g S o b o l e v i n e q u a l i t y . we have
If w
E
W2”(Rn)
then
(see (5.8))
(5.137) Furthermore, we can f i n d a n e x t e n s i o n o f W
w
E
W2”(OR) and
extends
2
”(OR) t o
such t h a t i f
W2”(Rn)
w, t h e n
It is s u f f i c i e n t t o u s e t h e system ( 5 . 1 3 4 ) t h e c o n s t a n t b e i n g independent o f R . of l o c a l maps and t o e x t e n d by r e f l e c t i o n (as i n LIONS-MAGENES Ell, v o l . 1, p . 4 3 ) . Applying (5.137) t o
it can b e v e r i f i e d t h a t i f w
t h e c o n s t a n t b e i n g independent o f R . 2 ’
W2yp(C+R)
we have
We now t a k e i n (5.138)
w = v expC- v ( 1 + l x l )‘I. We n o t e t h a t
E
STOCHASTIC D.E.'s
122
(CHAP. 2 )
OF ORDER 2
& P.D.E.'s
and exp - v ( j x j
+ 1) 6 exp
- u(1+
1 x 1 ~ )6' exp - v j x j .
Consequently it follows from (5.138) t h a t i f v
E
W2'pyv(eR) then
and hence for a l l A > 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.
(*)
<
u
Let us assume f o r t h e moment t h a t we have:
Let
LEMMA 6.1.
u be given i n W ( O , T ) , and suppose t h a t (6.101)holds.
Then
(6.102)
r e l a t i o n (6.102) i s meaningful since i f u (6.103) Then
U-
f L2(0,T;V)
n
E
W(0,T) then
Co([O,T];H)
2
(6.9) gives, v v f L (0,T;V)
.
:
(6.104) Taking v = u- i n
(6.104) and u s i n g t h e lemma, we t h e n deduce t h a t
s i n c e u-(T) = 0 if t h e n deduce t h a t
j:
(6.105)
2 0 , and s i n c e from (6.100) we have
.
o
a(t;u-,u-)dts
T(f,c)dt 2 0
s o
, we
can
Hence u- = 0.
Proof of t h e L a m a We f i r s t show t h a t t h e mapping u +. u- i s continuous ( a c t u a l l y L i p s c h i t z 2 c o n t i n u o u s ) from W(0,T) +. L (0,T;V) n C o ( C O , T I ; H ) .
1)
I f u,v
E
W ( O , T ) , we have:
.
lu-(xJt)
hence
Ilu-(t)
(*)
- v'(x,t)
- v-(t)\lHr
T h i s i s t h e c a s e i f V = {vlv on
T
I
E
( o r with capacity > 0 ) )
(u(xtt)
I\'(t)
- v(xJt) I
- v(t)llH
(11 1,
1 H ( O ) , v = 0 on
.
=
To c
1 1) r , To
w i t h measure > 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
.
Comparing (6.106) and (6.110)and by i n t e g r a t i n g by p a r t s , it can e a s i l y be shown t h a t
v ( t , ) = G*(t,,t2)y Taking we have
7
= Y
E
B(Rn),we
and hence G * ( t l , t 2 ) a d m i t s
.
n o t e t h a t from t h e d e f i n i t i o n of t h e d i s t r i b u t i o n
the representation
P(x,tl ,S,t2)Y(x)b and consequently (6.100)implies t h e r e l a t i o n
(6.111)
(*)
I n t h e sense of t h e d u a l i t y V , V’
p
STOCHASTIC D.E.'s & P.D.E.'s
150
OF ORDER 2
( C W . 2)
The problem which i s then posed r e l a t e s t o t h e r e g u l a r i t y of t h e d i s t r i b u t i o n p. I n o t h e r words, t h e question i s whether p i s i n f a c t a f u n c t i o n ( n a t u r a l l y f o r tl < t 2 ) . To examine t h i s question we s t a r t by estnablishing a r e g u l a r i t y r e s u l t concerning parabolic P.D.E.'s of second o r d e r i n R (Cauchy problem). THEOREM
6.14.
Suppose that, in addition to (6.92), the coefficients
a i j , a i , a. are bounded, continuous and m-times continuous2y differentiable with
respect to x, a22 their derivatives being bounded. Let then the solution v of f E L2(0,T;8-') (*) , m > 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
<
( 6.1 1 6)
u(t),f(t)
>
dt =
v Let write (6.1 1 7 )
'p
E b(Q)
j i < v,f,
<
- d'
+
+<
dt
.
>
v(T),g,
f t ~~(0,~;Hm-l) In t h e d i s t r i b u t i o n a l sense, we can
where Q = Rn x ]O,T[.
I'
>
Consequently,
A(t)u,cp
>
dt =
j
<
u,
2 + A*(t)cp
>
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
"<
+.
But then
H-m.
f,,v
>
bt
hence u
is
Integrating by parts, we have dt
E
S O
=
lo'
-
<
- fi + A(t)upv > 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 <
T
G(T)
P
I
;
if we introduce t h e domain D ( A k ) of t h e power Ak of A i e n t s of
(*)
A
being assumed ' s u f f i c i e n t l y r e g u l a r ' ,
See t h e d e f i n i t i o n o f Wm
(**I This
( k integer) ( t h e coeffics e e below), we have:
given e a r l i e r .
may be adapted t o t h e case where t h e c o e f f i c i e n t s depend on t ,
(SEC. 6)
PARABOLIC P.D.E.'s OF ORDER 2
(6.1 24)
t
+
G(t) is continuous from
155
.
ts T -C(D(Ak);D(Ak))
By transposition, we note that (6.124) is also valid for k < 0. since the semigroup G(t) is a m z y t i c (see YOSHIDA C l J we have: t
(6.125)
-f
(T-t)(G(t)) is continuous from t S T -&(D(Ak);D(A
Furthermore, k+l
))
and consequently (6.1 26)
t
+
(T-t)'G(t)
is continuous from t
$
T -&?(D(Ak);D(Ao2))
.
It follows from (6.124), (6.126) and from the theory of interpolation of linear operators (see J.L. LIONS and J. PEETRE Cl1)that (6.127)
t
+.
(T-t)sG(t) is continuous from t S T * d(D(AY);D(Ay*s))
and this applies with We take
y S 0
y
and
s
in (6.127).
arbitrary
-
in particular, non-integer.
We note that
( 6.1 28)
D(A~)= H~~ at least ZocaZZy if the coefficients of A are sufficiently regular. If we choose
we can take
y
-
y >
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 +.
<
z
zc *
'5
T~
p
zAvbx(t))dt
v I
u
and t h u s Av = Au = f ,
zf ( y x ( t ) ) d t
we o b t a i n by applying Lebesgue's theorem
t o simplify t h e n o t a t i o n .
.
(*), yx(7 A T) -yx(ZA T)
U ( ~ ~ ( TT )~) A
We s h a l l now show t h a t
(*) T =
0
, hence
,
and s i n c e t h e functions
I,
-E
v(x)
A T, y x ( t ) E &Vc
0 , and hence
m a x IuI and T m a x If 8 8
I
remain bounded by
160
STOCHASTIC D.E.'s .& P.D.E.'s
(7.17)
E 7 <
(CHAP. 2 )
OF ORDER 2
-. -
-
y1X-X I which was considIn fact, we consider the function w(X) = 2 exp ered earlier in Remark 5.4. We recall in particular that Awo2 C > 0. By applying Ito's formula, it can be shown (in a manner analogous to (7.16))that
E
w(yX(r
A T)) =
W(X)
hence c 6 7 A T S w(x)S 2
-E
Aw bx(t ) )dt
.
If we let T increase to infinity and apply the monotone convergence theorem, we then deduce (7.17). But it then follows from (7.17) that a.s. 'I <
-.
When
T
f
+m
, it
follows from the continuity of u
u(yx(r A T ) ) z ( r A T) 9 0
.
on
8 that
Application of Lebesgue's theorem then gives (7.11).
Remark 7.1. Assumption 7.9 allowed us to confirm the existence of a strong solution of equation (7.11). We could dispense with this assumption by working with a weak solution (which is permissible since under assumption (7.1) g is continuous and bounded). This observation will be crucial for the investigation of stochastic control. We shall return later in more detail to the use of this technique, as a discussion at this point would unnecessarily complicate the presentation. If u, satisfying the regularity assumptions ( 7 . 3 ) , were a solution Remark 7.2. of the non-homogeous Dirichlet problem
Au= u/r where
0
7.2
f
=
is continuous on
Q
r,
then we could prove by the same techniques that
Elliptic problems in Rn
We now adopt the following assumptions
(7.18)
161
PROBABILISTIC INTERPRETATION
a E c~(R") ij
, ai
E c'(R")
2
(7.20) (7.21)
bai 5,OXi&
k
bounded,
>
0
da. .
A
E c'(R~)
bounded
bXk
"k
ao(x) 2
, .a
,
- bounded
being sufficiently large;
"k
(7.22)
Theorem 5.10 allows us to confirm the existence of a unique function u that
u E C2(Rn)
,
such
1
u E H,
(7.23) AU
the weighting
I I
= f at any point of R"
(cf. (5.102)) being chosen such that f
E
Li.
By virtue of the assumptions, the functions g and u defined in the previous section satisfy
(7.24)
We are thus within the conditions for application of Theorem 3.1, so that we can define the (unique) process yx(t) = y(t), the solution of (7.81, (7.9). We shall now prove the following theorem: THEOREM 7.2. Under the asswnptions (7.18)to (7.22), t h e soZution ( 7 . 2 3 ) is given by
u(x) of
(7.25)
Proof. First we show that the right-hand side of (7.25) is well defined. We saw in Corollary 3.1 that the process yx(t) possessed moments of all orders and that Elyx(t)/
2k
s (1x1
2k
+
C , t ) exp C , t
,
Cl
>
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
<
constant.
satisfies w = g
E
LE then we have
(7.30) This r e l a t i o n i s proved as i n Theorem 5.9.
-.
E , ( w , ~w+) ~ = and t h e n we make R w
I
UN
+.
+
- YN
We w r i t e
k , e R w+) We next apply ( 7 . 3 0 ) with and
g
= fN
I f yN s a t i s f i e s t h e r e l a t i o n ByN L N , g
- BoyN 5
None-
.
0 and (7.30) then shows t h a t
(SEC. 7 )
(% +
It can s i m i l a r l y be shown t h a t Let
163
PROBABILISTIC INTERPRETATION
I 15. I s
OR I
, where
R]
yN)- = 0 i f ByN L C , which g i v e s ( 7 . 2 9 ) .
R w i l l tend towards
+ -, s o t h a t we can always belongs t o 8,. Let T, be
assume t h a t t h e i n i t i a l d a t a x on t h e process y , ( t ) t h e e x i t t h e from 6,. We have
(7.31)
a.s.
T~
<
+
which i s proved as i n t h e previous theorem, by means of a f u n c t i o n
We a l s o have
(7.32)
a .s. T,
-+
+
-
when R
-+
+
a,
Otherwise, i n f a c t , t h e r e would e x i s t a s e t no with P ( n o ) > 0 and T ~ ( w )5
A(u)
< +=
V R
9
W
E Po
-
But then
which i s impossible i n view of t h e c o n t i n u i t y of t h e process y , ( t ) . I t o ' s formula t o %(x) we o b t a i n
By applying
However
Making R (7.28).
-+
+
We now make N
(7.34)
-t
and using Lebesgue's theorem we deduce from (7.33) t h e e q u a l i t y
+
-.
It i s c l e a r from formula (7.28) t h a t we have
uN(x) t v ( x ) = E
:j
f ( y x ( t ) ) exp[-
st
ao(yx(s))ds]dt
.
But formula ( 7 . 2 8 ) shows t h a t
The left-hand i n e q u a l i t y i s easy t o o b t a i n ; t h e right-hand i n e q u a l i t y r e s u l t s from t h e estimates f o r v e s t a b l i s h e d a t t h e s t a r t of t h e proof. It then follows
164
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
(CHAP.
2)
2 t h a t uN l i e s i n a bounded subset o f LT and t a k i n g account of (7.34) we then deduce that
%+v
(7.35)
2 weakly i n LT
We s h a l l now e s t a b l i s h an estimate f o r
ax -
We have
I
I+II
.
We have
Furthermore, i f z ( s ) = Yx, ( s )
we have dz
= (g(Yxl)
z(0)
=
XI-
-
YX(S)
- g(Yx))dt
+ (U(Y~,)
- dYx))dw
x
from which we deduce, by applying I t o ' s formula, t a k i n g account o f t h e upper bounds of ( 7 . 2 4 ) and using Gronwall's lemma:
(7.37) We t h u s have
i.e. finally
Elyx,(t)
- yx(t)
I2 s
1x1-
x12 exp
cP
t
(SEC. 7 )
PROBABILISTIC INTERPRETATION
165
Moreover, i f we w r i t e
hence
so t h a t f i n a l l y
From t h e estimates ( 7 . 3 8 ) , (7.39) we deduce t h a t
or, s i n c e
%
is differentiable,
which implies t h a t
3 stays
i n a bounded subset of Hi
and t a k i n g account of ( 7 . 3 5 ) ,
3
-t
v weakly i n
4.
We proceed 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 sense, i n ( 7 . 2 7 ) and t h i s gives t h e r e s u l t t h a t v = u ( s o l u t i o n of ( 7 . 2 3 ) ) .
R m a r k 7.3.
The r e s u l t of Theorem 7 . 2 i s t r u e if we have
without having f L -C. The proof i s i n t h i s c a s e somewhat more complicated. I n s t e a d of f we consider N
166
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
if
-M5 f 5 N
N
if
f 2N
-M
if
f 5 - M
f
fm
of
I
We note t h a t
bfN
WlPp
2 n urn E c ( R )
Aum
I
,
.
and hence we can uniquely d e f i n e
loc
bX
(CHAP. 2 )
%M, t h e
solution
urn E Hl
fm
*
We next have, s i n c e f N Mi s bounded,
Now
urn(x) = E
Jy f r n ( y x ( t ) )
IfmW I
lfb)I
5
IfrnM(4 - f*b') I s
J:
expl-
If(d
.
ao(~x(s))ds]dt
- f(x') I
2.
so t h a t t h e estimates obtained during t h e proof of t e previous theorem remain v a l i d ; t h e r e f o r e %M s t a y s i n a bounded subset of
,
.
We f i x N and make M + - -, then f m 4 f N and u 1 UN Also 1 + uN weakly i n HT, 3 i s a s o l u t i o n of (7.27) and i s given by ( 7 . 2 8 ) . estimates obtained f o r %M
remain v a l i d f o r
1 uN t u weakly i n HT, and hence t h e r e s u l t .
Remark 7.4.
If f
E
Co(Rn),
with
f
3.
We next make N
€
bounded and
r e s u l t of t h e above theorem i s v a l i d without f follows from t h e a n a l y s i s given i n t h e proof.
W:il
+
+
m,
The
and
, then
the
.
This
n e c e s s a r i l y being C
1
Remark 7.5. Under t h e assumptions of Theorem 7.2 ( i n which ( 7 . 2 2 ) i s I n f a c t , from replaced by (7.4011, we can o b t a i n t h e r e s u l t of Corollary 6 . 1 t h e e s t i m a t e s obtained, it t u r n s out t h a t t h e s o l u t i o n u of (7.23) s a t i s f i e s
.
bll l,zl 5 CM(1+lxlm) . But then
A
~
u
=
s~ a t i saf i e i s
and t h u s Consequently A
Remark 7.8.
choice
of
S.
u
E
2
L
~
Alu E L,'
if
.
I t i s c l e a r t h a t i n formula ( 7 . 2 5 )
u
8
>
m+l
.
i s independent of t h e
(SEC. 7) 7.3
PROBABILISTIC INTERPRETATION
167
I n t e r p r e t a t i o n of parabolic problems i n Q = 6 x 10,TC
Let 6 be a bounded open subset of Rn such t h a t r = a6 i s of c l a s s C2 and l e t 9 = 6 x lO,T[. We consider t h e family of o p e r a t o r s A ( t ) defined by
where t h e c o e f f i c i e n t s a i j , ai, a
(7.42)
2
(7.44)
z . =i j s i c
aij=aji
b a A
satisfy
ib
2 a
I
2 a . iz ~ ~a > o
j i
Let
I n t h e l i g h t of Theorem 6.5 and Remark 6.5, we can then d e f i n e one and only one s o l u t i o n u of t h e nonhomogeneous p a r a b o l i c problem 1
E CO(P)
(7.47)
Furthermore, we know t h a t
u
satisfies
1
168
STOCHASTIC D.E.'s & P.D.E.'s OF ORDER 2
(CHAP. 2 )
Analogously t o t h e e l l i p t i c c a s e ( s e e ( 7 . 9 ) ) we adopt t h e supplementary assumption
(7.51) We then d e f i n e u ( x , t ) and g ( x , t ) by t h e formulas
(7.53)
Hence
dg
- € Co(aand .?%E Co(a "k
dxk
. we can assume t h a t
Thus by s u i t a b l y extending t h e f u n c t i o n s g and u o u t s i d e IdXDt)
- g(x',t)
/+la(x,t)
-
u(x',t)
I
C I X d
I
(7.54) Igl
D
101
We can t h u s d e f i n e i n unique fashion t h e process y x t ( s ) = y ( s ) , Let t E C0,Tl. t h e 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
dY = g b , s ) d t
+
dy,s)dw(t)
(7.55) Yb) = x and l e t
(7.56) If x
'cxt
= T be t h e e x i t time from 7
xt
c 8 ,Txt
- 7 = inf 2t 8
lY(S)
8 ,i . e .
$ a1
.
= t.
Under the assumptions (7.42) t o ( 7 . 4 6 ) together w i t h (7.511, the THEOREM 7.3. solution u ( x , t ) of (7.47) i s given by
(SEC. 7 )
169
PROBABILISTIC INTERPRETATION
-
Proof. This is analogous to that of Theorem 7.1. It is no longer necessary to prove that T < + a.e. We apply Ito's formula to the process u(yxt (s), s) e d -
s
ao(yXt ( h ) ,h)dh
I
between the instants t and
T
h(T-~)(t < T-E), which is permissible since
(yXt(s),s)
E
Q and u
We next make
7.4
E
+-
E
C2''(Q).
0, and we use the continuity of u on
Parabolic problems in Rn
x
5 to
finally obtain (7.57).
l0,TC.
We use the same notation as in Section 6.3.1. ai(x,t), ao(x,t) satisfy
We suppose that the coefficients
(7.58)
(7.59) 2
% bXkdXl
(7.60)
E Co(Rn x [O,T])
(7.61)
ao(x,t) 2
(7.62)
If
p >
0
,p
sufficiently large (this is not a restriction in the parabolic case),
da
bounded
(7.63)
Y 6 Co(Rn
(7.64) g =
-
+
A(t)Y
x.[O,T])
satisfying the same assumptions as f
(i.e. (7.63)).
170
STOCHASTIC D . E . ' s
2
1
u c L (O,T;Hn)
- -b t
bU
(7.65)
u(x,T)
Proof.
Remark 7.3.
(7.68)
+
9
A(t)u = f e
U(X)
.
c
& P.D.E.'s
OF ORDER 2
n Co(Rn x [O,T])
x ]O,T[)
C2"(Rn
(CHAP. 2 )
a t any p o i n t of Q
The proof i s analogous t o t h a t f o r Theorem 7 . 2 , supplemented by F i r s t , we have f o r a l l k:
2k
E ~ Y ~ ~ Ls~C )( 1 I+ l x l
2k
)
8
E [ t , T ] (**)
so t h a t t h e right-hand s i d e of ( 7 . 6 7 ) is w e l l defined. We s h a l l now prove ( 7 . 6 7 ) i n t h e c a s e of a p p l i e d t o Y we have
= 0.
i n f a c t , from I t o ' s formula
From (7.68) a p p l i e d with k = m and from assumption (7.64) concerning t h e growth
( * ) We apply t h e extension method described i n Remark 6.10 t o u - Y which i s a s o l u t i o n of a p a r a b o l i c problem with i n i t i a l condition 0. (**)
All t h e c o n s t a n t s n a t u r a l l y depend on t h e horizon T.
(SEC. 7 )
of
ay . ax, it
171
PROBABILISTIC INTERPRETATION
i s permissible t o t a k e t h e mathematical expectation of t h e s t o c h a s t i c
i n t e g r a l , which i s t h e r e f o r e equal t o zero.
E
u(yxt(T))exp
-E
Consequently we have
sods] = Y ( x , t )
[-
g(Yxt(s),s)exp
[-
5:
;
ao(y)dh]ds
but then ( 7 . 6 7 ) i s i d e n t i c a l t o u(x,t)
- y(x,t)
=E
jtT
(f-g)(Yxt(s),s) erp
[-
and hence t h e problem reduces t o proving (7.67)with = 0 , with f replaced by f - g (which has t h e same p r o p e r t i e s ) and with u replaced by u - Y, a s o l u t i o n of (7.65) corresponding t o f - g and 0 . We t h e r e f o r e assume
= 0.
We now have t o prove t h a t
We s t a r t by considering t h e bounded case.
o n Remark 7.3. Since f m f c (R x [O,T]) , of uniquely d e f i n e u ~ t h~e solu-fion u
NM f L2(0,T;HL)
, urn
and
as i n
We approximate f by f,, .*. afm bfm
, TE goO(dwe dXi t
E Co (Rn x [O,T]),
can
urn € C 2 ” ( Q )
(7.70)
However, %M
i s bounded.
This can be seen, f o r example, from t h e i m p l i c i t
i n a manner analogous t o Theorem 7 . 2 , scheme (6.641, by recurrence over t h e A s t h e bound obtained i s independent of n, t h i s i s preand by using (7.30). served on proceeding t o t h e l i m i t . Let i.e.
eR=
151 i R} and l e t -rR be t h e e x i t time from
We have, from t h e c o n t i n u i t y of t h e process, a.s. T Hence
> 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) <
w
e and Fatou’s lemma finally gives (8.58).
We then have the following theorem: THEOREM
-bQ bt
A(t)Q
Proof.
8.5.
L e t O(x,t) be a continuous function on
€ L2(~,T;L;’oc(Rn))
.
Q
= Rn
x
[O,Tl,
such t h a t
Then we have fusing the notation o f Lema 8.1)
By regularisation, we can approximate 0 by a sequence Pn such that
bt - A(t)Qn
- E-
A(t)@
2 in L ( O ~ T ; ~ , C ( a ” ) ) or Lp(O,T;LToC(Rn))
.
We apply Ito‘s formula to On and to the process y. It is permissible to proceed to the mathematical expectation, as the stochastic integral can be written as
and
186
STOCHASTIC D.E.’s & P.D.E.’s OF ORDER 2
We obtain (8.62), or (8.63) with On instead of 0. taking account of the fact that
(CHAP. 2)
We can proceed to the limit,
and
Remark 8.5. We can use Theorem 8.5 to obtain a probabilistic interpretation of the solutions of P.D.E.’s (as in Section 7 ) , by somewhat weakening the regularity This will be very useful in assumptions (in particular for the right-hand side). the investigation of control problems. However, the regularity of the coefficients is essential for the interpretation of the kernel representing the Green’s operator.
CHAPTER 3 OPTIMAL STOPPING-TIME PROBLEMS AND VARIATIONAL INEQUALITIES
INTRODUCTION This chapter is devoted to an investigation of stationary and evolutionary variational inequalities and to their probabilistic interpretation in the form of optimal stopping-time problems.
A typical optimal stopping-time problem is as follows: equation
we take a state
and a criterion
where
where e is a stopping time and T~ is the first exit time from an open set L). The problems which arise are then as follows:
- the analytic characterisation of the function u - the regularity of u - the existence of an optimal stopping time
-
the properties of an optimal stopping time.
If for example a = I, g = problem
(iv)
1
-TAu=
us
f
0,
the function u(x) is a solution of the following
a.e. in 6
J,
187
188
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP. 3 )
as long as Au is meaningful as a function. We can also rewrite (iv) in the form (V
1
e(u,v-u) 2 (f,v-u)
v
1
v E H~ ( O ) , v $ U S J,
J,
a.e.
a.e.
where
The form (v) is what is called a variational inequality (V.I.), a concept which was introduced by STAMPACCHIA 111 and LIONS-STAMPACCHIA [ll. The form (iv) demands greater regularity than does (v), which represents for (iv) exactly what the variational formulation represents for a boundary-value problem. We can thus say that (iii) is the analogue of the probabilistic interpretation of the solution of a boundary-value problem. Furthermore, the problem (iv) is linked with free boundary questions since 6 splits into two regions, the region where u = JI and the region where u < $I. The boundary between these two regions is clearly one of the unknowns in the problem. An analogous situation arises in the Stefan problem, and more generally in numerous problems in Physics and Mechanics, in which the boundary constitutes the unknown. In the case of the Stefan problem, this boundary represents for example the interface between the water and the ice, and this moves in time as the ice melts into water.
The very brief outline given above illustrates the large degree of commonality which exists between problems and applications which are a p r i o r i very different. The case of evolutionary problems is even richer in the variety of possibilities than is the stationary case. Finally, new questions arise in connection with differential games in which stopping times are involved. We shall investigate all these questions as extensively as possible in the present chapter, whilst following the spirit of Chapter 11. In Sections 1 and 2 , we present the theory of stationary V.I.'s, then that of evolutionary V.I.'s. We give numerous new results, in which the motivation of optimal stopping-time problems has played a significant part. These sections do not require any great understanding of probabilities and they make recourse only to that part of Chapter I1 which is concerned with boundary-value problems. Section 3 deals with stationary optimal stopping-time problems, and Section 4 treats evolutionary optimal stopping-time problems. Finally, Section 5 is devoted to stochastic differential games with stopping times (stationary and evolutionary) and with the corresponding V.I.'s. From the standpoint of solving optimal stopping-time problems, the V.I. technique is new. It introduces numerous new results. First, certain results are obtained relating to the regularity of the solution, under very weak regularity assumptions for the data. Furthermore, this technique provides a very convenient characterisation of the solution, this being much more tractable from the algorithmic point of view than the previously known characterisations. Finally, it allows evolutionary cases to be treated; hitherto these have been very little discussed in the existing literature, apart from a number of very simple cases.
STATIONARY V . I . ' s
(SEC. 1)
1.
189
STATIONARY VARIATIONAL INEQUALITIES
1.1
Various f o r m u l a t i o n s o f t h e problem
Let @ b e a n open s u b s e t of Rn which we s h a l l i n i t i a l l y assume t o be bounded. Let r d e n o t e t h e boundary of B , assumed t o be s u f f i c i e n t l y r e g u l a r ( * ) . The n o t a t i o n u s e d i s t h e same a s i n Chapter 2 , S e c t i o n 5. For u , v E H1(@) we p u t
where (1.2) The o p e r a t o r A a s s o c i a t e d w i t h (1.1) i s g i v e n by
(1.3) The o p e r a t o r i s assumed t o be, e l l i p t i c i n t h e f o l l o w i n g s e n s e
Remark 1.1. Except i n c a s e s where we e x p r e s s l y s t a t e o t h e r w i s e , we shall n o t a s s m e i n t h i s s e c t i o n t h a t a 1. 1. = a i i , as t h i s would n o t s i m p l i f y t h e p r o o f s . -" "However, we s h a l l always have a . . = a . . i n t h e a p p l i c a t i o n s a t which t h i s work i s 1J
aimed.
31
I n t h e f o l l o w i n g , we s h a l l work w i t h one of t h e f o l l o w i n g supplementary assumptions:
(1.5)
or
a (x) 2 a
(1.6)
(*I
> o
a.e.
r is a C1 m a n i f o l d of dimension n - 1, 6 b e i n g l o c a l l y on o n l y one s i d e o f r . We a r e c e r t a i n l y n o t l o o k i n g f o r minimal assumptions on r t o ensure t h e v a l i d i t y of t h e following r e s u l t s .
For example,
(**)/I
11
1 = norm i n Ho(6).
OPTIMAL STOPPING PROBLEMS & V.I.'s
190
Remark 1 . 2 .
(CHAP. 3)
We n o t e t h a t
if
so t h a t ( 1 . 5 ) holds i f ( 1 . 4 ) holds and i f a
(1.9)
0
I 2
bai
- + h
---Z
bXi
1
1 0
-
where A1 i s t h e p r i n c i p a l eigenvalue ( > 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<-(ui<)) since 6
5
fdOS 2 fo > 0
0 , we again have
$4, $42
0
.
u i b 5 $4I: v< ( 1 . 5 4 ) , (1.55), and two s o l u t i o n s
-
and
=
(aoS,v-ui) 2 (faoS,v-S-(ui-t;) ui<
U i 2
0
1
;
We t h u s have an analogous problem, s t i l l with 2
0.
We introduce A > 0 which i s l a r g e enough so t h a t (1 . 5 6 )
It 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 u1 # u2 and we introduce t 0 such t h a t
the largest a which i s real and
(1.57)
au
1-
a.e.
2
;
we have, i f we do not have u1 = u2 :
(1.58)
O s a < l
;
We introduce 6 such t h a t
(1.59)
a
1
, pf
+ Apul
5 f
+
hu2 a . e .
;
(SEC. 1)
STATIONARY V.I.'s
201
Such a 5 exists, s i n c e (1.59) w i l l hold i f
Bf + hBul 5 f + haul
(l-B)f,
and hence i f
2 h(S-a) SUP u1
,
and we can f i n d 5 with a < B < 1 such t h a t t h i s i n e q u a l i t y holds. satisfies the V.I.
a(pul ,BV-PU, )
(1.60)
v
Bv I
(Bf+hBu, ,Pv-Bu,
+ ~ ( P u, B, V - B U ~)
We note t h a t Bul
1
BJ,
and t h a t u2 s a t i s f i e s t h e V . 1
We a r e w i t h i n t h e s e t t i n g of Theorem 1 . 4 s i n c e we have ( 1 . 5 6 ) and
j3f+Apul I f+hu2,BC 5 yl
,
Bu, 5 u2 B > a we must t h e r e f o r e have u1 = u2.
( s i n c e J, 2 0 ) and consequently
i s i n c o n t r a d i c t i o n with t h e choice of a:
, which m
Proof of existence. We proceed as i n Chapter 2 , Theorem 5.2. I n t h i s case sequence un(we could a l s o , s t a r t i n g from uo = 0 , c o n s t r u c t we c o n s t r u c t a decreasi? an increasing sequence u ) . We s t a r t from uo i n H1(6) - not necessarily in V - such that a(uo,v) =
(1.62)
( f " , ~v) v,
E V
, f"rf ,
0
u
z
0
i?,
on
uo E L"(0)
For e z m p l e we can t a k e uo = constant > 0 , s u f f i c i e n t l y l a r g e . un s t e p by s t e p , a s follows:
.
We then d e f i n e
which admits a unique s o l u t i o n i n view of Theorem 1.1 ( A i s chosen such t h a t (1.56) holds )
.
We have
(1.64)
u
0
zu1
2...2u
n-1
2un;E...2-c
,
where c = s u i t a b l e constant 2 0. 1 1 I n f a c t i f i n (1.63)l we choose v t8 be given by V-U = -(Uo-U 1- i . e . r , and i f we choose Y = inf(uo,ul) I @ we have : v = i n f ( u , 0 ) I 0 on v = (uo-ul )- i n (1.62),we o b t a i n by a d d i t i o n :
,
202
(CHAP. 3)
OPTIMAZ, STOPPING PROBLEMS & V . I . ' s
which g i v e s a((uo-ul s i n c e ?-f
5
I-, (uo-ul)-) + A I(uo-ul 1-1 + 1 -
0 , we then deduce t h a t (uo-u )
(r-f, (uo-ul 1-1 5 o
;
= 0. n-1
n
We next choose v t o be d e g n e d i n ( 1 . 6 3 ) by v-un = -(u -u ) (1.63)n- by v-un-' = (un"-u ) - ; t h e s e cgoices a r e permissible; deduce t k a t
(un-l -un)-) + L(un-l -un, (Un-l-Un)')
a(un-'-un,
2 h(un-2-un-1
-
and i n we then
,(un-l
-un)-)
from which we deduce by recurrence t h a t un-l 2 un We next show t h a t choose c 5 0 such t h a t
Taking v = vo
s i n c e un i s that
-
E
u" 2
-C
,
with some s u i t a b l e c
0 , by recurrence.
We
K i n (1.63)n, we have:
from (1.64) - bounded i n L"(@),
We deduce from
5
(1.64), (1.66) t h a t
we deduce i n p a r t i c u l a r from (1.65)
(SEC. 1)
STATIONARY V . I . ' s
u
V p finite
weakly i n V and s t r o n g l y i n L p ( 6 ) ,
+ u
,
u E L"(6)
uo z
.
-c
u
I t can e a s i l y be shown t h a t
,
a
i s a s o l u t i o n of t h e V . I .
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 8.
1.8
Consider 6 c b c &' a i j , a . , a.
, 6'
E
bounded open s e t ;
suppose t h a t t h e c o e f f i c i e n t s
of A and t h e f u n c t i o n s f and $, a r e given i n
J
(1.68)
203
=
a&v)
and we denote by e put a(u,v). W
j
z U' ai5
ad.,.)
(f,v)o'
1.
2 dx.
J
v dxd x+z i
jD, g; Bj
8'; dx
we put:
+ 16'
. a
lJV
dx
t h e b i l i n e a r form which u n t i l now has been designated
=
fv d x , ( f , d 0 '
Jo, fv
dx
.
We denote by u' t h e s o l u t i o n o f t h e V . I .
i s coercive, as i s t h e f o m
Suppose that t h e f o m aol(v,v) Suppose that
THEOREM 1 . 6 .
ao(v,v).
(1.70)
Then u '
f > 0 , 9 1 . 0 a.e. 2
0 a . e . ( s e e Remark 1.10).
(LIP,
Proof. We denote b 5 the v-u' = i.e. we t a k e (1.15) we t a k e v t o be defined v = 0 on r ( s i n c e u' 2 0 ) . This
hence
in
6'
.
We have:
extension of_ u t o d by 0 i n t o 0'- 8 ; i n (1.69) v I s u p ( u * , u ) +' 5 and v = 0 on r ' = ad; i n by v-u = -(u-u ) 1.e. v = i n f ( u , u ' ) 2 $ and gives
i
+
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
204
and hence
aO' from which we deduce t h a t (ff-u')'
0
=
.
m
The conditions are taken a s those THEOREM 1 . 7 . ( P e n a l i s a t i o n of t h e domain). of Theorem 1 . 6 on a e ( u , v ) and a d ( u , v ) . Consider t h e c h a ra c t e ri st i c f u n c t i o n 5 of @ - 8.. For E > 0 , denote by uE the solution of on
uEI J, (1 .72)
a
6'(uE ,V'UE)
v v s 6 on Suppose t h a t JI
2 0.
01
,
uC E H~(o') ,
+ $uE,v--uE)61 1 61
,
2 (f,v--uC)61
.
v E H:(P)
Let:
= r e s t r i c t i o n of uE t o 8.
w
Then
(1.73)
1
T
Remark 1.11. 5uE.
weakly i n Ho(6) 1
W'U
as
c
-
0
I n problem ( 1 . 7 2 ) we have "penalised" t h e domain by t h e f a c t o r
N a t u r a l l y problem ( 1 . 7 2 ) admits a unique s o l u t i o n i n view o f t h e
preceding theorem.
8
Proof of the Theorem. ao,(UE'uC)
Taking v = 0
+ $wE'uE)o, 1
5
JI i n (1.72), we o b t a i n
s (f,u&
from which it r e s u l t s t h a t
(1.74)
(1.75) We can t h e r e f o r e e x t r a c t a subsequence, a i s o denoted by u E , such t h a t uE
and from
-.
(1.75) we
0
=
weakly i n
have
o
on
6'- 6
H;(O~)
,
.
v E H'(O), v 5 (L and we l e t D = extension of We t a k e by t a k i n g v = "v in'(1.72) we then 'lave:
v
to
by 0 o u t s i d e 8. ;
(SEC. 1)
STATIONARY V.I.’s
0 = u and we t h u s have (1.73).
so t h a t
...
Remark 1.12. Using t h e same kind of technique, we can prove t h e following: 8.i s a sequence of open domains with 8 C Oj , d j 3 dj+, 2 , and ,J d i s t a n c e (dB b b ) 0 , and i f u. ( r e s p . u) i s t h e s o l u t i o n of t h e V . I . i n
-
if
(resp.
3’ 8 )then
u u
4
=
r e s t r i c t i o n o f u. t o Q s a t i s f i e s
lo
-.
u
j
ej
J
J
w b i ~ yi n
.
~’(6)
See U. MOSCO [l] f o r a systematic i n v e s t i g a t i o n of p r o p e r t i e s of t h i s kind.
1.9.
8
Regularity of t h e s o l u t i o n
We have not e x t r a c t e d all t h e information p o s s i b l e from t h e proof of Theorem I n f a c t we have:
1.3.
THEOREM 1.8. The asswnptions are those of Theorem 1 . 3 . is t h e solution of (1.15)
.
Au E L2(d)
(1 .76)
.?roof.
I n f a c t bu
lausl S c
E
from which
We then have, if
+ -1 (uE- .&)+ = f E
and
(1.76) then follows.
& (uc- +)+I
I lf-Aib\
so t h a t
8
a . , E W””(O)
If, in the asswnptions f o r Theorem 1.3, t h e c o e f f i c i e n t s ( i n particuZar E C1(8) then the solution u of (1.15)s a t i s f i e s
(1.77)
u E
COROLLARY 1.1 1J
Au
Proof. E
H2(d
.
This is a c l a s s i c a l r e s u l t o f r e g u l a r i t y , s i n c e u
L2(S) ( s e e Chapter 2 ) .
We s h a l l now extend t h i s t o estimates i n L p , p > 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
<
I
.
f i n i t e , and we apply t h e Sobolev theorems.
m
Remark 1.13. I n t h e f o l l o w i n g we s h a l l g i v e a number o f r e g u l a r i t y r e s u l t s , but it i s important t o p o i n t o u t s t r a i g h t away t h a t , i n g e n e r a l , it is not p o s s i b l e t o go beyond estimates o f order 2 : t h e d e r i v a t i v e s o f o r d e r 3 do not i n g e n e r a l have any semblance of r e g u l a r i t y . m We deduce from (1.84) that i f 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- <)+)w+
(1.87)-)with w+ ( r e s p . -w+) and
, = (Us-
G)+)W+
= (uE- 4J
-(ac-
€
C
O
O
5
0
-6 -k ) 2 0
C
E
4J))w'
0
and i f
( s i n c e we can l i m i t
uE 2 dE + k 2 j + k 2
ourselves t o t h e p o i n t s where w b 0 and hence
= (u -6 -k +k -(&))(u
$)+W+2
O
.
$)
Thus
or alternatively a(w+k,w+)
i.e. a(w+,w+)
and hence
+ + a ( w ,w )
5
+
(aok,w+) I 0
0 , so t h a t w+ = 0 , giving
u - Q E ( k
.
Since we can interchange t h e r o l e s of uE and COROLLARY 1.4.
a',
we then deduce
The assumptions are those of Theorem 1-10,with
we also assume t h a t
(1.88).
rn
(SEC. 1)
STATIONARY V.I.'s
+
(1 .go)
€
C0(N
209
.
Then u, a soZution of (1.15), i s continuous i n
8.
(a)
2 Proof. (thus satisfying i n particular Let J , . be a sequence of f u n c t i o n s E C t h e analogue of ( 2 . 7 9 ) ) such t h a t I). + J, i n C o a ) , Let u. be t h e s o l u t i o n corresponding t o J , . o f t h e V . I . , evdrything e l s e being t h d same. I n view of u Remark 1.14, uj E 6'(6) (by v i r t u e of ( 1 . 8 9 ) ) and from (1.86) (with 3 = cO(6). ~j converges i n 0
$.I,
The following question now a r i s e s q u i t e n a t u r a l l y : i f J, E $ ' " ( S ) , do we have We s h a l l now prove t h e following theorem. t h e same p r o p e r t y f o r u? THEOREM 1.11.
(1.5) o r ( 1 . 6 ) .
u
W ~ . ~ ( B and )
E
(1.91
Suppose t h e o e f f i c i e n t s t o be constant with a . = 0 and with E W1*"(O.) with J, z10 on r , we have Then if f E Wfya'(S, and i f
1
i n which we have put
and where C denotes a s u i t a b l e constant.
Proof. F i r s t s t a a e ( * ) . The generaZity w i Z Z not be r e s t r i c t e d i f we a s s m e that f = 0 . I n f a c t , we introduce w, a s o l u t i o n o f t h e D i r i c h l e t problem w = O o n r
Aw=f, Then u
-
w satisfies the V.I.
A(u-w)
0
,
u-w-($J-w) 5 0 , (A(u-w)) (u-w-($J-w)) and u-w = 0 on We t h u s have
= 0
r. f
replaced by 0 and J,, by $-w, and s i n c e
can therefore a s m e that f = 0 .
We s h a l l now see t h a t we can assume t h a t
Indeed, s i n c e we have assumed t h a t f = 0 , we have: (*)
This s t a g e was pointed out by H. BREZIS.
w
is in
$'"(a, We
OPTIMAL STOPPING PROBLEMS & V.I.'s
210
, us
and t h e r e f o r e , s i n c e a (x) 2 0
(CHAP. 3 )
0 in 0 .
But i n t h i s case t h e V . I . w i l l not be changed by replacing JI by and s i n c e $ = 0 on r , we see t h a t we can assume t h a t ( 1 . 9 2 ) holds.
3
= inf(Jl,O),
Second s t a g e Let 8 be t h e s o l u t i o n o f
(1.93)
A8
1
6
in
with 8 = 0 on
.
r
We have
u+k8 2 0 i n 8 f o r k s u f f i c i e n t l y l a r g e .
(1.94)
Suppose i n f a c t t h a t u i s t h e s o l u t i o n of (1.87) and l e t u s demonstrate t h e analogue o f (1.94) with uE i n s t e a d of u. Taking t h e i n n e r product of (1.87) ( r e s p . (1.93) m u l t i p l i e d k) w i t t , (u +k8)-(resp.-(uc+ke)-) and adding, we o b t a i n :
BY
+ T( 1 (U,-+)+,
a(uc+ke, (uc+ke)-) P
(f+k, (UL+k8)-)
m
-
- b)+(uE+k8)-
E
a( (uc+kB)-,
2
.
since f = 0
(k, (ue+k8)-)
But(u Jl)+(u +kB)- i s not zero i f u choose k E i n such'a way t h a t
we w i l l have (u
(uE+ke)-)
$ and i f u
5
-k8;
i f we t h e r e f o r e
= 0 and t h e r e f o r e
(uc+ke)-)
+ (k,(ut+k8)-)
=
0
and hence ( u +kB)- = 0 , which t h e n g i v e s (1.94). Consequently, p u t t i n g $,
= -k8, we can consider t h a t u i s a s o l u t i o n of t h e V . I .
(1.95)
Third s t a g e We introduce
fi = extension of u t o Rn by 0 o u t s i d e 6 ,
5
mBT
(ffb-h)-dlhl,ff(x)]
,
h
given i n
Rn
,
(SEC. 1)
211
STATIONARY V . I . ' s
We have:
x f
if
r,
j(x-h)-dIh/ 5 T(x)
U(x-h)-dlhI
fl(x+h)+dlhl 2 ?,b+h)+dIhl 2 and t h u s u-(x) = 0. h
+
,
0
$,(XI
P
and t h e r e f o r e u+(x) h = 0 and
0
-
We n o t e t h a t Eh, Eh
<,
if
x E
if
.BE;; .dxi
-(x) bU
if
x f
V
=
4
I f i n t h i s l a s t i n t e g r a l we r e p l a c e x by x
-
h , we o b t a i n
Therefore i f we t a k e (as i s p e r m i s s i b l e ) obtain
(resp.
V
=
and
We then have:
i.e.
U-)
h
i n ( 1 . 9 5 ) s we
OPTIMAL STOPPING PROBLEMS & V . I . ' s
212
Thus E(x-h)-dlhl
(CHAP. 3 )
5 a ( x ) , and s i n c e we can show i n t h e same way t h a t
,
ff(x+h)+d)hl 2 ff(x) we have:
lff(x)-flb-h)
1S
d Ihl
from which t h e r e s u l t t h e n follows.
m
I t i s very probable, though t h i s does not appear t o be s t a t e d e x p l i c i t l y anywhere, t h a t Theorem 1.11 i s s t i l l t r u e under t h e assumptions of Corollary 1 . 4 r e l a t i n g t o t h e (variable!) c o e f f i c i e n t s , with ( 1 . 6 ) . The above proof i s c l a r i f i e d by t h e following comments, communicated by H. BREZIS ( s e e a l s o BREZIS and KINDEBLHERER [ l l ) ; l e t u b e t h e s o l u t i o n of (1.95); we approximate t h i s using penalisation; we a l s o denote by u t h e s o l u t i o n of t h e equation
where 8, and y, a r e C1 functions: B E ( A ) yE(h) =
-,A
= EA i f A 2 0,
i f h 2 0 , BQ2
E,
yQ
5 -E
on R ,
0, > 0 f o r A > 0 , y, > 0 f o r A < 0. We d i f f e r e n t i a t e t h e penalised equation i n an a r b i t r a r y d i r e c t i o n s ; we put
-au = as
We o b t a i n , since
Let x
lJ
A
s
has constant coefficients:
us
JI
If x
E
If us-$, > 0 , s i n c e we s t i l l have B;(A)
so t h a t u
I f t h i s i s a boundary p o i n t , then
be a p o i n t a t which us i s a maximum.
/ u s /has an upper bound s i n c e Q1 inuous and zero a t t h e boundary.
-qlS <
u 2 m m ( $ ,$ s
Is
0.
,
5 being L i p s c h i t z cont1 0 t h e n Aus(xo) 2 0 and t h e r e f o r e
2
Jt and
E,
we t h e n deduce t h a t
S i m i l a r l y i f u -$ls > 0 then us-$, < 0 so t h a t
) , from which we a g a i n deduce t h e r e s u l t .
(SEC. 1)
STATIONARY V . I . ' s
213
The f r e e s u r f a c e
1.10
Let u s now r e t u r n t o a r e g u l a r i t y theorem of t h e t y p e of (1.84). I n t h i s c a s e t h e r e a r e , i n g e n e r a l , two complementary regions i n c ( f , $ ) = (XI AU(X) = f(X))
au,JI) =1.I
u(x) =
b)f
6:
,
,
(defined e s s e n t i a l l y t o within a s e t of measure zero; i f J, and u a r e continuous, a(f,$) is closed). For reasons which w i l l become apparent l a t e r , C ( f , J , ) i s designated t h e 'continuation s e t ' and Z ( f , J , ) i s designated t h e 'stopping s e t ' or
'coincidence s e t ' .
On t h e i n t e r f a c e S = S ( f , $ ) ,
can define traces we have:
au u = $, = an
2
if
S
i s a manifold of dimension n-1 on which we
(where n i s a normal t o S ) .
We t h u s have t o apply Cauchy c o n d i t i o n s on t h e unknown s u r f a c e S; we a r e now dealing with a problem with f r e e boundary S. An important problem is concerned with t h e i n v e s t i g a t i o n of t h e r e g u l a r i t y and t h e p r o p e r t i e s of S. We s h a l l c o n f i n e our a t t e n t i o n i n t h i s r e s p e c t t o t h e following theorem, due t o H. BREZIS and D. KINDERLHERER C11:
THEOREM 1.12.
The asswnptiws are taken t o be those of Corollary 1.2, with 3 $ f C (0) n C2(6)I f f C'(0), A b f & 0 in 0 .
Let x be the characteristic function of the stopping (or coincidence) s e t . The function x is locally of bounded variation ( i . e . i t s f i r s t d e r i v a t i v e s a r e
Radon measures on 6).
Remark 1 . 1 5 The s e t s E of which t h e c h a r a c t e r i s t i c f u n c t i o n x i s l o c a l l y of bounded vari a t i o n were introduced by R. CACCIOPOLI, E. DE G I O R G I ; s e e i n p a r t i c u l a r E. DE G I O R G I , F. COLOMBINI and L.C. PICCININI [ll and t h e bibliography of t h e Theorem 1 . 1 2 t h u s l a t t e r ; t h e s e we term s e t s with locally f i n i t e perimeter. provides information r e l a t i n g t o t h e coincidence s e t .
Proof of Theorem 1 . 1 2 . Let 'p(h) P A+
and l e t A2
.
'p
rl
( A ) b e a r e g u l a r approximation of
i f A 2 0 , - i f o 2 A 5 11, A - n/2 i f A 2 11). 211 and we introduce t h e penalised equation
9 1 1 ( i )=
o
(A)
( f o r example
We w r i t e 'p i n s t e a d of
Q
We s h a l l now show t h a t , f o r any d i r e c t i o n 5 and f o r any compact subset E c @, we have: 6
IE
(1 .96)
Since
A$
-
f
E
1
Ip+s
c
C (6)apd does not v a n i s h , t h e c h a r a c t e r i s t i c function of t h e
7
214
(CHAP. 3 )
OPTIMAL STOPPlNG PROBLEMS & V.I.'s
coincidence s e t may be w r i t t e n i n t h e form :
x z -
Au-f A4 f
so t h a t t h e theorem then r e s u l t s from ( 1 . 9 6 ) . To prove ( 1 . 9 6 ) , we t a k e t h e d e r i v a t i v e with r e s p e c t t o 5 of t h e g e n e r a l i s e d equation, i n which from now on we s h a l l w r i t e u i n s t e a d of uE; we t h u s have
b(o),
Let 5 E t = 1 on E, 0 5 6 5 1 on 8 ,and l e t y S ( A ) be a r e g u l a r approximation of t h e f u n c t i o n s i g n ( A ) (we s h a l l t a k e IY,(A)~
S 1
t
YA(~)2
0
t
We t a k e t h e i n n e r product of (1.97) with
Let us assume f o r t h e moment t h a t
(1.98) p fixed a r b i t r a r i l y . Then t h e above i d e n t i t y implies t h a t
However, it can b e shown t h a t
so t h a t we o b t a i n
but then (1.97) gives
i.e.
(1.96).
Y6(0)
= 0)
cy6($(u&)).
-
This g i v e s
(SEC. 1)
STATIONARY V . I . ' s
We w r i t e O(1) f o r any term bounded ( i n d e p e n d e n t l y of 6 ) by
Then
The second and t h i r d terms a r e 0(1), so t h a t
Y =
a - joaij xi<\-+c) &i(\-+c)(yl(\-+F;)~)~
But t h e f i r s t term i n t h i s expression i s 5 0 ( s i n c e y ' ( h ) m t h e r e f o r e Y 5 O(1).
Z
0 , 5 2 0 ) and
216
(CHAP. 3 )
OPTIMAL STOPPING PROVLEMS & V . I . ’ s
Remark 1.16. and $
E
C
2
It i s also proved i n BREZIS-KINDERLHERER C11 t h a t i f f
(0w i t h $
> 0
C
1
(0,
on T, and under t h e assumptions of Theorem 1.12 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 of A, we have
Remark 1.17.
E
ax.
dym(8j,
E
I t i s proved i n D.G.
m
SCHAEFFER Ell t h a t i f f o r
f
and
$J
in
( t h e c o e f f i p i e n t s _ o f A being C”(M) t h e f r e e boundary i s Cm, t h e n t h e same i s a l s o t r u e f o r f and $ i f t h e s e a r e s u f f i c i e n t l y near ( i n C ” ( 3 ) t o f and Y (see Schaeffer, l o c . c i t . , f o r t h e p r e c i s e a s s e r t i o n ) .
C”(61
1.11.
Unbounded open domain, bounded c o e f f i c i e n t s
We now consider t h e case i n which 0 i s an unbounded open domain ( w i t h , i n For t h e moment, we t a k e t h e c o e f f i c i e n t s p a r t i c u l a r , t h e p o s s i b i l i t y O= R ” ) . t o b e bounded; t h e c a s e i n which t h e c o e f f i c i e n t s a r e not a l l bounded aij* * w i l l be i n v e s t i g a t e d i n t h e next s e c t i o n . We t h u s have
a
(1.99)
a j , a.
ij’
c
~“(8).
By analogy with Chapter 2 , Section 5.5.2, we introduce ( s e e Chapter 2 , (5.121), (5.122)):
We t a k e (1.1 01 1
f,$
E
Hp
f o r p > 0 fixed. We d e f i n e (1.102)
K = {vlv
P
c
vP,
v i J,
8.e. on
el
.
We s h a l l be i n v e s t i g a t i n g t h e following problem: f i n d u
( 1 .103)
a(u,v-u) 2 (f,v-u)
vv E
(f,v) = ( f , d l , =
j9
where
(1 .I04)
Remark 1.18.
able t o take
(*)
E
K
u
such t h a t
K
P
mi f~
.
I n t h e a p p l i c a t i o n s which we have i n mind, it i s important t o be
We could write a ( u , v ) i f t h e r e were any ambiguity.
u
217
STATIONARY V.I.'s
(SEC. 1)
.
and we w i l l then have f,+ f somewhat a w t i l i a r y r o l e .
H& V p > 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<
4aa
0
.
We can g i v e another e s t i m a t e by t a k i n g
m \x) = exp
P
- p(,G2)' ,
and by assuming t h a t (1 .1 07)
ai' 'ij
€ Wl'"(0)
.
2 = 1x12
,
218
(CHAP. 3 )
OPTIMAL S T O P P I N G PROBLEMS & V . I . ' s
Then i f we assume t h a t (1.108)
I: a.
lrl
(XI
cisj 2
alF;12
, ao(X) 2
and (1.107), then (1.106) will hold f o r
and a ot x )
a.
-71: bXi bai
1
sufficiently small
a.
'
O
,
.
The uniqueness of t h e s o l u t i o n i s proved e x a c t l y as i n Proof of Theorem 1.13. The existence can be e s t a b l i s h e d i n numerous ways. We g i v e two of Section 1 . 2 . t h e s e methods below. F i r s t method The penalisation method described i n Sections 1 . 3 and 1 . 6 can r e a d i l y be adapted t o t h i s problem. Second method: We start by s o l v i n g t h e V . I . i n a bounded open domain, and t h e n proceed t o t h e l i m i t on t h i s open domain, namely
We now p u t :
Suppose that (1.108) holds.
We denote by
%
t h e s o l u t i o n of
We introduce:
%= Let w (1 .I12)
E
%:
extension o f
we t a k e v
v
% o u t s i d e 8,.
i n (1.111)t o be defined by
2 - 5 = mp(w-uB)
,
We have v 2 JI, so t h a t t h i s i s permissible. defined by (l.lOO), but i n t e g r a t i n g over choice (1.112), (1.111)becomes
0,
I f we denote by a
instead of
8,we
!J ,R
(u,v) t h e form
s e e t h a t , with t h e
219
I f now w
E
K
we deduce from (1.113)t h a t
lJ'
(1.114)
aP(%lw-%) 2 JOE :m
We have i m p l i c i t l y assumed t h a t K
w = w
0
E
lJ
f(w-u,)& i s nonempty;
t a k i n g , i n (1.114),
KIJy
we then deduce t h a t (1 .I 1 5 )
Ilql
V
I
C
*
P
We can t h e r e f o r e e x t r a c t a sequer. e , a l s o denot 3. by
%,
such t h a t
then gives t h e l i m i t
and t h u s 0 = u. The approximations % of u, constructed from (l.lll), a r e independent of l ~ ; we have t h u s i m p l i c i t l y proved t h e following theorem. THEOREM 1.14.
Suppose that (1.106)holds for aZZ LI With
.
P, Iws Po (Pl 2 0 ) Take both f and $ E H!J Let uJ! be t h e solution of (1.103); then ulJ does not depend on p f o r v [ul,u0?
.
The monotonicity properties of S e c t i o n 1 . 6 , which a r e v a l i d f o r a r e t h u s a l s o v a l i d f o r t h e s o l u t i o n u = u ( f , $ ) obtained i n Theorem 1.13. S m i l a r l y i f we denote by a t h e s o l u t i o n of
%,
Remark 1.21.
220
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
then 0.
4 U.
8
Let us now examine t h e "non-coercive"
case.
THEOREM 1.15. Suppose t h a t (1,2), (1.4), (1.6)hold. holds and t h a t
.
4I E L?O)
(1 ,118)
We asswne t h a t (1.53)
Then there e x i s t s one and only one function and is also such t h a t u E K n Lm(@ v
.
u
which is a solution o f (1.103)
The uniqueness i s proved e x a c t l y a s i n t h e " 8 bounded" case. As Proof. regards t h e e x i s t e n c e , it can be shown by applying t h e r e s u l t s of Section 1 . 7 , t h a t t h e r e e x i s t s %, t h e (unique) s o l u t i o n of (1.111); furthermore
l\uJiLmca,,
(1.119)
We again introduce
s c
% and
constant independent of R.
I
we have
(1.120)
P
we deduce from (1.120) t h a t
P
and we conclude as i n Theorem 1.13.
8
Regularity of t h e s o l u t i o n The l o c a l r e g u l a r i t y - i n t h e sense "over any 0," - is a consequence of Section 1.9. We can a l s o o b t a i n , by r e p e a t i n g t h e proofs of S e c t i o n 1.9, g l o b a l r e g u l a r i t y r e s u l t s . For example:
The asswnptions are those of Theorem 1.13, with (1.107)and Suppose t h a t u is s u f f i c i e n t l y small ( s e e (1.125) below)
THEOREM 1.16.
(1.108).
E W"p'p(0)
(1.121)
J,
(1.122)
f E Lpvqo)
Then if u
, A$
E LptP(6)
.
i s t h e solution in K of (1.103),we have lJ
(SEC. 1)
Proof. (1 .I 24)
STATIONARY V.I.'s
We f i r s t show t h a t t h e g e n e r a l i s e d equation
hut
+
3ue-+)+
I
f , uE P 0
admits a unique s o l u t i o n i n d * P y ' ( ~ ) . We next n o t e t h a t , if 'p = 0 on
We p u t
n in1
+
r:
on
r
221
222
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP. 3 )
From (1.125) we then deduce t h a t
from which we deduce t h a t , when
-1 (uE-
(1.126)
(L)+ l i e s
E
+. 0 ,
i n a bounded subset of
Lpyp(0)
.
rn
From t h i s we deduce ( 1 . 1 2 3 ) .
The asswnptions are those of Theorem 1.15 with
COROLLARY 1 . 5 .
aij
E
C'(M
.
Then t h e solution u of (1.105)s a t i s f i e s
Proof. 1.12
.
u E P,P,%)
(1 .I271
We apply (1.123) and Theorem 5.10, Chapter 2. P r o p e r t i e s of t h e support of t h e s o l u t i o n
THEOREM 1.17 The asswnptions are those of Theorm 1.13, with (1.107)and (1.108). Suppose t h a t $I = 0 and t h a t there e x i s t s E > 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
and hence from
(1.187):
Therefore (1.193) i s equivalent t o
(1.1 94)
S(@-U) 1 e(Q-J,)
.
Since 0 i s an i n c r e a s i n g mapping and s i n c e u we have (1.194) I
COROLLARY
addition that (1 .195)
1.6.
5
JI and t h e r e f o r e 0-u
The conditions ure those of Theorem 1.21.
inf(A+,f) E LP(oj, f E IP(o)
.
2 @-$,
Suppose i n
(SEC. 2 )
EVOLUTIONARY V . I . ' s
We then have (1
.
Au E Lp(S)
.I961
235
Remark 1.27.
I f we a l s o adopt t h e c o n d i t i o n s of C o r o l l a r y 1 . 2 , we o b t a i n
.
u E w"P(0) 2.
rn
EVOLUTIONARY VARIATIONAL INEQUALITIES 2.1.
The v a r i o u s f o r m u l a t i o n s of t h e problems
-~ Notation: We s h a l l s t i l l u s e 8 t o denote a n open subset of Rn which f o r t h e We s h a l l put moment may be bounded o r unbounded.
rrm, z=~x]o,T[,
P=OX]O,T[,
,
V = H:(b)
H
I
o < T < -
,
L2(0)
2
2 and we s h a l l u s e t h e spaces L (O,T;V), L (0,T;H) e t c . , a s i n Chapter 2 , S e c t i o n
6.
We c o n s i d e r t h e f u n c t i o n s a . . ( x , t ) , a . ( x , t ) , a o ( x , t ) w i t h 1J J
(2.1 a
1
aij,
a j , a.
E
L=(Q).
No a d d i t i o n a l assumptions w i l l be made on a ; we s h a l l a l s o be a b l e t o have = 0 without modifying t h e fg.llowing argumen?s. I n t h e f o l l o w i n g , we s h a l l d i s t i n g u i s h between t h e two c a s e s below:
(2.2)
aij # a . .
t h e 'non symmetric' c a s e
a . . = a.. 1J J1
v
Jl'
or
(2.3)
i , j , t h e c a s e 'with symmetric principaZ part'.
We s h a l l s e e t h a t c e r t a i n a p r i o r i e s t i m a t e s d i f f e r depending on whether we a r e considering c a s e ( 2 . 2 ) or c a s e ( 2 . 3 ) .
For u , v E V, we p u t
We p u t
we a l s o t a k e
(*)
Sometimes it would be s u f f i c i e n t t o t a k e f
E
2 L (0,T;V').
236
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & V.I.'s
(2.8)
.
i i f H
a
The problem - f o r t h e moment, formal - i n which we a r e i n t e r e s t e d i s as follows: we seek a f u n c t i o n u , i n a s u i t a b l e space, such t h a t
(2.9)
Au
- f)(u-c))
u(x,T) = z ( x ) , x
(2.1 0)
E
i n Q, u = 0 on C
= 0
,
8. a
We s h a l l f i r s t give some p r e c i s e formulations of t h i s problem. Strong V a r i a t i o n a l I n e c p a l i t L
u
We say t h a t
,
u E L'(o,T;v)
(2.1 1 ) (2.1 2)
'
such t h a t
u(x,t)
(2.13) and
i s a 'strong s o h t i o n l of an evolutionary V . I .
u
5
bU
c
L'(o,T;H)
v(x)
$(x,t)
if
,
a.e. i n
8
$(x,t) a.e. in Q
a
s a t i s f i e s (2.10).
Remark 2 . 1 . It follows from ( 2 . 1 1 ) t h a t u i s - a f t e r p o s s i b l e modification on a s e t of measure zero - continuous from C0,TI + H (and b e t t e r s t i l l ) so t h a t (2.10) i s meaningful.
Remark 2.2. (2.1 4)
I f we introduce
K ( t ) = {vl v
E
V, v ( x )
5
$(x,t) a.e. i n 8 1 ,
which we assme to be nonempty f o r almost a l l t , we can reformulate (2.12), ( 2 . 1 3 ) a s follows:
u(t) E K(t)
(2.15)
,
-(E(t), v - u ( t ) ) + a ( t ; u ( t ) , v - u ( t ) ) 2 ( f ( t ) , v - u ( t ) ) Y v E K ( t ) bU
1
,
i n tt. . aa. e. e. . in
The f a c t t h a t t h e family of convex s e t s K ( t ) depends on t i s one of t h e b a s i c d i f f i c u l t i e s of t h e problem. The proofs which follow simplify q u i t e s i g n i f i c a n t l y i n t h e s p e c i a l case when $ ( x , t ) = $ ( x ) , i n which case K ( t ) = K does not depend on t . M
Remark 2.3.
satisfies
When it i s p o s s i b l e t o o b t a i n a s t r o n g s o l u t i o n which a l s o
(SEC. 2 )
EVOLUTIONARY V.I.'s
Y=
(2.1 7 )
v E L2 ( 0 , T ; V ) v(x,t)
5
t
237
,
" E L~(o,T;v~)
J i ( x , t ) a . e . i n Q};
we have
(2.1 9)
Since we can t a k e v = v ( . , t ) i n ( 2 . 1 2 ) , t h e f i r s t i n t e g r a l on t h e right-hand s i d e of ( 2 . 1 9 ) i s 5 0; t h e r e f o r e
xz j
T
1
d
o - 7 'dz Iv-uI
dt
and t h u s
Replacing u(T) by G, we a r e t h e r e f o r e l e d t o t h e following d e f i n i t i o n : we say t h a t u is a 'weak solution' of t h e V . I . i f
(2.20) (2.21
1
u
E
2 L (O,T;V),
u
5
Ji a . e . on Q ,
(2.22)
We s h a l l f i r s t study t h e ' s t r o n g ' problem, and t h e n go on t o t h e 'weak' problem. 2.2. Existence and uniqueness r e s u l t s f o r t h e s t r o n a s o l u t i o n s We g i v e h e r e d i f f e r e n t expressions depending on whether we a r e c o n s i d e r i n g case ( 2 . 2 ) o r ( 2 . 3 ) ; as is a p p r o p r i a t e , t h e r e g u l a r i t y assumptions on t h e d a t a a r e s t r onger i n c a s e ( 2 . 2 ) t h a n i n c a s e ( 2 . 3 ) . To c l a r i f y t h e c o n c e p t s , we s h a l l assume is bounded. that THEOREM 2 . 1 .
(2.23)
Suppose t?mt ( 2 . 1 ) , ( 2 . 2 ) hold, w i t h
Z a. (x,t)Si 5 . 2 a C 12 J
Si2
,
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
a.e. i n Q
and we adopt t h e assumption t h a t
For y > 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
-
<
k
sufficiently large that
J, a . e . ,
JI)+(uE+k 8 ) - = 0 a . e . and consequently I d - y;ii I (uE+ k0I-l 2 + a( (uE+ k0)-,
(u,+ kBl-1
- (k,
from which we deduce t h a t (uE+k 8 ) - = 0 , and hence t h e r e s u l t . assume t h a t
u
i s a solution of a V.I.:
(uE+ k0)-) = 0
We can t h e r e f o r e
OPTIMAL STOPPING PROBLEMS & V.I.'s
272
3)
We then proceed as in the proof of Theorem 1.11.
il = extension of u to Rn
x
(0,T) by 0 outside
= max.{ff(x-h,t)
- dlhl , U(x,t)]
q(X,t)
= min.{U(x-h,t)
+ djhl
Ei(t)
P
(XI.
f
Rn
We introduce
8,
%(x,t) +
Ei(t) = (xlx E Rn
{ and v =
h given in Rn,
, Ci(x,t)j
, U(x-h,t) - d/hl > U(x,t)) , ff(x+h,t) + dlh( < ff(x,t))
We take, as is permissible, v =
(CHAP. 3)
<
.
in (2.176).
To clarify the concepts, we assume that a(u,v)
=
jograd. u grad.
v dx
.
We obtain
and
Replacing x by x-h in the last inequality and adding the result to (2.177), we obtain b (u(x, t)-u(x-h,t))(u(x, t)-u(x-h, t)+ d ( h )dx
IE;(t)
-
I
(SEC. 2)
We then deduce, since u(x,t)
(2.1 78)
273
EVOLUTIONARY V.I.'s
1
d
'y 'Fit
jE:(t)
-
u(x-h,t) + dlhl = 0 on the boundary
- u(x-h,t)
(U(X,t)
But since u(x,T) = 0, E+(T) = h
0
]u(x,t) - u(x-h,t) from which the result follows.
Ei(t):
dx
and consequently (2.178) implies
0
from which it follows that E+(t)..= h
2
+ dlhl)
Of
I
V t; likewise Eh(t) = (d 6
and consequently
dlhl
8
For results of the kind stated in Theorem 1.12, we refer the reader to A . FRIEDMAN and D. KINDERLHERER [11.
2.13
Properties of the solution relative to the domain('%
There are some simple extensions of the results of Section 1.8 to the evolutionary case. For example, with the notation o f Theorem 1.6, and assuming that
(2.179)
r,Ji
L
t 0
o a.e. in a.e. in
0'
x
IO,TI ,
d
the solution of the evolutionary V.I. ( e i t h e r the strong we have, denoting by solution OF the maximum weak solution):
(2.180)
u
5
u a.e. in 0
x
l0,TC
.
Similarly, we can adapt the result of Theorem
1.7 to the evolutionary case.
8
274
OPTIMAL STOPPING PROBLEMS & V.I.'s
( C H A P . 3)
Remark 2.18 (non-cylindrical open domains). More g e n e r a l l y , we can consider: Q = non-cylindrical open domain i n Rn
and f u n c t i o n s a We define
a j , a.
u
E
Rt,
contained i n 0 < t < T
L"(Q), with t h e assumption (2.23).
t o b e a weak s o l u t i o n i n Q o f t h e V.I., by:
V v € y
where
giE L'(Q),
y a ~ v l vE L?Q),
(2.182)
x
E L~(Q) (*I
,VS
(L
a.e.1
with v = 0 on t h e " l a t e r a l boundary" C of Q, with
€ L2(Q) axi
u, bU
(2.183)
,u=
0
on t h e " l a t e r a l boundary" o f Q
.
Using t h e same type of methods a s b e f o r e , it can be shown, a t l e a s t with a s u f f i c i e n t l y r e g u l a r l a t e r a l boundary X , t h a t there e x i s t s a maximwn s o h t i o n of (2.181), (2.382), (2.183). We then have extensions of Theorems 1.6 and 1 . 7 with Q c non-cylindrical open domain contained i n 0 < t < T.
2.14
6
c
Q', Q' being a
I n f i n i t e horizon
So f a r w e have considered evolutionary problems f o r t < T , t h e horizon T being finite. The case i n which "T = + -I' a r i s e s i n t h e a p p l i c a t i o n s considered ( s e e Section 3) and i s i n t e r e s t i n g i n i t s own r i g h t . We t h u s assume t h a t
(2.184)
Q=o
The f u n c t i o n s a i j ,
.
IO,=C
... a r e given
i n Q as before.
assume t h a t (2.
(2. (2. We introduce y with
(2.1 88)
O < y < a
2
. 2
We w r i t e L2(V), L ( H ) f o r L (O,-;V), L2(0,-;H). (*)
This condition can be weakened.
To c l a r i f y t h e concepts, we
EVOLUTIONARY V.I.'s
(SEC. 2)
275
We s h a l l now prove t h e following theorem:
Suppose t h a t (2.184)
THEOREM 2.17.
(2.189)
e-Ytf
. . .,
(2.188)hold.
Take f,$ with
c L~(H),
, e-yt(-
e'Yt+ E L2(V)
(2.1 (2.193) 93)
-- ((EE,,v-u) v-u) ++ a(t;u,v-u) a(t;u,v-u) 22 (f,v-u) (f,v-u) vv vv EE VV YY 22 $$ aa. e . e. .
.
bJ,+ dt A ( t ) + ) E L2(H)
(2.1 90)
i inn tt..
such such t thhaat t
I I
Remark 2.19. There i s no " i n i t i a l condition"; t h i s is replaced by t h e growth m condition a t i n f i n i t y (2.191)
Proof of uniqueness. Let u and ti be two p o s s i b l e s o l u t i o n s . Taking v = ti ( r e s p . v = u) i n t h e V . I . (2.193) ( r e s p . i n t h e analogous V . I . r e l a t i n g t o 5 ) and adding, we o b t a i n on t a k i n g u - YI = w: I T
d
Idt) 1
hence
d xlw(t)j Since
2
llvll 2 I v ( d
2
- a(t;w,w) 2
0
- 2allw(t)ll 2 2 o .
( * ) , it t h e n follows t h a t
Iw(t)
l 2 - 2alw(t) l 2
2 0
hence
dt
IW(t) l21
and t h u s
However e-2atlw(t)12 Z c e-2(a-Y)t f o r e c o n t r a d i c t t h e c o n d i t i o n emYtw
E
e
-2at
IW(t)I
2
2
C
i s not summable on (O,+
. m),
L2(H)9 u n l e s s w = 0 (**).
so t h a t we t h e r e m
( * ) This r e s u l t i s n o t t h e b e s t p o s s i b l e ; we can thus take y s l i g h t l y larger than a i n (2.188),a t l e a s t when Ois bounded.
( * * ) In f a c t we have uniqueness w i t h i n t h e c l a s s of f u n c t i o n s
e-Ytu E
L~(v)
,
e-Yt
5 E bt.
L'(VI)
.
u such t h a t
dU
n)
-(+v-u
dt
(2.1
uns
94)
+
a(t;un,v-u ) 2 (f,v-u ) n n
w , un(d
un E L2(0,n;v)
We approximate u
v
v s
+,t
3)
,
= 0
.
We s h a l l now e s t a b l i s h a p r i o r i e s t i m a t e s on u t o t h e limit i n n . 2)
(cm.
OPTIMAL STOPPING PROBLEMS & V.I.'s
276
which w i l l allow us t o proceed
by p e n a l i s a t i o n ; w e thus l e t u
nE
We t a k e t h e i n n e r product of ( 2 . 1 9 5 ) with u--- JI.
b e t h e s o l u t i o n of
This gives
(2.1 96) Then by i n t e g r a t i n g from t t o n:
from which we deduce, s i n c e y bounded),
a (which can be improved somewhat when 6 i s
We t h u s have t h e same e s t i m a t e f o r u extension of u
3)
(making E
-f
0 ) and i f we denote by En t h e
by 0 for t > n , we have:
We can o b t a i n a supplementary e s t i m a t e by t a k i n g t h e i n n e r product of ( 2 . 1 9 5 )
a
with-E(unE
-
jl).
We o b t a i n , using t h e n o t a t i o n o f t h e proof of Theorem 2.2:
EVOLUTIONARY V.I.'s
(SEC. 2 )
hence
Using (2.199) we t h e n deduce:
The sane e s t i m a t e s h o l d f o r un and t h e r e f o r e we have
4)
We r e a d i l y deduce from ( 2 . 1 9 9 ) , ( 2 . 2 0 1 ) , (2.202) t h a t , p o s s i b l y a f t e r e x t r a c t -
i n g a subsequence, ii
+
2
u weakly i n L (V), and
(2.203) Remark 2.21.
-+
all
weakly i n L
2
(H), u being
m
t h e s o l u t i o n of t h e problem.
Remark 2.20.
aa
atn at
The following supplementary information i s o b t a i n e d :
e-Yt u
c L=(V)
.
m
I t i s a l s o p o s s i b l e t o o b t a i n : e s t i m a t e s i n t h e spaces
dyp(S);
Lp(@, r e s u l t s analogous t o t h o s e of Theorem 2.1; p r o p e r t i e s of monom t o n i c i t y and r e g u l a r i t y of t h e s o l u t i o n .
278
(CHAP. 3)
O P T I M A L S T O P P I N G PROBLEMS & V . I . ' s
To prove Theorem 2 . 1 6 , we can a l s o c o n s i d e r t h e following penalRemark 2.22. i s e d e q u a t i o n on (0 + m ) ,
This problem admits a unique s o l u t i o n , which decreases as We say t h a t
u i s a weak s o l u t i o n on e+
jy
(2.205)
V v E
,U
S $
[-e-2yt(g,v-u)
+
u E L2(V)
y
with
ye
O x
lo,-[
if
a . e . and
- e -2yt ( f , v - u ) ] d t
e-2Yt a(t;u,v-u)
{ V I E L2(V),
e -yt
decreases
E
at
E
~
~
(
~
v1 s1
+,
2 0 a.e.1
Then (as i n Theorem 2 . 6 ) :
Suppose t h a t we have (2.185) (2.188) (2.189), 2 converges weakly i n L ( V ) , from above, emYt$ E L ~ ( H ) ,Y # @; then when E $, u t o t h e m a x i m solution u of ( 2 . 2 0 5 ) . rn THEOREN 2.18
2.15
Unbounded open domain. bounded c o e f f i c i e n t s
We use t h e n o t a t i o n of S e c t i o n 1.11.
We now p u t , f o r u,v
t h e open domain B b e i n g unbounded ( w i t h t h e p o s s i b i l i t y
fopormulation" of t h e e v o l u t i o n a r y V . I . i s : f i n d u w i t h
(2.208)
u
E
8.=Rn).
V :
The "strong
6 J, a . e .
We e x p r e s s t h e "weak formulation" s i m i l a r l y , by r e p l a c i n g , i n ( 2 . 2 2 ) , t h e i n n e r product i n H by t h a t i n H and a ( t ; u , v ) by a ( t ; u , v ) . We now have established
a l l t h e statements analog8us t o those given Yn t h e preceding sections, with V replaced by Vp, H by Hp, and using t h e same methods. 2.16
Supports
In t h i s s e c t i o n we g i v e a r e s u l t which extends t o t h e p a r a b o l i c c a s e t h e "compact support property" which was proved i n S e c t i o n 1 . 1 2 f o r t h e e l l i p t i c c a s e . The following r e s u l t i s due t o H. B R E Z I S and A. FRIEDMAN [I1 which may be A number of v a r i a n t s o f t h e following r e s u l t a r e consulted f o r f u r t h e r r e s u l t s . rn i n d i c a t e d , u s i n g extremely d i f f e r e n t p r o c e d u r e s , l a t e r on i n S e c t i o n 3.
(SEC. 2 )
FVOLUTIONARY V.I.'s
us
(2.21 1 )
0
,
, - g + A u - f s O
bU
u(-E+
AU
- f) =
.
u(x,T) = E(x)
o
279
t
for
<
T
,x
E R"
,
Then (2.214)
u = O f o r t < T = T
Remark 2.23.
p > 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
t h a t we have ( 2 . 2 1 4 ) .
-
T
fO
= To, it follows from (2.218) and from u
M/f
s 0
m
Theorem 2.19 gave a p r o p e r t y of t h e support “ i n t “ ; we s h a l l now g i v e a property of compactness of t h e support “ i n x”.
The conditions are those of Theorem 2 . 1 9 ) .
THEOREM 2.20
(2.21 9)
U
=
Proof. (2.221
1
c and R
for 1x1
r
2
0‘
u denoting t h e m m i m weak solu t i o n :
We then have, (2.220)
o
Also suppose t h a t
u(x,t) =
o
Ix I 2 ro .+ (- 1
for
2fo
id .
E)3
.
W e consider a p r i o r i t h e function
w(x,t) = w(x) =
being constants
lo -
C(R~-
r)2
if if
1x1 =
rs R 0
r 2 Ro
’
,
0 t o be chosen.
2
We have, i f r = Ro:
so that
- -+dwd t
I f we choose R
i f we choose 2c = f
Aw I f
0‘
such t h a t
- c(R,-
2
ro)
s 3x1 v
x
,
(which i s p e r m i s s i b l e ) , we have: w(x,T) = w(x) 5 u ( x ) . Then from Theorem 2.9 2.17
we have w
5
u, from which we then deduce (2.220).
Unbounded open domain, unbounded c o e f f i c i e n t s
We now consider an unbounded open domain 8 and an operator of which t h e f i r s t With n o t a t i o n analogous t o t h a t of Section order c o e f f i c i e n t s a r e unbounded. 1.13, we put:
We s t i l l assume t h a t
m
(SEC. 2)
( 2.224)
EVOLUTIONARY V.I.'s
E Bxk
ai, ba.
I
lA(Xlt)l bai -TZ-
+z
(2.230)
dXi
so
,
C O W
m(x)
f o r a l l t , with
z p o m - c,
,
ai(x,t) a
v E Fc
bU
s cp d~ i
+ EII(t;u,v)
+
J'.
m
.
8 , > 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 ) = <(x)
A(t)UE
- -1 (uE- dJ, 1- = f,
1 +r (uE- +2)*
E
.
E
>
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
<, a
,
we can t h e r e f o r e see t h a t t h e soZntion u given by Theorem 2.25 coincides with t h e r e s t r i c t i o n t o l0,TC o f t h e bounded solution fi ( o r t h e s o l u t i o n with "growth" eYt) of t h e V.I.
2.20
on R,.
Estimate f o r
- at
+ Au.
I n t h i s s e c t i o n we s h a l l e s t a b l i s h some estimates which a r e t h e "evolutionary analogues" o f those i n Section 1 . 1 5 .
THEOREM 2.26. additionally:
The assumptions are taken t o be those of Theorem 2.6, with
Suppose t h a t
4 E H'(Q)
(2.313)
Let
(2.314)
u
, P+ E R ( Q ) , $ 2 0
on
Z
where
P =
be t h e mmimm weak soZution 'of ( 2 . 2 0 ) , ( 2 . 2 1 ) , (2.22). id. (~4f , ) sP U S f
.
-bm + A We have:
.
OPTIMAL STOPPING PROBLEMS
296
&
(CHAP. 3 )
V.I.’s
Pro0f. 1)
We introduce
a s o l u t i o n of
w
pw = f , w = Considering u
-
G
for t = T, w =
o
on C.
w i n s t e a d of u , t h e problem then reduces t o t h e case:
We then reduce, as i n Section 1 . 1 5 , t o t h e case where
(2.31 6 )
+ = O
on
Z
. -
We next introduce a sequence p o f r e g u l a r functions i n Q, f o r example i n C2(z) such t h a t 3
(2.31 7 ) Let J I . be t h e s o l u t i o n of J
(2.318)
PGj = v j
9
+j(~,~ = )o
, cj
=
o
on
.
L
By v i r t u e of (2.312) we have
We can .therefore apply Theorem 2.1, t h e s o l u t i o n u o f t h e V.I. corresponding t o f = 0 , u = 0, JI = J I . i s a strong s o l u t i o n . From thejtheorem of s t a b i l i t y of t h e maximum s o l u t i o n s , theJproblem reduces t o showing t h a t
for
u
a s t r o n g s o l u t i o n o f t h e V.I., satisfying (2.28).
We denote by 8( +) t h e maximum s o l u t i o n of t h e V.I. r e l a t i v e t o f = 0 , 2) and t o JI which may o r may not be r e g u l a r . We know t h a t 8 i s an increasing mapping. Let 4 be t h e s o l u t i o n of
(2.320)
rn
= (P+) A 0, @(x,T) =
o ,Q=o
on
L
,
Everything then follows from t h e following formula which is analogous t o formula (1.289) of J . L . JOLY:
u= 0
29 7
EVOLUTIONARY V.I.'s
(SEC. 2 )
d@ -
We put Y I St+)) and we note t h a t u = S($) s a t i s f i e s ( 2 . 2 8 ) . Since we a r e i n t h e "symmetric" case ("1 t h e maximum s o l u t i o n a c t u a l l y t u r n s out t o be a s t r o n g s o l u t i o n i n t h e sense of Theorem 2.2. Proving t h a t Y = 9-u amounts t o showing t h a t
(2.322)
2
=
y-@+u) + a(~-Q+u, Y-@+u)]dt I 0
J ,T [(-~(Y-@+U),
a l l t h e i n t e g r a l s being meaningful. We then conclude more o r l e s s as i n Section we have Z = Z + Z2 + Z3, with 1.15;
a s we can t a k e
g, = 9-u i n t h e V . I .
then
and t h e problem amounts t o showing t h a t Z3 S 0. S i n c e , by d e f i n i t i o n of u, we have
(2.323)
(-
6U
v-u)
+
a(u,v-u) 2
o ,v s
J,
9
t h e r e s u l t w i l l be e s t a b l i s h e d if we can show t h a t we can t a k e v = 0-Y i n (2.323), i . e . 9 - Y 5 v, i . e .
But g(O-$) = O - $ equivalent t o
8 ( @-
(*I
P(@-J,)
since U)
2
8 ( @-
I
(P$)
0
- P$s
0
, so
t h a t (2.324) i s
$)
Theorem 2.26 has been proved without using t h e symmetry assumption on t h e a i J , These authors with a i j E C l ( C ) , by P. CARRIER and G.M. TROIANIELLO [l]. use e l l i p t i c r e g u l a r i s a t i o n and a r e s u l t of HANOUZET-JOLY [ll, TROIANIELLO [11 r e l a t i n g t o t h e extension o f t h e r e s u l t s of Section 1.15 t o mixed problems.
298
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.’s
s i n c e 0-u
8 i s i n c r e a s i n g , we then have t h e r e s u l t .
and
5 0-$
The above then allows c e r t a i n r e g u l a r i t y theorems t o b e deduced.
F o r example:
The asswnptions are taken t o be t , b s e of Theorem 2 . 6 , w i t h
THEOREM 2.27.
and
Then t h e solution u of the V . I . s a t i s f i e s
Proof.
From Theorem 2.25, we have
pu E ~ ~ ( 0 ) and we then apply Theorem 6 . 5 of Chapter 1.
m
An extension of Theorem 2.27 By using ordering r e l a t i o n s i n spaces of t h e type H - l , it i s p o s s i b l e ( P . CHARRIER, B. HANOUZET, J . L . JOLY C 1 l ) t o extend t h e r e s u l t of Theorem 2.27 t o t h e case i n which J,
,
E L~(o,T;H’(Q))
f
E L~(o,T;H-’(P))
,
i d . (p+,f) E L ~ ( o , T ; H ” ( Q ) )
,
t h e i n f . being taken i n a s u i t a b l e sense, and t h e c o e f f i c i e n t s s a t i s f y i n g
aiJ
E C’(q),
ao, ai
sense t o U(T) and 2.21
E co(q) ;
U(T)
=
it i s t h e r e f o r e p o s s i b l e , i n t h i s way, t o give a
7.
Maximum weak s o l u t i o n a s an upper envelope o f sub-solutions
we t h u s have t h e “minimum” We now r e t u r n t o t h e s i t u a t i o n of Theorem 2.6; assumptions on t h e d a t a (when t h e operator A i s given i n divergence form; see Section 4 l a t e r , f o r o t h e r c a s e s ) . We say t h a t w i s a weak sub-solution i f
29 9
EVOLUTIONARY V.I.'s
(SEC. 2 )
[Formally, we t h u s have
We then have t h e following theorem.
THEOREM 2 . 2 8 . The assumptions are taken t o be those of Theorem 2.6. Then the maximum weak solution of ( 2 . 2 0 ) , ( 2 . 2 1 ) , ( 2 . 2 2 ) is the maximum element o f t h e s e t o f weak sub-solutions.
Remark 2.29. This r e s u l t , which i s a v a r i a n t of Theorem 2 . 6 , w i l l be u s e f u l i n I t h i s form i n S e c t i o n 4.
Proof. 1)
The maximum weak solution i s a weak sub-solution.
2 I n f a c t we know t h a t u i s t h e l i m i t , f o r example i n L (O,T;V), of u e , t h e s o l u t i o n of ( 2 . 3 L ) , ( 2 . 3 5 ) . I f 'p i s a r e g u l a r t e s t f u n c t i o n such t h a t g(o) = 0, we deduce from ( 2 . 3 4 ) , (2.35) t h a t
Hence i f 9 2 0 we have
jt [(us, 2)+
a(t;uE,rp)
- (f,(p)~dt - ( c p ( ~ ) , ~ ) s o
which g i v e s t h e r e s u l t , by proceeding t o t h e l i m i t , t h a t 2)
If
w
u
s a t i s f i e s (2.328)
i s a weak s u b - s o l u t i o n , t h e n
Assuming f o r a moment t h a t ( 2 . 3 3 0 ) h o l d s , it t h e n follows t h a t w 5 u and t h e r e f o r e t h a t Sup. w ( s u p . i n t h e s e t o f weak s u b - s o l u t i o n s ) 5 u , which g i v e s t h e r e s u l t , s i n c e from 1) t h i s sup. i s 2 u. I f we p u t z = w-uE, we deduce from ( 2 . 3 2 8 ) and ( 2 . 3 2 9 ) , by s u b t r a c t i o n , t h a t
which i s i d e n t i c a l t o ( 2 . 8 5 ) . 2.22
Hence, from ( 2 . 9 0 ) , z 5 0 , i . e .
(2.330).
8
S t a b i l i t y of t h e maximum weak s o l u t i o n
We s h a l l now show t h a t t h e maximum weak s o l u t i o n "depends continuously on t h e d a t a " ; t h i s type o f r e s u l t h o l d s (under t h e " n a t u r a l " assumptions) f o r t h e S o l u t i o n o f a l l t h e V.I's encountered i n t h i s volume, b u t it i s only f o r t h e maximum weak s o l u t i o n t h a t t h e r e s u l t i s not obvious and it i s t h e r e f o r e only t h i s c a s e which we
OPTIMAL STOPPING PROBLEMS & V.I.'s
300
(2.331)
ar (x,t), ai(x,t), at(x,t) E L ~ ( P ) , P = v x 1 ij 8 bounded, 2 ar ( x , t ) c . c . 2 a z ci2 , a > 0 lj
( a independent of
(2.332)
aij
-
aij
, ari - a i
fr, f , G r , fq,
We a l s o t a k e
(2.333)
(CHAP. 3 )
fr
-
f
in
d
-
0~~1
1 J
,
r)
: aro - a ii
in
LYQ)
8
t
v Ei E R
.
with
L~(Q) , u-r -ii
L~(o) ,
in
t h e f u n c t i o n s Jlr and JI s a t i s f y i n g t h e following c o n d i t i o n s :
f i r s t we p u t
we assume t h a t
(2.335)
if
Vr S
then
and we denote by
ur
Qr
V S4
a.e.
.
,V r
,and
t h e mareinnun solution of
if
Vr
-
V
weakly i n
L2(Q)
8
jz [(-XI
v-ur) + ar(t;ur,v-ur)
bv
(2.337)
- ?I2=
1
0 V v €
+~lv(T)
and by
301
EVOLUTIONARY V.I.‘s
(SEC. 2 )
- (fr,v-ur)]dt $ , ur
E
$
,
u t h e m a r i m solution of ( 2 . 2 2 ) (”limit” V . 1 . ) .
We have t h e following r e s u l t due t o MIGNOT and PUEL C11:
...
THEOREM 2.29. We adopt the USSz.mptiOnS ( 2 . 3 3 1 ) , (2.335). Let be t h e marimwn solution of (2.337) ( r e s p . of ( 2 . 2 2 ) ) . We then have
(2.338)
u’
-
u
weakly i n
L2(0,T;V)
ur
(resp.
U)
.
Pro0f. 1)
We s t a r t with a preliminary reduction
LEMMA 2.3.
(2.339)
The generality w i l l not be r e s t r i c t e d i f we assume t h a t
a r ( t ; v , v ) B al llv[12 , a,‘*> 0
,vv
€ V
, a 1 independent
In f a c t , l e t u be t h e maximum s o l u t i o n of (2.22); of (2.341, ( 2 . 3 5 ) ; i f we put
u = e then ii
of r.
l e t uE be t h e s o l u t i o n
k( T-t ) bE
satisfies
consequently i f we put
u = ek(T-t)
ff
we see t h a t il is t h e maximum s o l u t i o n of t h e weak V . I . with a ( t ; u , v ) replaced by a ( t ; u , v ) + k ( u , v ) , J, by e 2)
-k(T-t) $ , a n d
f
b y e -k(T-t)f.
We then deduce Lemma 2.3. B
A p r i o r i estimates ( I )
By v i r t u e o f ( 2 . 3 3 4 ) , we can t a k e , i n (2.337), v = vo independent o f t h a t we can deduce t h a t
(2.340)
r
so
302
OPTIMAL STOPPING PROBLFNS & V . I . ' s
(CHAP. 3 )
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 r , such t h a t
(2.341)
Ur
-. d
weakly i n
L2(0,T;V)
,
We s h a l l now show t h a t ii i s one weak s o l u t i o n of (2.22) ( t h e whole d i f f i c u l t y w i l l then l i e i n proving t h a t it is i n f a c t t h e m a x i m s o l u t i o n ) . Since ur
5 (I~ it,
follows from (2.341) and (2.335) t h a t
.
fiS+
r.
Let v be given i n Then from ( 2 . 3 3 4 ) , which can be w r i t t e n i n t h e form
J*
[-(x,v-ur)
Bv
+
+
1:
a ( t ; ur ,v-u r )
[ar(t;ur,v-ur)
, and
E $V r
we have (2.337),
- (fr,v-ur)]dt + ~1 l v ( T-) u-r I 2 - a(t;ur,v-ur)]dt 2 0 .
Using ( 2 . 3 3 2 ) , we then r e a d i l y deduce t h a t 3)
V
fi s a t i s f i e s (2.22)
A p r i o r i e s t i m a t e s (11).
We s h a l l now show t h a t
I
there exists
w(r)
0
i .
if
r +
m
such t h a t
. .
where
ur
( r e s p . u E ) i s 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 a s s o c i a t e d with (2.337)
( r e s p . of ( 2 . 3 4 ) ) .
In f a c t we have:
zj'. l e t us t a k e t h e i n n e r product o f (2.343) with (uc - u examining t h e various p o s s i b i l i t i e s ) t h a t
x(u,We t h e r e f o r e o b t a i n
r +2 0
UE)
.
It can be shown (by
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
v)
L'(o,T; r
+ m
5 C
, so
303
t h a t by v i r t u e of (2.332) t h e r e e x i s t s w l ( r )
+
0 if
such t h a t
We have an analogous e s t i m a t e f o r t h e right-hand s i d e o f ( 2 . 3 3 4 ) . e x i s t s 0 2 ( r )+ 0 i f r + m w i t h
Hence t h e r e
and hence ( 2 . 3 4 2 ) .
4)
Proof o f t h e Theorem 2 s t r o n g l y i n L (Q) ( r f i x e d ) when E + 0 , so t h a t (uE-Uz)+
We know t h a t u -ur + u-ur E
E
2 s t r o n g l y i n L (Q).
-. ( u-')
We t h e r e f o r e deduce from (2.342) t h a t in
particuhr
(2.345)
But from ( 2 . 3 4 1 ) and t h e convexity of
lim. i d .
V-
E j I(u-ur)+12dt
ST I'
I(U-V)+12dt
2
0
, we
have:
l(u-tI)+l2 dt
which, i n c o n j u n c t i o n w i t h ( 2 . 3 4 5 ) shows t h a t
(u-6)' and hence u 5 fl; m theorem.
3.
since
0
u
0 i s t h e maximum s o l u t i o n we have u = Q , and hence t h e
OPTIMAL STOPPING-TIME PROBLEMS. 3.1
STATIONARY CASE.
Synopsis
We s h a l l now reexamine t h e s i t u a t i o n touched on b r i e f l y a t t h e beginning o f t h e book, i n Chapter 1. Our aim i s t o s u b s t a n t i a t e t h e h e u r i s t i c arguments developed i n Chapter 1, and i n p a r t i c u l a r t o i n t e r p r e t t h e s o l u t i o n s of t h e v a r i o u s 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 v e s t i g a t e d i n t h e preceding s e c t i o n s , on t h e b a s i s of optimal stopping-time We t h u s have t o perform f o r t h e i n e q u a l i t i e s what was performed i n problems. Chapter 2 f o r t h e c a s e of e q u a t i o n s . A s i n t h e e q u a t i o n s c a s e , t h e i n t e r p r e t a t i o n i s made e a s i e r t h e more r e g u l a r t h e s o l u t i o n becomes. F o r t h i s r e a s o n , we s h a l l begin by studying t h e r e g u l a r c a s e corresponding i n t h e f i r s t i n s t a n c e t o s t a t i o n a r y problems, t h e n a f t e r w a r d s t o e v o l u t i o n a r y problems. Subsequently, t h e assumptions w i l l be weakened a s much as p o s s i b l e .
304
(CHAP. 3)
OPTIMAL STOPPING PROBLZMS & V.I.’s
3.2
Regular case
- bounded open
domain
Let 6 be a bounded open domain of R n , with t h e second order e l l i p t i c o p e r a t o r
(5.1)
a
d
A = - Z
Fi
ij
‘ij
S J. f +
=
a & of c l a s s
C
2
.
We consider
b ai
+
‘0
where
(3.2)
We s p e c i f y f and jl such t h a t
Clearly JI
E
CO
(5).
By a p p l i c a t i o n of Theorem 1.9 and o f Corollary 1.3, t h e r e e x i s t s one and o n l y one f u n c t i o n u such t h a t
, u E C’(W , a . e . , us $ on 5
(3.5)
u E WqO)
(3 . 6 )
Aus f
(3.7)
u/r
= 0
.
, (u-J,)(Au-f)
I
0
a.e.
If a 3 a ( x ) denotes t h e matrix with elements a i J , we d e f i n e u ( x ) as i n Section 7 , Chapter 2, by
(3.8)
a(x) =
a 2 tx )
2
and
(3.9)
gj(x) =
n ba. i=l
Aa. , dX. J 1
j
= l..n
.
( * ) A s i n Chapter 2 , Section 7 , we s h a l l simply make use of t h e f a c t t h a t g,o a r e uniformly Lipschitz-continuous i n Rn by s u i t a b l e extension o u t s i d e 5 .
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
t
305
,
We s p e c i f y a system (G,O,p,Z w(t)) and we consider 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
(3.10) (3.10)
=
dy
g(y(t))dt
+
&(t))dw(t)
Y ( 0 ) cz
whose s o l u t i o n i s denoted by y x ( t ) . we also w r i t e y(t),r).
Let T~ be t h e e x i t time from @ ( f o r shortness
I f 6 i s a stopping time with r e s p e c t t o
where
xa<@
Zt , we
put
= 1 i f a < @ , o r 0 i f a'@.
Then we have THEOREM 3.1. Under t h e assumptions ( 3 . 2 ) , ( 3 . 5 ) , ( 3 . 6 ) , ( 3 . 7 ) i s given by t h e fomnuZa
(3.1 2)
U(x)
= Inf.
e
( 3 . 3 ) , ( 3 . 4 ) , t h e soZution
U(X)
=
Jx(ex)
V X
of
.
Jx(e)
Moreover, there e x i s t s an optimal stopping time e
(3.13)
u
E 5
such t h a t
t
and Ox i s defined by
(3 -1 4) Proof.
ex = i d We extend
=
(8
20
/
d y X ( 8 ) ) = JI(Y,(s))
I
.
u by 0 o u t s i d e s a n d J1 by continuation. € R"
/
u(E)
<
If
$(<)I
then 8 i s t h e e x i t time from t h e open domain C ( s i n c e u,$ a r e continuous) and i s t h u s axstopping time. We put
thus
We apply t h e g e n e r a l i s e d I t o formula (8.48), Theorem 8.3, Chapter 2 , t o t h e functional
@ ( x , z , t ) = u(x)z
306
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & ‘4.1.‘~
and t o t h e p r o c e s s ( y ( t ) , z ( t ) ) , t o which t h e r e corresponds t h e d i f f e r e n t i a l operator
a =-
Z-
thus
ff@= ( A d z If we p u t we o b t a i n :
‘aT
a
a
a
.
= 8 A T , t h e n for a l l s t o p p i n g t i m e s ?’ and f o r a l l f i n i t e T ( * )
7 . 1 , Chapter 2 ) t h a t + 1, t h e n
We know ( s e e Theorem if
cp
E
L’(s),
a
axi ai j Fj + z a i ~ i + a o e ~
p >
-
I n f a c t , l e t us assume t h a t
(Ao
+
A,)w
92
T
i s a.s. f i n i t e .
But we n o t e t h a t
0 and c o n s i d e r t h e boundary problem
‘P
= 0 (where A.
+ A1 = A
-
a o ) . which p o s s e s s e s a s o l u t i o n w E W * ” ( S ) ,
and w
E
Co(&).
A s f o r (3.15) we have
:1
t h u s s i n c e T i s a. s . f i n i t e
lim E T-rm
cp(Y,(t))dt
=
W(X)
and by a p p l i c a t i o n o f t h e dominated convergence theorem, it t h e n f o l l o w s t h a t
( * ) The a p p l i c a t i o n o f Theorem 8.3, time i n t e r v a l .
(‘E**)
Chapter 2 , r e q u i r e s us t o work on a f i n i t e
I n t h e i n t e g r a l we have chosen a r e p r e s e n t a t i v e o f t h e equivalence c l a s s
9.
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3)
307
( V representatives of the class 9). The general case is derived from the posi(9' and y-.
tive case by decomposing 'p into Then we can let T
+
= in (3.15); on the basis of the above arguments, and since
thus
(3.1 6) Furthermore if x ; l ( x ) denotes the characteristic function of C , we have x^(x)(Au-f) = 0 a.e. Rn;
C
thus
Moreover, for s < Ox A
T ~ , we
have x^(y(s)) = 1; thus C
Taking into account (3.17) and also the fact that
we obtain u(x) =
Jx(ex)
which together with (3.16) proves the theorem. Remark 3.1.
The optimal stopping time 6
has a particularly interesting
structure. In fact at eacn instant, it is possible to ascertain whether it is necessary to stop or not, solely by considering the present state of the system We continue if y ( s ) , without making use of past information y ( h ) , h < s . This is why is termed the continuation set. y ( s ) E C; we stop if y(s) { E . In the case of stochastic control (where the control variable is a "function"
e
308
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP
.
3)
and not a stopping t i m e ) we can, a t each i n s t a n t , s e l e c t t h e optimal s t r a t e g y without using past information; t h i s i s w e l l known, see FLEMING-RISHEL 113; we have j u s t seen t h a t this result extends to stopping times. Another explanation
of t h i s property w i l l a r i s e out of t h e f a c t , proved l a t e r , t h a t optimal stoppingtime problems a r e limiting cases of c o n t r o l problems using a "function" v a r i a b l e . The above i s t r u e for t h e case where all t h e information on t h e present s t a t e of t h e system i s a v a i l a b l e . We s h a l l s e e i n Section 4 how t h i s r e s u l t i s modi f i e d i n t h e case o f partial information.
3.3
Non-homogeneous problems
Suppose we have h function 0, i . e .
(3.18)
h
Suppose t h a t
Co(r).
,
I
We now consider
(3.1 9) 9) (3.1
E
(*)
where
P E
h
i s t h e t r a c e of a r e g u l a r
k?'P(a) , h
l 4 ~ / ~
.
u , a s o l u t i o n of t h e non-homogeneous i n e q u a l i t y
Wqqaa)) uu EE W (u-+)(Au-f) == 00 aa. .ee. . UUSS4J+ ,, AAuuSS ff ,, (u-+)(Au-f) uIr
I
.
h
I f we p u t (3.20)
u - o = w
t h i s then reduces t o t h e homogeneous case
wl4J-P
,
A w S f -A@
(W-$+@)(AW-f+AP)
w/r
0
= 0
,
,
7
a problem which c l e a r l y has one and only one s o l u t i o n . Thus, t h e r e e x i s t s a s o l u t i o n of ( 3 . 1 9 ) . A proof analogous t o t h a t of Theorem 3.1 l e a d s t o $/,
THEOREM 3.2. Under the assumptions of Theorem 3 . 1 , wi t h t h e asswnption 2 0 replaced by (3.18), we obtain t h e same resuZts a s i n Theorem 3 . 1 ,
taking f o r Jx (8) the functional (3.21)
We s h a l l now g i v e a v a r i a n t o f t h e preceding r e s u l t .
We suppose t h a t @ i s a
( * ) Following LIONS-MAGENES c11, we s h a l l d e s i g n a t e a s "non-homogeneous" problems t h o s e V . I . problems i n which t h e boundary c o n d i t i o n s i n t r o d u c e non-zero
data values.
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
309
We denote by 8’ t h e r i n g and t h a t i t s f r o n t i e r i s s p l i t i n t o two p a r t s r 1, r2. rl and @“= 6 u 19‘u rl. We term :T t h e e n t r y
open domain contained w i t h i n
-
time i n t o 9’ and :T t h e e x i t time
.
from
Naturally,
Suppose t h a t h = hl on (3.22)
.
1 2 1 2 = min(zx,Tx) = z A z x x
z
=
hl
Q/r
rl,
and h = h
ha =
t
1
Q/r
2
t
2
on Q
E
r2,
where
2”
The f u n c t i o n a l J ( 0 ) i s w r i t t e n as (3.23)
J,(e)
= E
jenrx
f(yx(t))exp
[-
0
8
h S
1;
C/r
*
ao(yX(s))ds]dt
- j: a o ( y x ( s ) ) d s 1 ao(yx(s))ds + E hl ( Y x ( z1x ) ) x z t z 2 A e exp P x + E h 2 ( ~ x ( ~ ~ ) ) ~ z 2 s , lexp Ae ao(Yx(s))ds . x x Note t h a t t h e event = is-impossible . x x +
E
c(YX(e))xe<,+;
T’
3.4
Extensions I .
exp
T~
Weakening of assumptions concerning t h e c o e f f i c i e n t s
A f i r s t s t r a i g h t f o r w a r d extension i s a s follows.
(3.24)
f
:l l$
E LP(0)
>+ .
,p
We r e p l a c e ( 3 . 3 ) by
The f u n c t i o n a l (3.21) remains meaningful s i n c e t h e f i r s t i n t e g r a l i n t h e right-hand s i d e of (3.21) i s independent of t h e r e p r e s e n t a t i v e of t h e c l a s s chosen(*). Furthermore (3.19) remains v a l i d . The function u Sobolev‘s Theorem. 8 shown.
1
f
(s),
does not belong t o C b u t remains i n Co(8),according t o The proof o f Theorem 3.2 remains v a l i d , a s can r e a d i l y be
A second extension r e l a t e s t o t h e assumption ( 3 . 2 ) concerning t h e c o e f f i c i e n t s . We r e p l a c e ( 3 . 2 ) by
(3.25)
a a
ij
E
C1(6),
a . and a
measurable and bounded,
2 0
aiJ = a J i ;
a. ij
6 6
i j i j
2 c1 Z
Sj2 VCl,
...,En.
j
The weakening ( 3 . 2 5 ) has no e f f e c t whatever on ( 3 . 1 9 ) . On t h e o t h e r hand, it i s c l e a r t h a t we can no longer pose 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 n We s h a l l use t h e weak formulation described i n Chapter 2 , Section s t r o n g form. 4. We n o t e t h a t a ( x ) remains L i p s c h i t z continuous s o t h a t for (fi,a,P,’st,w(t)) (*)
See proof of Theorem 3 . 1
OPTIMAL STOPPING PROBLEMS & V . I . ' s
310
(3.26)
dy = o ( y ( t ) ) d w ( t ) Y(0) = x
(3.29)
dy(t) = g(y(t))dt Y(0)
(*)
(**)
We t a k e 3" =
+
o(y(t))dfl(t)
=x
a ,to
simplify slightly.
I n o r d e r t o a l l o w t h e lemma t o be s t a t e d i n a g e n e r a l manner.
(CHAP. 3 )
(SEC. 3 )
and
0
5
OPTIMAL STOPPING : STATIONARY CASE
8 5 8'
9
T fmo stopping times,
8;
311
= Min (8',8+h); we have
The constants do not depend upon R.
Proof. ications.
We use again the proof of Lemma 8.1, Chapter 2 , with certain modifIf (for a representative of the class $ ) we put
we observe that Let
% = BR x
5 is 5T measurable, and thus 2 = T . [O,TL.
We can again assume that J, c . 8 ( Q R ) . J, t 0.
(see Lemma 8.1, Chapter 2)
and
But
and therefore since
(*)
a
2 '
is bounded, we see that (E C T ) I S C
We use C to denote several constants.
.
We have
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & V . I . ' s
312
Consequently
if s
have
E" Xh s
7
C
4
which proves ( 3 . 3 3 ) .
'"ll
2
h we
We s h a l l now prove ( 3 . 3 4 ) .
'
We can f i n d a > 0 such t h a t l + a > 2 + 1 ( * ) . We put f3 = 1t a and
= y.
Following HBlder's formula, we have
2 By an argument i d e n t i c a l t o t h a t used f o r t h e c a l c u l a t i o n of E ST, we can show t h a t E $'S
C ; thus
A s y > ;+l, 2
we see t h a t t h e i n t e g r a l i s convergent.
we then deduce (3.34).
By making
h
tend t o 0,
8
From t h e preceding lemma, we can d e r i v e a formula due t o I t o , analogous t o Theorem 8.3, Chapter 2. I n order t o s t a t e a r e s u l t which i s a l s o o f use i n t h e nonstationary c a s e , we consider A ( t ) , as i n ( 3 . 1 ) , where t h e c o e f f i c i e n t s may depend on time t E C0,Tl and s a t i s f y
(3.36)
aij
aji
;
fjaidSiCj 1
a
;E j , 2
v 5 ,,...,5 E
R, a
>
0
a . . bounded, ai measurable and bounded, a measurable and bounded. 1J
(*)
=
a
2 0 .
We s l i g h t l y modify t h e proof h e r e a f t e r i f p > n/2.
313
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
We have THEOREM 3.3 We adopt the asswnptions ( 3 . 3 5 ) , ( 3 . 3 6 ) . Let t h e re be 0 (x,t) - A(t) 0 E L2 (O,T;Lfoc(Rn)); then we beZonging t o Co (Rn x C 0 , T l ) such t h a t have ( i n the notation o f Lema 3 . 1 )
We put
r
We can thus apply the usual Ito formula to On(x,t)z where On is a reguiar approximation of 0, and to the process (y(t),z(t)). Using Lemma 3.1, and arguing as in the proof o f Theorem 8 . 3 , Chapter 2, w e obtain the result sought. We can now state the optimal stopping time problem relating to ( 3 . 2 9 ) . all stopping times 8 we put
(3.39)
We then have
We adopt the asswnptions ( 3 . 2 5 ) , ( 3 . 2 4 ) , ( 3 . 1 8 ) and
THEOREM 3.4. (3.40)
+
E W”P(0)
,p
>+ .
For
I
314
OPTIMAL STOPPING PROBLEMS & V.I.'s
Then the solution
(3 . 4 1 )
(CHAP.
3)
of ( 3 . 1 9 ) i s given by
u
u ( x ) = Inf 9
Jx(e), y t o (3.29).
where J x ( 9 ) is given by ( 3 . 3 9 ) , E corresponding t o (3.28) and Moreover, there e x i s t s an optimal stopping time 6
defined by ( 3 . 1 4 ) .
We w r i t e BT = 9 A T , f o r T a r b i t r a r y . Proof. Let 8 be a stopping time. We can apply formula ( 3 . 3 8 ) with O ( x , t ) = u ( x ) , 8 = 0 , 8 ' = B T and TR = T (*I; thus:
i s a.s. P f i n i t e ( s e e Theorem 7.1, Chapter 2 ) .
We know t h a t T
n e c e s s a r i l y follow from t h i s t h a t T
It does not
i s a . s . P f i n i t e , because P i s not absol-
utely continuous with r e s p e c t t o P. However, we can confirm t h i s d i r e c t l y using an argument analogous t o t h a t used i n Theorem 7 . 1 , Chapter 2. We consider x 1 0 , the function w(x) = 2 - exp - ylx-xo[ and we have ( A
+ A )w(x)
0
1
2
We can apply (3.42) with
w(x) =
-
thus
:1
C > 0 a.e. x [il
s"
€ 0 ,where .A + A1 = A
= 0 and 8 = T .
i n p l a c e of u and a
b(y(s)>ds
-
a
. We o b t a i n
+ s" U(Y(TAT~))
C E TArx Iw(x) C 2 so t h a t , l e t t i n g T t e n d t o
+
we have
m,
We n o t e i n a d d i t i o n t h a t i f
(3.43)
'p
I'p(y,(t)
E $(O),p
Idt
< +
-
T~
>
<
+
m
+
2 then we nave
and hence
T
i s a.s.
P finite.
.
I n f a c t we can show, a s with Theorem 3.1, t h a t
i s a s o l u t i o n of
1 I
(Ao+ A , ) w = w/r
'p
= 0
.
( * ) We can r e p l a c e e R by Qwhich i s bounded i n t h e statement of Lemma 3 . 1 and of Theorem 3.3.
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3)
We conclude the proof as for Theorem 3.1.
3.5.
315
8
Extension 11. Weakening of the assumptions concerning interpretation of the 'penalised'problem.
We are now going to weaken the assumptions concerning $ and 0 . that
(3.44)
(1, f C 0 ( W
We put h =
$
and @ and We assume
I
We consider u to be the unique solution of
(3.45)
where, with regard to f and the coefficients in the form a(u,v), we have made the assumptions (3.24), (3.25).
Co(a).
We saw in Section 1.9 that the solution u of (3.46) belongs to We shall now give a probabilistic proof of this result and interpret u as the solution of an optimal stopping time problem. We observe at the outset that 'ihe functional (3.39) is meaningful under the assumptions mentioned above. We then have
THEOREM 3.5. We make t h e assumptions (3.24), (3.25) and ( 3 . 4 4 ) , (3.45). Then t h e s o l u t i o n u o f (3.46) belongs t o Co(0). Moreover
(3.47)
u(x) = Inf Jx(e)
e where Jx ( 8 ) i s d e f i n e d by (3.39) Proof.
We consider functions Qk and
-+
@k/r
and qk Let
+
@
and (1,
5 be
k
qk
E
2 C (8)such that
'k/r
in Co (5).
defined from P k , Jik
4s
is u
from a,$.
Then
\
E
W2 "(0). If we
put h = @ k / r and if we define J ( 8 ) using (3.39), with $,h changed to k%X then theorem 3.4 applies, and therefore
Now we have
r
I),hk, ,
316
OPTIMAL STOPPING PROBLEMS & V . I . ' s
Thus, s i n c e f o r 8 <
T
y, x ( 8 )
~
(CHAP. 3)
€ 0 ,we have
and t h e r e f o r e
Since i n a d d i t i o n
\
u, f o r example i n L2 (o),( 3 . 4 7 ) t h e n f o l l o w s , and s i n c e
-+
u i s a uniform l i m i t o f continuous f u n c t i o n s , u
E
co(B).
I
N a t u r a l l y , it follows from t h e preceding proof t h a t i f u ( r e s p . t h e s o l u t i o n of t h e V . I . r e l a t i n g t o $ , f , h (resp. t o $,f,fi), then
Auc +-$uc-+)+ = f Auc +-$uc-+)+ = f
u
=
h
4u4 =
,
h
,
6) denotes
OPTIMAL. STOPPING : STATIONARY CASE
(SEC. 3)
317
Then we have THEOREM 3.6.
Under the assumptions of Theorem 3.5, t h e s o l u t i o n
uE of
the
penalised problem (3.48) s a t i s f i e s
(3.50) where J:(v)
u (x) = Inf J:(v), v
i s given by (3.49).
Moreover, there e x i s t s an optimal control given by
Proof.
The function u
E
Co(i)f,
so that GE is clearly an admissible control.
Furthermore, from Ito's formula (3.38) we have, as for Theorem 3.4,
for any admissible control v.
Thus, using equation (3.48), it then follows that
Suppose furthermore d(x)
1 if uE(x) 2 $ ( x ) I
0 otherwise
then
(uE- $ ) + = d ( U E
and
YEW Taking v =
- $),
= d(y(s)).
in (3.52) we obtain
,
318
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & V.I.'s
and hence t h e r e s u l t . ( * )
I
We have proved, i n Section 1, t h a t u
-t
1
u i n H (8).
We have a l s o given an estimate of t h e d i f f e r e n c e u - U. another t y p e of e s t i m a t e and prove uniform convergencg o f u
We s h a l l now g i v e t o u.
Under t h e asswnptions of Theorem 3.5, we have
THEOREM 3.7.
uE(x) + I n f J x ( e ) i n c0(a.
(3.53)
e
Thus u ( x ) = I n f Jx(e) e
E
I f we put
Co(8).
(3.54)
then
ix
-
i s an optimal stopping time for
v x c u , $: Proof.
gx
We put w(x) = I n f Jx(e)
e
Jx(8)
and
a.s.
.
We observe t h a t
We have, applying I t o ' s formula,
Thus
( * ) Since t h e information on t h e c u r r e n t s t a t e i s complete, t h e optimal c o n t r o l Since t h e p e n a l i s e d problem approximates t h e depends only on t h e c u r r e n t s t a t e . stopping time problem, we can now see why t h e optimal stopping time depends only on t h e c u r r e n t s t a t e ( s e e Remark 3 . 1 )
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
But f o r
s
<
ii A
, we
T~
uE(x) =
Jx(ez),
u
w(x).
have
uE(y(s))
<
JI(y(8))
319
, so
that
and t h e r e f o r e (3.55)
(XI 2
Now, l e t e be a r b i t r a r y .
v
p =
We a s s o c i a t e with it
O i f s < e 1 i f s 2 8.
-
We now e v a l u a t e t h e d i f f e r e n c e J:(ve)
i.e. 111 5
(3.56)
I1
E
i[
E
i z , - [4J(Y
(8)
1 4 Y ( eA7) 1 3 (exp
TA( Wr+6 ) -+(y(s))(exp
, enr
We have
J (0).
-
j:
aodh
-
- exp
a dh)exp-
* &BAT
dh)expYdds
ds]
+
320
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.’s
+ E ” J‘
~A(Bhz+6)
We have
1
‘
-+(y(s))(exp
-
5”
a.
ddexp
-- s-ehs
ds
OPTIMAL STOPPING
(SEC. 3 )
321
: STATIONARY CASE
We put
Certain precautions need t o be t a k e n , s i n c e T-6 i s not a stopping time. have
We
C .C-&S.c
7-&&7
and i f Nk = T
(k+6Im k
S C T - .
m
Therefore
Choosing
T =
1 61/2, m=+,
~ ’ / l o6
k = 6 , h
we o b t a i n (3.57)1.
Consequently
113 < C[ 6”‘
6 + exp
+ p(6’”’)+
--166
,
Likewise
.
111, 15 C [ S 1 l 2 + p(6l’”)l
C o l l e c t i n g t h e r e s u l t s t o g e t h e r , we o b t a i n
s o t h a t we have
(3.58)
lut(x)
- w(x)I s C[a1/2
a
+
p(6l/lo)
+ e q -++
c
.
322
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP. 3 )
By making f i r s t E, t h e n 6 , t e n d t o z e r o , ( 3 . 5 3 ) can t h e n be d e r i v e d s t r a i g h t forwardly.
As u
-+
u i n H1, we c l e a r l y have u = w and u
E
Co(8).
Furthermore, we have
.
ex s e*x *E
(3.59)
hence ( 3 . 5 9 ) . If that
ex = 0 , it follows t h a t i z = 0 and t h u s ex > 0 and l e t t h e r e be 6 such t h a t ex >
t h e convergence i s e v i d e n t .
Suppose
6.
Then we have
<
u(y(s))
W(Y(S))
v
8
E r0,"-
61
We n o t e t h a t we may s t i l l assume t h a t u +. u uniformly i n R", which t o g e t h e r w i t h t h e c o n t i n u i t y of u and J1 as w e l l a6 t h e a . s . c o n t i n u i t y of y , allows us t o assert that
<
uc(y(s)) .
and t h u s
e:
.
I
2
ex .
e:
+(y(s))
6 , for e .
S
v
e0(x,6;w)
E r 0 , ~ ~63 -
.
We have c l e a r l y proved t h a t
A
+ .
ex
a.s
We now prove t h a t 6 i s - o p t i m a l . If u(x) = $(x) (which can only occur on r a t p o i n t s where h = $ ) , d e n Ox = 0 a . s . which i s c l e a r l y o p t i m a l . We may t h e r e f o r e suppose t h a t u(x) < $(x). Let 6 > 0 be such t h a t Then bx > 0 a . s . U(X)
<
-6
+(XI
and
e: = i d thus
Let c 6 be such t h a t
16
2 olu(Y(s)) 2 +(Y(a))
-9
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3)
323
then ( n o t i n g t h a t c6 i s not random and t h a t 0' i s a stopping time independent of we have
E S cd
and
ii2 e:
A Tx
8
E [O,~:AT~]
. )
u
S
6
u +-<
4-
J,
6
and t h e r e f o r e
We l e t
E
+
,
0 , and by v i r t u e o f t h e uniform convergence we o b t a i n
Taking account again o f t h e uniform convergence, we have A
which means t h a t
6
- - +2 - 4
E)
OPTIMAL STOPPING PROBLEMS & V.I.'s
324
u(x) =
(CHAP. 3)
Jx(ex)
and completes the proof.
8
Direct proof of the continuity of inf J x ( e ) with respect t o x. We have shown amongst other things in Theorems 3.5 and 3.7 that the functional w(x) = Inf Jx(e)
e
belongs to Co(8). The proofs given are not direct, being based either on a regularisation procedure, or on a penalisation procedure. One might ask if a direct proof is possible. The answer is affirmative, though at the price of stronger regularity assumptions.
We assume (3.25) and additionally
(3.60)
gj
cO(U)
(3.61 )
a.
E ~'(6)
(3.62)
f
E
cO(U)
Then we shall show d i r e c t l y that u(x) = Inf J x ( 6 ) i s continuous on d (*).
...,(3.63) the functional Jx(e) is well defined.
Under the assumptions (3.60), Letting x
+
x
, we
use the notation
( * ) We shall also make a more precise assumption on the regularity of 8. (see Lemma 3.2).
(SEC. 3 )
OPTIMAL STOPPING
:
STATIONARY CASE
2
- erp -
325
We have
We again consider the function w(x)
so that
t
lxJoI;
thus
326
OPTIMAL STOPPING PROBLEMS & V.I.’s
(CHAP. 3 )
Furthermore
where C i s independent o f xn, x, 8 , T.
- Y(t)
Y,(t)
= xn
-x +
Then we have
1:
(u(yn(s))
- a(y(s)))dw(s)
so t h a t it can e a s i l y be deduced, s i n c e u i s L i p s c h i t z continuous, t h a t
We s h a l l prove i n Lemma 3.2 t h a t using a more p r e c i s e assumption on t h e r e g u l a r i t y of t h e open domain 8 ,we a l s o have
(3.66)
z
-
z
a.e. (P)
(SEC. 3 )
We may assume t h a t
OPTIMAL STOPPING : STATIONARY CASE
g
i s uniformly continuous on R”.
We d e f i n e
I E 5 ’ Is 6 so t h a t
and by Gronwall‘s i n e q u a l i t y we o b t a i n
327
OPTIMAL STOPPING PROBLEMS
328
&
(CHAP. 3 )
V.I.’S
We t h e n p u t
We cannot o b t a i n f o r I1 a n e s t i m a t e which i s uniform w i t h r e s p e c t t o 8 . We t a k e 6 > 0 and V6, a c l o s e d n e i g h b o w h o o d of 2 0 such t h a t i f 5 5 6 t h e n f E V6 , and we l e t
d(5,aO )
O6 =
{c
E O\d(E,,a)
>
6) = 0
- Vg .
We w r i t e T~
= i d Is 2 01y(s) ,l 06]
.
With any s t o p p i n g t i m e ‘2 we a s s o c i a t e
We have
which a g a i n d e f i n e s a s t o p p i n g t i m e .
os
T A
e6
- T A es T
T
6
’
If i n a d d i t i o n we i n t r o d u c e t h e boundary problem
(Ao + A l l ” = 1
dr = whose s o l u t i o n z
E
,
W2”((P), and hence i n p a r t i c u l a r
Iz(E,)-z(c1)
we have
0
t
1s
C l W ’ IY
v E,,S’ E d
,
E
8 and
OPTIMAL STOPPING
(SEC. 3)
We now evaluate the difference Jx(e)
and from (3.69)
Jx(e)-Jx(e6)
and
2
- CbY
:
STATIONARY CASE
-
Jx(eG).
We have
329
330
OPTIMAL STOPPING PROBLEMS & V.I.’s
and t h u s
(3.70)
Jx(e)-Jx(06) 2
+
C[-aY-
4 +
Y
A (6 7) - h3(6 ) - 6 ]
4
where
A4(6)
SUP
15-5’ k6
(CHAP. 3 )
-
1+(5)- + ( C t ) I
We r e t u r n , t h e n , t o t h e estimation o f 11, and we work with B 6 i n s t e a d o f 8 . note t h a t i f e 6 # + a, then e 6 < T ~ . Moreover
(3.71)
if
sup lyn(s)-y(s)I I 05s ST
6 2
then
(3.72)
But i f ( 3 . 7 1 ) i s s a t i s f i e d then we must have
n 1 76
7
We
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
331
(3.73) and
(3.74)
We have t h u s f i n a l l y proved t h a t
F i r s t we l e t n h we have
-+
-
and we n o t e t h a t by v i r t u e o f ( 3 . 6 6 ) and t h e c o n t i n u i t y o f
so that
S u b s e q u e n t l y we l e t 6 t e n d t o 0 , and t h u s
l i m sup w ( x n ) 5 w(x) and f i n a l l y , l e t t i n g T +
(3.76)
+
a,
+CT
we o b t a i n
lim sup w(xn) s
W(X)
.
We now s e e k t o o b t a i n a n e s t i m a t e i n t h e o p p o s i t e d i r e c t i o n . We d e f i n e -rn r e l a t i v e t o yn t o g e t h e r w i t h 8; and w i t h y,. 6We a g a i n have
a s s o c i a t e d w i t h each s t o p p i n g t i m e 9
332
OPTLWL STOPPING PROBLEMS & V.I.
and s i n c e f , g , o a r e bounded
Jx (8)
n
- Jx
Y
(0;)
n
2
C[-
(CHAP. 3 )
's
Y
Y
t - A3(6 T -i6 ]
h4(6 )
6'-
.
The e s t i m a t e f o r 111 i s s t i l l v a l i d , s i n c e it was uniform w i t h r e s p e c t t o t h e and h a r e bounded s t o p p i n g time. The e s t i m a t e f o r 11111 remains v a l i d s i n c e and s i n c e z n ( T ) - z ( T ) i s independent o f t h e s t o p p i n g t i m e . I n p l a c e o f (3.71), we have
+
(3.77)
sup
0 ss ST
- Y(S) I s -62
I+)
. )
n 1z
7
6
.
If t h e i n e q u a l i t y on t h e l e f t - h a n d s i d e o f ( 3 . 7 7 ) is s a t i s f i e d , t h e n
so t h a t I1 remains v a l i d , as does I13. Therefore (3.75) remains v a l i d w i t h 8; i n p l a c e o$ e 6 , so t h a t
and t h u s dXn)
5 w(x)
- c[ 3 - TC -CTi
1
from which it can r e a d i l y be deduced t h a t
lim i d w(xn) 2
W(X)
which, t o g e t h e r w i t h ( 3 . 7 6 ) , completes t h e p r o o f .
It now remains t o prove t h e r e s u l t ( 3 . 6 6 ) .
LEMMA 3.2 (3.78)
1
I
We have:
Suppose t h a t t h e open domain 6 is d e f i n e d by 0
=
{xiQ(x)
<
01
Q
where
E C2(Rn)
,
t h e n p r o p e r t y ( 3 . 6 6 ) follows. Proof.
Let C be t h e complement o f
6 and
E
t h e i n t e r i o r o f Z.
We p u t
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
Ax = inf ( t 2 z x / y x ( t ) E
t
)
333
.
We s h a l l now prove t h a t , by v i r t u e o f t h e assumption ( 3 . 7 8 ) we have
zx a . s .
Ax
(3.79)
-
We s t a r t by assuming t h a t ( 3 . 7 9 ) i s t r u e and we prove t h a t t h i s must imply (3.66) a.s. We d i s t i n g u i s h two cases. We know t h a t T~ < 1 s t case:
= 0.
T
From ( 3 . 7 9 ) , Ax = 0, t h u s V
E
>
E
!.
i s a compact subset o f
8.
such t h a t 0 < t E 5
0, 3 t
E
and yx(t,)
But then 3 N such t h a t
n> N ThUS T n
-t
= yx (t,,) E
Yn(tc)
0
n
zns
c
.
0.
2nd case:
> 0.
T
which
Let 0 < E < T~ andK = { y ( t ) , t E [O,zxLet d be such that
c E K E , l c q l < d ; m c E b . There e x i s t s N
such t h a t for n 2 N E , we have
- y(t)I
IYn(t) and t h e r e f o r e for t (3.80)
E
<
dE
9
V t E [O,z]
we have y n ( t )
[O,T-E]
E
8 ,which
implies
.
n 2 N E e z 2 ~ - E
n
Moreover, we have
IT,
and t h u s t h e r e e x i s t s t
E
e x i s t s a l s o N1
for n 2 N
such tha: n z N~
E
4
T+E]
1
such t h a t y ( t E ) E we have y n ( t )
z 5 t IZ+E
which with (3.80) shows t h a t
~
T~
E.
1,
Consequently, t h e r e so t h a t
E
+. T .
I t remains f o r us t o prove ( 3 . 7 9 ) .
(3.81)
E
We s h a l l i n f a c t prove t h a t
T A A = T A zx V T a . s .
Let 6o < 6 , where 6 i s defined i n assumption ( 3 . 7 8 ) .
We put
334
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
a@
which i s m e a n i n g f u l , s i n c e
,(y(t))
d@
\ \ G ( Y ( T ) ) ~ ~2 6
i s c o n t i n u o u s and
(since
Y(T)
aB ) .
E
We t h e n p u t
A T
T ' = T A T , A ' = A A z 60
which a r e a l s o s t o p p i n g t i m e s .
B(t) = and
t
For X > 0 we p u t
.
x,,(t)(l-X,,(t))@(y(t))~(y(t))
We now show t h a t I B ( t ) l is u n i f o r m l y bounded. I n f a c t , i f 6 # 0 , t h e n A ' > T' T ' = T n e c e s s a r i l y , t h u s [ T ' , A ' I c [T,AI, which i m p l i e s t h a t y ( t ) E 5 f o r
E
[T',A']
/ B ( t ) / 2 C since
and t h e r e f o r e
E exP [
50'
1 -7
B(t).dw(t)
8.
a r e bounded on
u,
le(t)l2dt]
Consequently,
= 1
i.e.
(3.82) But from I t o ' s formula we have
If A' > T ' , we have s e e n t h a t T' = T and y(t) P ( Y ( T ' ) ) = 0 and @ ( y ( A ' ) )S 0 , it f o l l o w s t h a t
I n addition, s t i l l with A ' >
drn
l\~(Y(t))l2 \ b0
T',
.
E b V t E [T',A'],
t E [*',A']
we have
Consequently, it f o l l o w s t h a t
which, w i t h ( 3 . 8 2 ) , i m p l i e s
E exp(hM
If no
= {tu[A' >
TI),
- h$l
ao)(A'-
21)
2 1
it t h e n f o l l o w s t h a t
.
. )
t
E[7,'E
so s i n c e
]
6
and t h e r e f o r e
OPTIMML STOPPING : STATIONARY CASE
(SEC. 3 )
Letting A
-f
-, it
335
can t h e n be deduced t h a t
P(f2-no) = 1 which proves t h a t
A' = Now
T' a s s .
> 1 2 T A T , t h u s a l s o proving
T&
(3.81).
a
0
Remark 3.2.
I n Theorem 3.7 we have given an e s t i m a t e for uc - u ( s e e ( 3 . 5 8 ) ) .
We can g i v e a b e t t e r e s t i m a t e i n t h e c a s e where $ is r e g u l a r . b e s i d e s ( 3 . 4 4 ) we have A$
E
Lp(o)and
Thus suppose t h a t
h 2 jl/ r '
Putting
5 = u
-+
and
EE ,=
UE
-$
-
If we a g a i n t a k e
?(ye)
S
i
ehr
and t h e r e f o r e Ib,(X)
with
l + a
>
-
* f -.A+ , h h dr and JI - Jx(e), i n an obvious n o t a t i o n , we o b t a i n
we a r e l e d t o an e q u i v a l e n t problem where f
- u(x)j
(if p =
-,
( * ) We can a l s o e s t i m a t e u
(f-&)
-
dh exp
0.
a-BAT -e de
8
s CJlf-A+JJ
El-
LP
we can t a k e a = - )
-
exp
+
(*).
m
u i n t h e g e n e r a l c a s e , beginning by r e g u l a r i s i n g $,
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
336
3.6
A d d i t i o n a l weakening o f t h e r e g u l a r i t y assumptions on
E x t e n s i o n 111. $ and 0
We a r e now g o i n g t o f u r t h e r weaken t h e a s s u m p t i o n s on $ and 0 .
I n p l a c e of
( 3 . 4 4 ) , ( 3 . 4 5 ) , we suppose
II
(3.83)
I
J,
measurable and bounded,
t h e r e e x i s t s a sequence and
,
5 C
J,j -. J,
J,j €
P € Co(6), A@ 6 L p ( 0 ) , p
Co(6),
Jlj/r
a t each p o in t of 0
2 h
P
>
4+ 1
P/r
, Gj 2
J,
on
0
;
.
The problem ( 3 . 4 6 ) s t i l l has one and o n l y one s o l u t i o n u, b u t t h e a d d i t i o n a l p r o p e r t y u E c O ( & i s no l o n g e r t r u e . F u r t h e r m o r e , t h e f u n c t i o n a l ( 3 . 3 9 ) remains m e a n i n g f u l . THEOREM 3.8.
(3.84)
We s h a l l now prove
For asswnptions ( 3 . 2 4 ) , ( 3 . 2 5 ) and ( 3 . 8 3 ) , we again have u(X)
J,(e) ,
= Inf
e
a.e. x
where Jx(e) i s defined by ( 3 . 3 9 ) and where
u i s a so l u t i o n of ( 3 . 4 6 ) .
Proof. Let u . ( x ) b e t h e f u n c t i o n c o r r e s p o n d i n g t o I$.. c o u r s e , a p p l i c a b l d , so t h a t J
,
u ( x ) = ~nf?(el
e
j
Theorem 3 . 7 i s , o f
where J J ( e ) i s d e f i n e d by ( 3 . 3 9 ) w i t h $ . i n p l a c e . o f $. We n o t e t h a t t h e f u n c t i o z a l s JJ o r J o n l y b r i n g i n t h e v d l u e s of $J o r JI a t p o i n t s i n t e r i o r t o 8. S i n c e J, b $ on 8 , we s e e t h a t
j
e
.
P r o v i s i o n a l l y we p u t
w(x) = Inf Jx(e)
e
so t h a t
u . ( x ) 2 w(x) J
.
Furthermore j
(x) S J i ( 6 )
and t h e $J converge p o i n t w i s e on P t o $ , and b e i n g bounded, it i s c l e a r t h a t
so t h a t
-
Jx(e)
lim sup u (x) 5 j
Jx(e1
(SEC. 3 )
337
OPTIMAL STOPPING : STATIONARY CASE
i.e.
lim sup u.(x) Iw(x) J and t h u s
Furthermore u
j
i s a solution of
P u t t i n g u . - @ = z . , we s e e t h a t z . i s a s o l u t i o n of J J J
F u r t h e r m o r e , s i n c e t h e sequence $ . i s bounded by a c o n s t a n t J remembering t h a t
a(v,v) it t h e n f o l l o w s t h a t
+ h/vl
I/ z j / I
2
2
5 C.
~lIv112 ,
P
>
0
12.
J
I
5 C
and,
v
By e x t r a c t i n g a subsequence and r e a s o n i n g a s
i n t h e p r o o f o f t h e s t a b i l i t y theorem ( s e e S e c t i o n 2 ) , it may e a s i l y be shown t h a t
u j
-t
u i n H1(B and t h u s w =
Remark 3.3.
U.
m
When t h e f u n c t i o n JI i s n o t c o n t i n u o u s , t h e R.V.
8x =
infit \u(y(t)) = d Y ( t ) ) f
at'")
which i s a s t o p p i n g t i m e ( a t l e a s t f o r t h e f a m i l y i s no l o n g e r n e c e s s a r i l y a n o p t i m a l s t o p p i n g t i m e , a s i s shown by t h e f o l l o w i n g counter-example, s u g g e s t e d by DYNKIN,YUSHKEVICH [11. We t a k e
0=]0,1[,
f=o, h = o , g = o
,
aij-a,,=l,
ao-O
.
The problem i s t h e n
- u" 5
0, u
-$
0,
Ufl(U-C)
= 0
,
and u ( o ) = u(1) = 0 . Thus u i s a convex, p i e c e w i s e - l i n e a r f u n c t i o n i f u # $; i f J, i s c o n t i n u o u s and if t h e g r a p h of $ p o s s e s s e s any i s o l a t e d e x t r e m a l p o i n t s , we obtain t h e following picture
OPTIMAL STOPPING PROBLEMS & V.I.’s
338
(CHAP. 3)
F i g u r e 1. where t h e f u n c t i o n Ji i s r e p r e s e n t e d w i t h a full l i n e and u w i t h a d o t t e d l i n e . If we i n c r e a s e t h e v a l u e o f $ ( a ) , Ji = { a ] U (b} The c o i n c i d e n c e s e t becomes d i s c o n t i n u o u s a t a. Then t h e c o i n c i d e n c e s e t r e d u c e s t o {b}. This i s t h e r e f o r e a n unstable s i t u a t i o n . I f we a p p l y t h e s t r a t e g y c o n s i s t i n g of s t o p p i n g o n l y when we r e a c h t h e p o i n t b , we o b t a i n a c o s t which i s g r e a t e r t h a n o r e q u a l Wow u i s c l e a r l y l e s s t h a n $(b) a t c e r t a i n p o i n t s . rn t o Q(b).
.
{.=+I
3.7.
Extension I V .
O p e r a t o r s which a r e n o t i n d i v e r g e n c e form
Synopsis Let us c o n s i d e r t h e o p e r a t o r A d e f i n e d i n ( 3 . 1 ) . To s i m p l i f y s l i g h t l y , we t a k e a. = 0. With t h e n o t a t i o n ( 3 . 9 ) we w r i t e A i n t h e form
I n t h e c a s e where 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 , we a s s o c i a t e w i t h t h e o p e r a t o r A a n I t o e q u a t i o n ( i n t h e s t r o n g s e n s e , t o make t h e i d e a s d e f i n i t e ) d? = g(y”)dt $40)
+
,
&)dw
= x
and i f 6 i s a bounded open domain o f R n , we d e n o t e by y ( t ) = ? ( tA t h e process_ s t o p p e d on e x i t from 8 . values i n 6 . We now c o n s i d e r t h e V . I .
(A++a)us f,
(a t
us
The p r o c e s s y ( t ) i s a Markov p r o c e s s w i t h
0 )
+ ,
To a v o i d w r i t i n g t h e n u l l p r o d u c t , we may s t a t e : t h e s e t of w ( * ) s a t i s f y i n g
(*)
u
i s t h e maximum element i n
S i n c e t h e arguments p r e s e n t e d i n t h i s s y n o p s i s a r e o n l y i n t u i t i v e , we s h a l l not define t h e functional spaces.
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
339
However, ( i )l e a d s t o
o r w r i t i n g t h i s a n o t h e r way
I f , f o r 'p measurable and bounded on @(t)'p
we can s a y t h a t
u
(v 1
W S +
(vi)
w S
b , we
put
= ET(Y(t))
i s t h e maximum element o f t h e s e t o f
{:
, w/,=o e-at
@(t)f
b
dt
+
@(h)w
w
satisfying
.
Now, ( i n a s e n s e which needs t o be d e f i n e d ) A a p p e a r s as t h e i n v e r s e o f t h e Moreover, on t h e b a s i s o f t h e i n f i n i t e s i m a l g e n e r a t o r o f t h e semi-group O ( t ) . t h e o r y o f STROOCK-VflRN)€iAN(mentioned i n Chapter 2 , S e c t i o n 4), O ( t ) i s w e l l d e f i n e d w i t h c o e f f i c i e n t s a i j l g . which a r e merely continuous. But t h e n t h e o p e r a t o r A can no l o n g e r be w r i t t e n i n a i v e r g e n c e form ( t h e c o e f f i c i e n t s would t h e n b e d i s t r i b u t i o n s ) and t h e r e f o r e t h e v a r i a t i o n a l f o r m u l a t i o n which we have c o n s i d e r e d up t o t h i s p o i n t i s no l o n g e r a p p l i c a b l e . On t h e o t h e r hand t h e "maximum element o f t h e s e t o f w s a t i s f y i n g (v), ( v i ) " f o r m u l a t i o n s t i l l remains m e a n i n g f u l , a s does t h e o p t i m a l s t o p p i n g t i m e problem ( * ) . Our o b j e c t i v e i s t o p r o v e t h a t t h e s e t o f w ( i n a f u n c t i o n a l space t o b e d e f i n e d ) s a t i s f y i n g (v), ( v i ) p o s s e s s e s amuximwn element u ( t h i s i s no l o n g e r o b v i o u s , a s we can no l o n g e r c a l l on v a r i a t i o n a l t h e o r y ) and t o p r o v e t h a t under s u i t a b l e assumptions u a l l o w s u s t o s o l v e t h e o p t i m a l s t o p p i n g t i m e problem. Although, a s we have i n d i c a t e d , t h e problem f o r m u l a t i o n i s v e r y similar t o t h a t o f DYNKIN and GRIGELIONIS-SHIRYAEV, o u r approach i s f u n d a m e n t a l l y d i f f e r e n t ( * * ) We s h a l l f o l l o w , and a d a p t , t h e t e c h n i q u e s o f t h e v a r i a t i o n a l c a s e ; i n particular, we s h a l l i n t r o d u c e a penalised problem.
.
We s h a l l employ a g e n e r a l f o r m u l a t i o n , our p o i n t o f d e p a r t u r e b e i n g a n " a c c e p t a b l e " Markov semi-group. I n t h i s s e c t i o n we c o n s i d e r two examples: d i f f u s i o n i n ( * ) T h i s f o r m u l a t i o n i s v e r y s i m i l a r t o t h a t i n t r o d u c e d o r i g i n a l l y by D Y N K I N [ 2 1 and a d o p t e d by GRIGELIONIS-SHIRYAEV El1 t o s t u d y o p t i m a l s t o p p i n g t i m e problems. We s h a l l r e t u r n t o t h i s t o p i c i n Remark 3.9. (**)DYNKIN and GRIGELIONIS-SHIRYAEV f i r s t s o l v e a d i s c r e t e - t i m e problem, t h e n approximate t h e continuous-time problem by a sequence o f d i s c r e t e - t i m e problems.
340
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
Rn and d i f f u s i o n stopped on e x i t from a bounded open domain. examples e x i s t ( s e e BENSOUSSAN-LIONS [ e l , R O B I N E l ] ) .
3.7.1. Let
Assumptions
-
P l e n t y of o t h e r
Notation
(E,B) be a t o p o l o g i c a l space endowed w i t h t h e Bore1 o-algebra. C
We put
= C(E) = space o f f u n c t i o n s continuous and bounded on E ,
B = B ( E ) = space of f u n c t i o n s measurable and bounded on E .
For t h e sup norm, B i s a Banach space and C i s a Banach subspace of B. we s h a l l a l s o have t o work w i t h a weak topology on B , d e f i n e d as f o l l o w s :
.
(E,B)
M = space of s c a l a r measures on
In fact let
We p u t B and M i n t o d u a l i t y v i a t h e i n n e r product
p(f)
e
lE
,
f ( x ) dp(x)
if
f
We t h e n d e f i n e a weak topology on B by
f n -. f
p(%)
if
+
v
p(f)
,
E B
P E M
.
The weak convergence i s i n f a c t e q u i v a l e n t ( * ) t o t h e sequence
I[ f n 11
( 1 1 11
i s bounded.
p E
fn(X)
= sup norm).
M
-
.
f(x) V x E E
,
and
Let t + H ( t ) be a mapping w i t h v a l u e s i n B. We s h a l l s a y , a s i n DYNKIN [ll, p , 3 3 ) , t h a t it i s weakly continuous (measurable) i f f o r a l l p E M
t -. p ( H ( t ) ) i s continuous ( m e a s u r a b l e ) . The mapping i s weakly i n t e g r a b l e on an i n t e r v a l l a , b E , i f p ( H ( t ) ) is i n t e g r a b l e and i f , i n a d d i t i o n , t h e r e e x i s t s K E B such t h a t
We t a k e a family t + O ( t ) of mappings E d ( B ; B ) ( B being endowed w i t h t h e topology o f a Banach s p a c e ) s a t i s f y i n g t h e f o l l o w i n g p r o p e r t i e s
(3.86)
8-
b-
if
f
E B, f 1 0, I 1
IIQ(t)Il,p(B;B)
c- O ( t ) l = 1 , t Z 0 d- Q ( o ) = I
(*)
Q ( t ) f 2 0, t Z 0
then
t Z 0
;
;
; Q(t) ~ ( s ) v
e- Q ( t + s )
I
f-O(t):
c-c,
8,
V t l O
See, f o r example, DYNKIN [ll, p. 50.
t
.
z o
;
;
(SEC. 3 )
O P T I W STOPPING
341
STATIONARY CASE
:
Then we s h a l l s a y t h a t @ ( ti)s a Fellerian MarkoV semi-group ( t h e F e l l e r p r o p e r t y b e i n g more p r e c i s e l y t h e p r o p e r t y f ) . With such a semi-group we can a s s o c i a t e a c o n c e p t o f a strong infinitesimal generator and a concept o f a weak infinitesimal generator, t h e l a t t e r b e i n g t h e t h i s i s t h e r e f o r e t h e o n l y one we s h a l l i n t r o d u c e . more u s e f u l for our p u r p o s e s ; We p u t
Bo = {f E BlO(t)f
f w e a k l y , when
4
t
+
01
.
Then we d e f i n e t h e o p e r a t o r A ( i n f a c t t h i s i s t h e inverse o f t h e weak i n f i n i t e -
simal g e n e r a t o r ) by
f- @ ( t ) f t
Af = weak l i m t l 0 and we u s e t h e n o t a t i o n
(BAi s
&A = domain o f A = s e t of f weakly dense i n
Iff E Bo
,t
-
f o r which t h e l i m i t e x i s t s
Bo).
@ ( t ) f i sweakly r i g h t c o n t i n u o u s .
We d e n o t e by t h e t e r m "weak
equation" t h e equation
Af For g formula
E
+
a f = g,
f,.€ Ba
.
Bo t h e weak e q u a t i o n p o s s e s s e s one and only one solution, g i v e n by t h e
We assume t h a t
We n o t e t h e f o l l o w i n g p r o p e r t y , which i s a consequence o f ( 3 . 8 7 ) .
O(t)m(t)dt
m ( t ) E Co([a,b];C) ; t h e n t h e i n t e g r a l
Suppose
i s meaningful and moreover
(3.88) In f a c t @ ( t ) m ( t ) because
E
C for all
O(t+h)m(t+h)
See DYNKIN 111, p.
40.
and t
- @(t)m(t) +
(*)
t
(@(t+h)
+
@ ( t ) m ( t ) i s weakly r i g h t - c o n t i n u o u s
= @(t+h)(m(t+h)
- Q(t))m(t)
- m(t))
342
OPTIMAL STOPPING PROBLEMS & V . I . ' s
(CHAP. 3 )
and t h e f i r s t ( r e s p . second) term tends t o 0 s t r o n g l y ( r e s p . weakly) i n B s i n c e (@(t+h)
- @(t))m(t)
and s i n c e Q ( t ) m ( t ) E C C Bo i f x + x i n E , then
.
Q(t)m(t)(x,)
= (Q(h)-I)@(t)m(t)
The i n t e g r a l s (3.88) a r e meaningful.
+
v
Q(t)m(t)(x)
Further,
t
and from Lebesgue's theorem we have
which c l e a r l y implies ( 3 . 8 8 ) . With such a semi-group, we can a s s o c i a t e ( s e e DYNKIN C11) a Markov t r a n s i t i o n p r o b a b i l i t y defined by
We now assume t h a t E i s a closed subset of a Banach space, and
P
is the
t r a n s i t i o n p r o b a b i l i t y of a continuous s t a t i o n a r y Markov process with values i n process
y(t)
,
E
a family of p r o b a b i l i t i e s
y(t) is
if
3 P, Zt
i.e. on
Px
adapted
3" such t h a t
pX a.s. continuous,
8 5 8 c 6 a r e two stopping times such t h a t 1 2
e,
- e 1 <- A,
a.s.
pX
,
(**)
We s h a l l note t h a t (3.90) implies ( 3 . 8 7 ) , because ence i f m E C , when t -t 0.
(*),e
, an
,
E
with values i n
Q(t)Ic-m
f o r simple converg-
= c h a r a c t e r i s t i c function o f t h e s e t G .
( * * ) This assumption, which is t r u e in t h e case of d i f f u s i o n s , may be l a r g e l y dispensed with ( s e e R O B I N [ll) a t l e a s t i n t h e finite-dimensional case.
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
Suppose we have L , II f o r x E E we put
E
B and l e t 0 be a stopping time with r e s p e c t t o
We s h a l l adopt t h e following assumptions concerning L , II;
Do of B closed for t h e weak topology, such t h a t O ( t )
L E B~ n D~ ; ( v - d + E c n D~
(3.93)
343
e-Bt O(t)Ldt
E C
,
V !3
>
:
,vv 0
.
Do
+
at;
we t a k e a sub-space
Do, and we have
E c n D~
Moreover, we s h a l l need one o r o t h e r of t h e following assumptions:
(3.94)
1
i s uniformly continuous on E
or
, 3 an open subset e o f t h e Banach space i n which E i s imbedded,E,and "yt) i s obtained from a continuous Markov process y ( t ) , by stopping t h e process on e x i t
(3.94;
from
$Ir'
=.(v E B / v l r = 01 (where r = $0 1 JI i s uniformly continuous on E and
0; furthermore Do
and I = 0
%$, where *
Examples Example 1: Let O b e a bounded open domain o f R n , s a t i s f y i n g t h e assumptions ) a continuous f u n c t i o n o n e w i t h values i n Z ( R n ; R n ) , and l e t ( 3 . 7 8 ) ; l e t ~ ( x be g be a measurable and bounded f u n c t i o n from B + Rn (we assume by s u i t a b l e extension t h a t O,g a r e defined on R"). We t a k e
0 = Co(O,=;Rn) t o be a Frdchet space of f u n c t i o n s continuous on CO,-C with values i n #,endowed with t h e topology of uniform convergence on a l l compact s u b s e t s . If
and
w c u(t)
st =
we put
g(t;u)
= w(t)
0-algebra generated by w ( s ) , 0
5 s 5
t.
From t h e theory of Stroock-Varadhan, t h e r e e x i s t s one and only one family Px f o r which t h e following property i s s a t i s f i e d :
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & V . I . ' s
344
w(t)
t h e r e e x i s t s a Wiener p r o c e s s such t h a t
(3.95)
F(t)
Let
T
=
X
a.s.
Px
f:
+
y(t) = Y(t A
a
y(t)
.
f:
+
g(g(s))ds
be t h e e x i t t i m e o f
with respect t o
we have a(S;(s))dw(s), V t 2 0
8 . We d e f i n e
from
7)
i . e . t h e p r o c e s s stopped on e x i t from t h e open domain. We put
at
I
{A E
-t
a ,An
Iz>tI = a
.03
I
which i s c l e a r l y an i n c r e a s i n g sequence i f
it i s ,
and
3"
E
?
.
The s e t (n,$, S t , y ( t ) ) d e f i n e s a Markov p r o c e s s w i t h v a l u e s i n E = denote by P ( x , t , G ) t h e t r a n s i t i o n f u n c t i o n of y ( t ) and
8
.
We
Stroock and Varadhan have proved t h a t t h e family Px depends c o n t i n u o u s l y on x , i n t h e following sense: i f xn + x t h e n Pxn
+
9
.
weakly
Now, a proof very similar t o t h a t f o r Lemma 3.2 shows t h a t t h e f u n c t i o n a l f ( g ( t A r ) ) on C o ( O , - ; R n ) i s Px a .s. continuous i f f i s continuous on 8 and t h e r e f o r e by e x t e n s i o n on R n . Consequently t h e c o n d i t i o n s ( 3 . 8 6 ) a r e c e r t a i n l y satisfied. By c o n s t r u c t i o n ( 3 . 9 0 ) i s s a t i s f i e d .
y
+
t
I f el, 8 a r e two t i m s w i t h r e s p e c t t o 3 t h e n 0 A T , 6 A T a r e two s t o p p i n g 2 1 times w i t h r e s p e c t t o 8' and (3.91) i s immediately s k t i s f i e d a s a consequence of p r o p e r t i e s analogous t o t h o s e of 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 . E
Co(6)
L ( x ) = f %(x),
x E
I n a d d i t i o n , we t a k e f
Then L
E
Bo.
E
6
E
ss
,
b
+Ir
Co(6) t(x)
=
then
8f b ( Y ( t A 7 ) ) T
-
= 8f(g(t))Xt<,
i s Px a . s . > 0 and y ( t )
f(Y(t))xt<7
a.s.
f(x)
It t h e n follows t h a t
O ( t ) L ( x ) -. L(x)
v
x E 0
*
.
5: 0 and we p u t
(J
In fact
Q(t)L = Now i f x
and (J €
+
a
.
x Px a.s . when t
-+
0, thus
(SEC. 3 ) Since f o r x
r,
E
E
Bo.
vt,
we have @ ( t ) L ( x )= 0
@(t)L
2nd t h u s L
34 5
OPTIMAL STOPPING : STATIONARY CASE
-f
t h e n c l e a r l y we have
L weakly i n B ,
We t a k e
D =
{V
E BlvIr = 0 )
which i s c l e a r l y closed for weak convergence i n B. Furthermore
j-;
at
e-b(t)L
1;
= E?
from t h e c o n t i n u i t y o f t h e family x Finally i f v
E
C n
3 0 , then v I r
-t
e-Ptf(g(t))at
E c
E
Bo n Do.
,
Px seen above.
= 0 so t h a t
(v-L)+ = (v- +X ) + =
8
We indeed have L
(v-+)+ E c n D
and t h u s ( 3 . 9 3 ) i s s a t i s f i e d . L a s t l y , (3.94)1 i s s a t i s f i e d by c o n s t r u c t i o n a l s o . The weak equation
A v + a v = L ,
.. E B*
v
i s i n t e r p r e t e d i n t h e form
Av + av = f at a l l p o i n t s of
(3.96)
v I r = 0, v
and t h u s v
E
E
0,
BA.
C.
We s h a l l now prove t h a t we have C2(& then
c
Bn and
t h a t furthermore i f v
The f a c t t h a t Avlr = 0 follows from t h e r e l a t i o n
@ ( t ) v l r = Ex v(g(tA.c)) Now l e t x
€0.We
set
H(x) =
-T
t
2
1
d V
f j aij
From I t o ' s formula, we have
ZZTi j
.
vIr
' ri dV
i
gi
*
E
C2( B ) ,
(CHW. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
thus
Since H
Co(8),
E
it i s c l e a r t h a t
v-@(h)v(x) h -,H(x) and hence ( 3 . 9 7 ) . The e q u a l i t y ( 3 . 9 7 ) j u s t i f i e s t h e n o t a t i o n A ( t h e i n v e r s e o f t h e i n f i n i t e s i m a l Thus g e n e r a t o r o f @ ( ti )s c l e a r l y an e x t e n s i o n o f t h e d i f f e r e n t i a l o p e r a t o r ) . i f t h e s o l u t i o n o f ( 3 . 9 6 ) belongs t o C 2 ( & ,
v i s a solution of
.
vlr = 0 The c r i t e r i o n ( 3 . 9 2 ) can be w r i t t e n
and consequently does not e n t a i l any c o s t a t t h e boundary. This i s only an apparent l i m i t a t i o n , f o r a s we s h a l l s e e , we can always reduce t o t h i s c a s e . Letting
h
be measurable and bounded on
Z(X)
.
Iii(g(r))e-"
r,
we p u t
Then we put
9" = t
u - a l g e b r a g e n e r a t e d by w ( X ) , X
by
xt
P
The p r o c e s s
bhl) E
[ t , s l , and PXt a measure on
rn) =
rl,...,dhn)c
(g(s),<,Pxt)
E
px(w(h,-t)
E
defined
t
rl ,...,w ( h -t)
E
r
i s a s t a t i o n a r y Markov p r o c e s s , for which i f
A € $ ,
-
ptA = ( o j ct w E A ] defines a translation operator:
.
Bt5 = S(ctw) I n p a r t i c u l a r we t a k e
5 = h(g(7))exp We s h a l l now c a l c u l a t e Bt5.
-
, where 103
St
(17
We have
.
c w
t
Iw
(t+s),
For t h e fj? measurable R.V.
.
5 we put
.
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
347
Consequently
Thus, i f w e adopt a s a c r i t e r i o n
and t h e r e f o r e from t h e preceding e x p r e s s i o n s , we o b t a i n
I n o t h e r words, t h e c r i t e r i o n ( 3 . 9 8 ) reduces t o the c r i t e r i o n ( 3 . 9 2 ) , when J, i s changed t o $ 1 - 2 , though with t h e a d d i t i o n t o t h e c o s t of a function independent of e . I f h E Co(T),h 2 0 , then z ( x ) possesses t h e same p r o p e r t i e s as $ J ( x ) . r e s u l t s obtained with h = 0 t h u s extend immediately t o t h e case
A l l the
348
OPTIMAL STOPPING PROBLEMS & V . I . ' s
h
E
(CHAF
*
3)
C c ( r ) , h L 0.
Example 2 . We c o n s i d e r t h e d i f f u s i o n Y ( t ) , d e f i n e d i n ( 3 . 9 5 ) and we p u t
We t a k e E = Rn.
st = 3
y(t) = ph),
.
P r o p e r t i e s ( 3 . 8 6 ) , ( 3 . 9 0 ) , ( 3 . 9 1 ) a r e s a t i s f i e d . We t h e n sT:e c i f y f E Co(Rn), We p u t I$ , u n i f o r m l y c o n t i n u o u s and bounded on R
.
f b e i n g bounded, and
We t a k e D
-
L = f,
= $.
Assumptions ( 3 . 9 3 ) and ( 3 . 9 4 ) a r e s a t i s f i e d .
B.
The penaZised probtem.
3.7.2.
We now i n t r o d u c e t h e e q u a t i o n which assumes t h e r o l e o f t h e p e n a l i s e d e q u a t i o n (3.48). We d e s i g n a t e a s t h e weak penaZised probZem t h e problem:
Our o b j e c t i s t o prove t h e e x i s t e n c e and u n i q u e n e s s of t h e s o l u t i o n o f ( 3 . 1 0 0 ) . It w i l l b e c o n v e n i e n t , however, t o i n t r o d u c e as an i n t e r m e d i a t e form a f u r t h e r weakened form o f ( 3 . 1 0 0 ) .
I
We s e e k a s o l u t i o n u (3.101)
u, E Do,
, then
1)'
Q(t)(u,-
We o b s e r v e t h a t i f u and t h u s a l s o u t o CE
of
eBA.
E
weakly m e a s u r a b l e w i t h v a l u e s i n B.
C , t h e n from ( 3 . 9 3 ) we s h a l l have
(u
- 1)+ E
C c B
T h e r e f o r e i f we f i n d a s o l u t i o n o f ( 3 . 1 0 1 ) , which b e l o n g s
it i s a s o l u t i o n o f ( 3 . 1 0 0 ) .
We b e g i n w i t h a theorem on 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
(3.100). THEOREN 3.9. Under the assumptions ( 3 . 5 6 ) , ( 3 . 9 0 ) , ( 3 . 9 3 ) , t h e re e x i s t s one and onZy one solution o f probZem ( 3 . 1 0 1 ) .
Proof. Existence Suppose we have A > 0.
(3.1 02)
We s h a l l b e s o l v i n g t h e problem
w E Do, @ ( t ) w , @ ( t ) ( w - 1 ) + w
I
:j
e-(a+h)t
We o b s e r v e immediately t h a t i f
weakly m e a s u r a b l e
Q(t)(L+?,w-i(w-J)')dt
w
.
i s a s o l u t i o n of (3.102) t hen u
= w is a
(SEC. 3 )
349
OPTIMAL STOPPING : STATIONARY CASE
I n f a c t , we put
s o l u t i o n o f (3.101).
z (u) = e x p
- ~(s-a)
We have
1
- zs(u)
5:
=
. .
A zs(p)dp.
We put
1
H = L --(w-L)+
;
thus
and
Let we Put
g
be such t h a t g
h =
Jz
E
e+
Bo n Do and
@(t)g
dt E
l o
e-@
cn
dt E C
Q(t)g
n D~
.
For 6 = a+X,
.
We s o l v e t h e a u x i l i a r y problem
We s h a l l now show t h a t (3.103) p o s s e s s e s a unique s o l u t i o n , a t l e a s t i f h i s For t h i s we d e f i n e t h e i t e r a t i v e scheme: sufficiently large.
w = h
w = h n
,
- [y
Q(t) (wn-l -l)+dt
The above scheme d e f i n e s a sequence w
E
C
naAn
.
Do, following ( 3 . 9 3 ) .
350
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
and t h u s i f A i s l a r g e enough (depending on -+ w i n C .
w
Therefore w
E
-1)' It t h e n f o l l o w s t h a t (w n-1 that
m(t)
(wn-l
-j)+
-. (w-1)'
.fi < u+A E
i n C; t h u s s i n c e
Q(t)(w-j)+
-. e
Consequently, we have found w E
have
1
and so
in
m ( t ) E &(C;C) , we
see
c.
is weakly measurable, being t h e l i m i t of weakly
But e @(t)(w-j)+ measurable f u n c t i o n s , so
Since ( w - j ) +
, we
.
E C n D
Do and (w-j)'
E)
E
C n
Do which s o l v e s t h e e q u a t i o n ( 3 . 1 0 3 ) .
C c Bo, t h e i n t e g r a l belongs t o b A , hence t h e r e s u l t .
The
uniqueness i s proved by u s i n g similar e s t i m a t e s . If
g6BonD
andhr
1"
m(t)g d t E C
e
n Do
(3.103) a mapping T A : subspace o f B
-+
C n
& ,
n BA n Do ,
we d e f i n e by
n Do.
We s h a l l prove t h a t t h e mapping: T i s increasing for t h e o r d e r r e l a t i o n on A B d e f i n e d by f,
1f
f,(x) 1 f2(x) V x E
Q)
2
To do t h i s we s h a l l i n t e r p r e t w For v adapted and v ( t ) problem. (3.104)
JZ"(v)
I
5"
E
E
E
.
with t h e a i d of a s t o c h a s t i c optimal control [ O , l l we put
~g(y(t))+-;-~(y(t))v(t)~exp 1
-(+
0
We s h a l l show t h a t we have w(x) = Inf
(3.1 05)
s
v(r)b+cc+h)t
J:"(V).
V
We p u t
H=
g
- 3 1w - j ) +
and we observe t h a t we have t h e r e l a t i o n (3.106)
w
I
j:
O(t)
H
dt
e
+ @(s)
w exp- (A+a)s, V s
It t h e n f o l l o w s t h a t t h e p r o c e s s [exP
- (h+a)s]w(y(a))
i s a Px m a r t i n g a l e .
+
j:
H(y(t))
exp
- (A+a)t
I n f a c t , we need t o prove t h a t
dt
.
at
.
(SEC. 3 )
OPTIMAL STOPPING
:
STATIONARY CASE
we s e e t h a t
Now
and t h e r e f o r e (3.107) i s equivalent t o
which i s c l e a r l y t r u e s i n c e , from (3.106) we have w(x) = ExS.
From Lemma 3 . 3 bel.ow it follows t h a t f o r a l l admissible c o n t r o l s v ( t ) , t h e
process
i s a l s o a P" martingale, and t h e r e f o r e we have
We l e t s
+
+
-.
Since
w
i s bounded, it follows t h a t
351
352
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & V.I.'s
o(t) =
1 if
>
W(Y(t))
L(Y(t))
0 otherwise.
w
w
0
=
5" 0
Q ( t ) L d t 5!!$ = K
e-"
,
= s o l u t i o n o f (3.103) w i t h g = L+AW,-~, n t 1.
The sequence is w e l l d e f i n e d because, a t each s t e p wn
ja
exp
- (a+A)t Q(t)wndt E C
E
C
nBA n Do
and t h u s
,
so t h a t g s a t i s f i e s t h e c o n d i t i o n s of s o l u t i o n of (3.103) We now show t h a t w1
5
w
.
We a c t u a l l y have
Since we a l s o have
it c l e a r l y f o l l o w s t h a t w1
5
w
.
But t h e n t h e sequence w
i s decreasing.
f a c t , we hirve
We s h a l l now show t h a t t h e sequence w Let K be such t h a t L 2 Kn and 1 1 We suppose t h a t w
i s bounded from below.
1 2 K (we may always assume t h a t K1
1
~ t- I?; ~ t h e n s i n c e (w
n-1
-1)'s
9-5 , we
see t h a t
5
0)
In
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
353
Thus i f we choose
Rs--(K, 1 weseethat
w
n-I
-5)
+ 51 Kl
ZR.
Z R a w
w = - -KtZ R ,
NOW
o
sothat
a
w
Z B , V n .
It f o l l o w s t h a t
,
wn(x) I W ( X )
.
w E B
Thus
L + hwn-1 - 41E w n-I
-1)+
-
L + w - 4 1w - 1 )
+
a t a l l p o i n t s , and t h e r e f o r e a l s o ( s i n c e t h e sequence i s bounded) weakly i n B and t h e l i m i t b e l o n g s t o D It follows t h a t
.
o ( t )[L +
hWn-l
- T(wn-l-j)+i I Q(t)w
weakly i n B , and a l s o t h a t
- at)
, Q(t)(w-j)+
From Lebesgue’s t h e o r e m , f o r e v e r y measure
p[Q(t)(L since w
+
w weakly i n B.
[L
u
w
- T1( w - i ) + ~
a r e weakly m e a s u r a b l e . we have
+ hw. - T(w-L)+]dt 1
It follows t h a t
+ ~w
,
= p(w)
,
is indeed a s o l u t i o n of (3.102).
Uniqueness. Using a p r o o f i d e n t i c a l t o t h e p r e c e d i n g o n e , we can show t h a t e v e r y s o l u t i o n u o f ( 3 . 1 0 2 ) can b e i n t e r p r e t e d a s t h e lower bound o f a s t o c h a s t i c c o n t r o l problem.
(3.109)
We p u t
JE(v) =
E”
fy [ L ( y ( t ) ) +T L ( y ( t ) ) v ( t ) ] e x p 1
- (at+$
1:
v(s)ds)
.
Then
I n f a c t , r e g u l a r i t y ( i . e . u E C ) i s n o t e s s e n t i a l f o r o b t a i n i n g (3.110), a l The r e l a t i o n though t h e p r o o f h a s been condkcted w i t h w E C , ( s e e a b o v e ) . (3.106) remains v a l i d , f o r i f H E B i s such t h a t O ( t ) H i s weakly m e a s u r a b l e , we a g a i n have
The remainder o f t h e p r o o f i s unchanged. We t h u s have u n i q u e n e s s . We now g i v e t h e lemma which was used i n t h e p r o o f o f t h e p r e c e d i n g theorem.
354
and
OPTIMAL STOPPING PROBLEMS
&
LEMMA 3.3. Let z ( s ) and ~ ( s be ) two processes adapted t o z is integrable. We suppose that z(s)
+
c(r)b
-
j:
v(A)dh
is a pX martingale. Proof.
at;
5 i s bounded
i s a Px martingale.
Then if v ( s ) i s an adapted process, 0 s v(s) p ( s ) = e(s)exp
(CHAP. 3 )
V.I.’s
+
r
5
1 ( * ) , t h e process
-
k[~(r)+v(r)Z(r)lexp
We need t o show t h a t ( w r i t i n g E i n p l a c e of Ex)
(3.111) Now
We have
and
Kow, from t h e assumption, we have
thus
( * ) The f a c t t h a t t h e upper bound is u n i t y i s not important
J:
v(lr)dw
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
355
But
so that
Consequently we s e e t h a t
,.
hence t h e r e s u l t .
We s h a l l now g i v e a r e s u l t c o n c e r n i n g t h e r e g u l a r i t y o f t h e s o l u t i o n o f (3.1011, i.e, u E C , which p e r m i t s us t o s o l v e ( 3 . 1 0 0 ) . For t h i s purpose we s h a l l approximate problem (3.101) by a sequence o f nonstatI n a d d i t i o n , t h i s w i l l p r o v i d e a new p r o o f o f e x i s t e n c e .
ionary problems.
356
OPTIMPL STOPPING PROBLEMS & V.I.'s
(CHAP. 3 )
3
We start by c o n s i d e r i n g a family of s t o c h a s t i c c o n t r o l problems of t h e t y p e We p u t , f o r s 2 t : = u-algebra g e n e r a t e d by t h e measurable,
(3.1091, indexed by time. R.V.'s B t C where 5 i s
zt PXt is a measure on
= ; 3
zt
1
Ext
d e f i n e d by
Let v(s) be a p r o c e s s adapted t o
3;
pt'p
= E?y V
'p
3"
measurable and bounded.
such t h a t Y(S) E [0,11; we put
We t h e n have
LEMMA 3.4.
Under t h e asswnptions o f Theorem 3 . 9 , we have us(xlt) = u,(x)
Proof.
We p u t
thus
Then (3.112) i s e q u i v a l e n t t o
(3.1 13)
v
t 2 0
,x
E E
.
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
By d e f i n i t i o n (3.113) i s equivalent t o proving t h a t f o r any R.V. measurable and bounded, we have
357 Sl-t
5, 3
which by d e f i n i t i o n o f t h e measure Pxt i s equivalent t o proving
Now
n
Thus (3.114) f u r t h e r i n d i c a t e s t h a t
a r e l a t i o n s h i p which w a s e s t a b l i s h e d during t h e proof of Theorem 3.9. From (3.112) and by an immediate extension o f Lemma 3 . 3 t o t h e nonstationary c a s e , we s e e t h a t
The proof i s completed as f o r Theorem 3.9, giving
Remark 3.5.
We e v i d e n t l y do not have J Z t ( v ) = J:(v)
f o r a l l v , which i n any
case does not have a p r e c i s e meaning s i n c e t h e admissible c o n t r o l s e t s vary with t . The r e s u l t of Lemma 3.4 i s a consequence of t h e Markovian c h a r a c t e r of t h e optimal c o n t r o l ( i . e . t h e optimal c o n t r o l a t each i n s t a n t depends only on t h e s t a t e a t t h a t 8 instant).
358
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
We now d e f i n e s t o c h a s t i c c o n t r o l problems, on a f i n i t e h o r i z o n . T > 0 and t 5 T , we p u t
jtT
JE,T x t (v) = Ext
Supposing
j:
~ L ( ~ ( s ) ) 1+ ~ I ( y ( s ) ) v ( e ) ] e x p - ( a ( s - t ) + ~v(h)dh)ds
.
We t h e n d e f i n e
T We s h a l l examine t h e f a m i l y u C ( t ) o f f u n c t i o n s of
t
with values i n B , defined
by
uT(t)(x) = uT(x,t)
0 s tS T
.
We immediately o b s e r v e
LEMMA 3.5.
Under t h e asswnptions of Theorem 3.19 we have T
us(t)
-+
uc strongly in 9 when T
-+
+
-
for any f i x e d t .
so t h a t
and t h u s by v i r t u e o f Lemma 3.4
Sup x EE hence t h e r e s u l t .
IUf(X,t)
- UE(x)l
-. 0
when
T
4 -
,
8
I n o r d e r t o show t h a t u
E
T
C , it i s t h u s s u f f i c i e n t t o show t h a t u ( t ) E
C.
To do t h i s we i n t r o d u c e t h e f o l l o w i n g problem, which i s t h e n o n s t a t i o n a r y analogue f i n d a f u n c t i o n w ( t ) , t 5 T such t h a t of (3.101):
OPTIMAL STOPPING
(SEC. 3 )
: STATIONARY
CASE
359
We have
, the
V t l T
woof.
s
Under the assumptions of Theorem 3.9 and
THEOREM 3.10.
e-'
~ ( h ) ~ Ed C h
probtem (3.115) possesses one and onty one sotution.
We d e f i n e a sequence w , ( t )
by
We n o t e t h a t
thus w o ( t ) Since L
E Co([O,T);B),
But wo
E
C.
E
Do, we a l s o have w o ( t )
Let us assume t h a t w
(Wn-l ( s ) - d + E
E
thus
.
E Co([O,T];C)
Do.
E Co([O,T1;C)and
n-1
w
w
n-1
(t) E Do
.
Then
C0([0,T];C).
We p u t
n- 1 We have
from (3.88), s i n c e
A
-
cpn(t+h)
so.ro
E CO([O,T-t];C)
.
But I (t) 6 Co([O,Tl;B), f o r ( t a k i n g t < T i n o r d e r t o c l a r i f y t h e concepts) we have T-t-h
In(t+h)-In(t)
I
@(h)e-"'
cpn(t+h+h)dh
T-t-h
=
Q(X)e-"'(qn(t+h+k)
-
:J
Q(h)s-"'
- cpn(t+A))dh
cp,(t+h)dh
360
OPTIMAL STOPPING PROBLEMS & V.I.'S
s i n c e qn(s)
E Co([O,T];C)
that wn(t)
i s uniformly continuous.
E Co([O,T];C),
Thus In(t)
(CHAP. 3)
i . e . w,(t)
E Co([OjT];C).
Furthermore it is c l e a r
Do.
E
We then have (8)
- ~~-~(s)IIds
I t e r a t i n g , we see t h a t
Now,
and t h u s
It then follows t h a t t h e s e r i e s
wo
+ w1
- wo + ... + wn+l - wn + --
converges i n C o ( [ O , T l ; C ) Co ( [ O ,TI ; C )
.
t o a function w ( t ) .
Thus wn +. w i n
The passage t o t h e l i m i t i n (3.116) i s immediate and l e a d s t o ( 3 . 1 1 5 ) . The uniqueness follows from using upper bounds i n s i m i l a r fashion. We then have t h e following i n t e r p r e t a t i o n of
Proof.
w(t):
We put
Then t h e process
w(y(s),s)exp
is a
Pt martingale
We put
on C0,Tl.
- a(5-t)
+
H(y(h),h)exp
- a(h-t)
I
OPTIMAL STOPPING
(SEC. 3 )
:
STATIONARY CASE
361
I n t h e same way a s f o r Lemma 3 . 4 , we can show t h a t t h i s amounts t o p r o v i n g t h e relation
s”[stla
Sl-t
3 = W(Y(sl-t),sl)
that is
We t h e n put
Thus it w i l l be s u f f i c i e n t to p r o v e t h e r e l a t i o n W(X,Sl)
=
2F;
or alternatively
(3.118)
~
(
8
=~
)
IQ(p)H(p+sl)exp
Now, from (3.115) we have
But
- ap dp + Q(s2-sl)w(s2)exp
- a(s2-sl) .
362
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
t h u s c l e a r l y proving ( 3 . 1 1 8 ) . A s i n t h e case o f Lemma 3 . 4 , we t h u s see t h a t
i s a PXt martingale and we t h e r e f o r e have (bearing i n mind t h a t w ( T ) = 0 )
A s i n Theorem 3.9, we deduce from t h i s t h a t
w ( x , t ) = Inf J,";'(v) V
m
hence t h e r e s u l t .
We can now s t a t e t h e p r i n c i p a l r e s u l t of t h i s s e c t i o n . THEOREM 3.11. Under the assumptions of Theorem 3.9 t h e penalised problem (3.100) possesses one and only one solution.
Proof.
We assume p r o v i s i o n a l l y t h a t , i n a d d i t i o n t o t h e assumptions o f Theorem
3.9, we have (3.119)
1;
e-='
Q(A)Ldh
E C
.
Then according t o Theorem 3.10 and Lemma 3.6 we know t h a t U T E C
.
Now from Lemma 3 . 5 , uT + u s t r o n g l y i n B ; for s t r o n g convergence. 'But From (3.100)
t h u s uE
E
C , s i n c e C i s closed i n B
and t h e r e f o r e , s i n c e C c Bo, it follows from equation (3.101) t h a t ue E b , . We s h a l l now show t h a t t h e assumption (3.119) i s not r e a l l y r e s t r i c t i v e . f a c t we put
so
In
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
and
z=uE-x
363
-
We s e e t h a t z i s t h e s o l u t i o n o f 2;
=:
Now i f v E C
--1 jy e-at
n Do , v + x
Q(t)(z-(L-X))+dt E C
( v - ( L - x ) ) +E
n
, thus
Do
cn
,
.
D~
Consequently, we a r e brought back t o t h e same problem where L .) 0 and 1 X , s i n c e t h e assumptions o f Theorem 3 . 9 , as w e l l as p r o p e r t y ( 3 . 1 1 9 ) , a r e s a t i s f i e d . ‘Fnus
L
0
-
zECnB,nD,
n
so t h a t u E Cn &A
Remark 3.6.
I
Do a l s o .
U
It i s p o s s i b l e t o p r o v e d i r e c t l y t h a t
Inf JZt(v) = Inf Ji(7) V
,
V t
V
w i t h o u t knowledge o f t h e e x i s t e n c e o f t h e f u n c t i o n uE, a s o l u t i o n o f ( 3 . 1 0 1 ) . Under t h e s e c o n d i t i o n s i f we p u t UE
= Inf <(v) V
,
it f o l l o w s from Lemma 3.5 (which remains v a l i d ) t h a t u s t r o n g l y i n C. Now, from Lemma 3 . 6 ,
uT(t) = =
jT
~ ( s - t ) e - ~ ( ~ -L ~ds )
jy @(s-t)e-a(s-t)xT(s)L
ds
-L E
-4/:
E
T
C and t h a t u c : ( t ) + uF
j’
t Q(s-t)e-a(s-t)(uT(s)-L)+
@(s-t)e
-a(s-t)
T
We o b s e r v e t h a t IIu (s)ll 5 C independent o f T and s ;
T
X,(s)(u,(s)-L)+
ds ds
.
p r o c e e d i n g t o t h e weak
l i m i t , we o b t a i n
(3.1 20)
uE 0 .)
J”:
@(s-t)e-a(g-t) L d t
J”: @(t)c?’atL
dt
- -1
:j
-L E
:1
Q(t)e-at(UE-L)+
which p r o v i d e s a f u r t h e r p r o o f o f Theorem 3.9.
Remark 3.7. of Example 1.
@(s-t)e-a(s-t)
(~~4 ds ) ’
dt
I
Let us now i n t e r p r e t t h e p e n a i i s e d problem ( 3 . 1 0 0 ) i n t h e c o n t e x t
364
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
We have s o l v e d t h e f o l l o w i n g problem:
,.
suppose we have
; $ E Co(6) $/r 2 0 and A , which i s t h e weak i n f i n i t e s i m a l g e n e r a t o r o f t h e semi-group a s s o c i a t e d w i t h t h e d i f f u s i o n ( a . . , g . ) stopped on e x i t from t h e open domain 8; t h e r e e x i s t s one and only one s o i d t i & of t h e problem
f E
Co(6)
(3.1 21
E ~'(6)n
uE a
u
u
4
1
+ A u +-(uE-+)+ = f a t any p o i n t of E
s
o
E
.
I f moreover, we know t h a t uE E C2(&,
3.7.3
o
then
Investigation of the inequality and solution of the optimal stoppingtime problem.
We now i n v e s t i g a t e t h e l i m i t of uc a s i n ( 3 . 9 2 ) . We t h e n put
E +.
0.
We r e c a l l t h a t
Jx(e)
was defined
We t h e n have t h e following theorem.
h d e r the assumptions of Theorem 3 . 9 and ( 3 . 9 1 ) , and also one
THEOREM 3.12.
or other of the assumptions ( 3 . 9 4 ) , (3.9411, w e have u
hoof.
+
u in C.
We f i r s t of a l l show t h a t we have
.
u 2 u
We know from t h e proof of Theorem 3 . 9 t h a t
i s a Px m a r t i n g a l e , where we have put
H = L --(u1 E
E
4)+
.
We now show t h a t t h e r e f o l l o w s from t h i s , f o r any s t o p p i n g t i m e 0 w i t h respect to
JL,
the relation
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
365
(3.1 23)
I n f a c t (3.123) further indicates
We approximate 8 by a sequence 8" d e f i n e d by
and 8"
+ 8
when n
Moreover
en
+
+
and 0 5 O n ,
m,
takes t h e values
k -,
e
3
so t h a t
c
en .
3
k 2 0.
We p u t
We n o t e t h a t Bk
z@dPx
-+
I
J B ~
Now, when K
PX(BK)
z(%)a$
I
IBk Sunning for
s
3n. S i n c e z ( s ) i s a Px m a r t i n g a l e , we have
E
*k+l
z (k+l)dPx +
.
z@$d)g
k between 0 and K-1 we o b t a i n
-t
+
I z ( ;K) I
we n o t e t h a t
m,
0 since On i s a.s. f i n i t e .
We can t h e n l e t x
+
property (3.124), i . e . OE = inf
+
a,
and s i n c e
(3.123).
remains bounded by a c o n s t a n t and t h a t
Thus, l e t t i n g K +
Iz(s)
1
C V
B
+
m
, we
i n (3.125), we o b t a i n
clearly obtain t h e
We t h e n p u t
{t 2 OIu&(y(t)) 2 t ( y ( t ) ) ]
which i s i n f a c t t h e e x i t t i m e from t h e open domain { x ( u E < k } ( t h i s i s e v i d e n t l y an open domain, from ( 3 . 9 4 ) o r (3.9411 and s i n c e uc y ( t ) i s continuous, ( * ) Since u
E
$
i s a stopping time.
E
Since t h e process
C n Do).
Applying ( 3 . 1 2 3 ) w i t h e =
*e
and n o t i n g
C , we s e e t h a t we may assume, w i t h o u t l o s s o f g e n e r a l i t y , 8 a.s.
f i n i t e ( o t h e r w i s e we can c o n s i d e r a sequence 8 A T , when T
+
+
m).
OPTIMAL STOPPING PROBLEMS
366
& V.I.’s
(CHAP.
3)
we s e e t h a t we have
t . is continuous, we n e c e s s a r i l y have
Since t h e mapping x + ( u E ( x ) - L ( x ) )
However, t h e s t r i c t i n e q u a l i t y i s i m p o s s i b l e , s i n c e o t h e r w i s e y ( e E ) would belong t o t h e open domain { x Iu BEand t h e f a c t t h a t
<
j),
which i s absurd i n view o f t h e d e f i n i t i o n of
i s continuous.
y
But t h e n we have
u E ( x ) = JX(BE) which from t h e d e f i n i t i o n o f
u c l e a r l y implies u
2
u.
I n o r d e r t o demonstrate an e s t i m a t e i n t h e o p p o s i t e d i r e c t i o n , we adopt t h e With any s t o p p i n g time 8 we approach used i n t h e proof o f Theorem 3.7. associate:
v
O i f s < 9
=
e( 8 )
1 i f s 2 8.
We have
J
and 111 5
If h 2 0 , we have
I1 = Ex[
+
f
cc.
s,
- at
e+h ’(Y(t)),xp
+-
t(Y0)exp
8+h
1
exp --(%@)at
- at exp -+t-O)dt]
- j(y(0)exp - a8 = 11,
+
and
111
2
I
5
c
exp -
.;h
If now t.he assumption ( 3 . 9 4 ) i s t r u e , then
II~S c
2
sup \i(y(t)) 6 s t SB+h
- d y ( e ) ) ( + c exp -Th
112
,
(SEC. 3)
OPTIMAL STOPPING : STATIONARY CASE
and if we put p(6) =
IdS)
sup
5,5‘€ E 15-5’ I 5 6
367
- e(s,)l
we see that on the basis of assumption (3.91) we have 112
c
2
Thus
uE(x) s
(3.1 26)
dx) +
C[E
+ h’ +
t) + exp - $ h
p(h
.
If the assumption (3.94). is satisfied, we have
We then conclude the proof as €or Theorem 3.7.
We shall now show that u inequalities.
rn
is a solution of a problem of non-variational
THEOREM 3.13. Under t h e a s s k p t i o n s of Theorem 3.12, t h e f u n c t i o n i n (3.122) is the m a x i m element o f the s e t o f t h e w s a t i s f y i n g
u
defined
(3.127)
Proof. Furthermore u satisfies the It is clear that u E C n Do and u 5 1 . second inequality (3.127). As u + u in C, we see that u satisfies the conditions (3.127). Now let there be w
E
w(x) 5
(3.1 28)
C n D , which is a solution of (3.127); we shall show that 0
Jx(e) v
0
from which it follows that w 2 u , which clearly proves that u element.
is the maximum
NOW, by an argument similar to that used in proving (3.123) (with the inequalities replacing the equalities), it follows from the second inequality (3.127) that we have W(X)
5
and since w
2
Ex[
L(y(8))erp
-
as d t
+ x(y(e))exp
L , (3.128) clearly follows.
THEOREM 3.14.
-a8],
m
Under t h e assumptions of Theorem 3.12,
ix= infisz
01u(y(s))
= .t(y(s))I
is an optimal stopping time for Jx(e) (defined i n (3.92)).
v
stopping times
8,
OPTIMU STOPPING PROBLEMS & V.I.'s
368
Proof.
i s c e r t a i n l y a stopping time, because
We note t h a t 0
<
ixl.
(CHAP. 3)
dl
We f u r t h e r have
i s an open domain, under t h e assumptions ( 3 . 9 4 ) o r (3.94)1. A
.
1
8'1.8
x
x
....
We can show a s i n Tneorem 3.7 t h a t B E x
+ 8
x
a . s . and t h a t
ex
i s optimal.
Application to diffusions. Additional results.
3.7.4.
We s h a l l now apply t h e preceding r e s u l t s ( p a r t i c u l a r l y Theorem 3.13) t o t h e s p e c i f i c s i t u a t i o n of Example 1, f o r which it i s p o s s i b l e t o give supplementary r e s u l t s , based on t h e theory of p a r t i a l d i f f e r e n t i a l equations. We s m a r i s e t h e s e t of information about t h e problem t o a s s i s t t h e discussion. 2
(3.129)
bounded open domain of R n , s a t i s f y i n g ( 3 . 7 8 ) ; o o f c l a s s C ;
u E C'(B;~(R~;R~)) g
E Co(6;Rn) ; f E
(3.130)
o,g
;
?=
a 2
PI , fi > o
,
a r e extended t o Rn i n a way which preserves t h e i r properties;
C0(B) , +
c0(6)
E
,
+p 0
.
Associated with t h e p a i r ( o , g ) i s a d i f f u s i o n f ( t ) ,described i n ( 3 . 9 5 ) , dependent on t h e parameter x E & , ( t h e i n i t i a l s t a t e ) . F i n a l l y , t h e r e i s asso c i a t e d with p,f,J, a c o s t f u n c t i o n a l J (0)which i s defined i n ( 3 . 9 2 ) , and X
u ( x ) i s defined i n (3.122). The process y ( t ) = p ( t i z T ) ( a d i f f u s i o n stopped on e x i t f r o m e ) i s a Markov process with values i n 6 ,t o which t h e r e corresponds O ( t ) and an operator A. Suppose H E Co(&; we know ( s e e ( 3 . 9 6 ) ) t h a t t h e r e e x i s t s one and only one s o l u t i o n of t h e problem
(3.131)
av + Av = H i n e ,
(3.132)
vlr
I
o , v E ~ ' ( 5n)
.
I n f a c t ( s e e e . g . MIRANDA [ll) there exists a function z such t h a t ( p being a r b i t r a r y but f i x e d ) :
I
E
w2,P(a) Z
(3.1 33) Since
2
ij p
a,,
,
b z - E gi xi+ az = f
2
1J bX.OX
1
dZ
a.e. i n 8 ,
J
i s a r b i t r a r y , we can always assume t h a t p >
and t h u s z
E
Co(b).
We s h a l l now prove t h a t we n e c e s s a r i l y have
z(x) = Ex
f(g(t))exp
- a t dt
I
v(x)
which w i l l prove t h e uniqueness of t h e s o l u t i o n of (3.133) and a l s o (3.132). We consider a sequence z
such t h a t
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
369
-
and n a t u r a l l y
F
-+
f in LPW. I t o ' s f o r m u l a g i v e s us
Suppose we have h > 0.
(3.136)
Now,
EX
z
We e x t e n d f by 0 o u t s i d e 8,so t h a t f E L p ( R n ) , and we c a n always assume t h a t i s e x t e n d d o u t s i d e 8, i n such a way t h a t
F~ - f
+.
o
in L ~ ( R ~ ) .
Now
-
and making use of t h e f a c t t h a t p ( t ) p o s s e s s e s a probability d e n si t y p ( x , t , S ) we have (See STROOCK-VARADHAN c11 p.h93) such t h a t f o r any B s a t i s f y i n g 1 < B <
we s e e t h a t
and t h e r e f o r e I1
+.
0 when n
-+
-.
But t h e n p r o c e e d i n g t o t h e l i m i t i n
(*I
If
T
= +
-,
n
i n (3.136), we o b t a i n
zn(f(~))exp - a~ = 0 by d e f i n i t i o n .
370
OPTIMAL STOPPING PROBLEMS & V.I.'s
EX e(Y(nrh))exp
(3.1 37)
z
+ Ex Then we let h tend to 0 ; if
and if
T
= +
T <
m,
e(f(zvh))exp
- a(rvh)
e(g(nrh))exp
- a(rvh)
m
(3.138)
- a(nrh) EX e(g(h))exp - ah - e-at f(g(t))dt ,
E
UE
A u
u
lr
, v
+Yo) + a uL
+-(uEE
1 0 .
Now, from Ito's formula, we have
Then however,
I
p 2 1
1
We know that
thus
-
then
- a7 = 0
e(f(7))exp
0V h
.
,
+)+ = f
a.e. i n O ,
(CHAP. 3)
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
371
thus a l s o
1
= f - -(u
so t h a t i f we put g But then u that u
+
E
remains‘within
u in
Co(B;t h u s u
W2”(0)
E
E
-
I))’, we see t h a t
g
remains bounded i n Co(B).
a bounded subset of W2”(&,
Vp
2
1.
Now we know
and uE + u weakly i n W 2 ’ p ( a ,
Now we have
.+ Au
au
5
f a.e. i n
8
thus
8.
au
+
(u
- +)-(Au~+ auE- f) = o
AU 5
f a.e. i n
Moreover E
Since (uE-
+)-
-
(u-+)’
i n c0(m
it follows t h a t
so NOW u
5
a.e.
(u-+)-(Au+ a u - f ) k = o
.
J I ; t h u s (u-JI)- = -(u-$), so t h a t
We have t h e r e f o r e proved:
THEOREM 3.14. Under t h e a s s w n p t i o n s (3.129), one and onlu one solution o f t h e oroblem
(3.140)
(3.130), (3.139), t h e r e e x i s t s
U€W29P(O) , v p 2 1 , U S + V X E b
au + Au
5
f
8.e. i n 8 ,
( a u + Au-f)(u-$)
= 0 a . e . i n 8 , u l r = 0.
,
372
OPTLMAL STOPPING PROBLEMS & V.I.'s
(3.141)
+' - +
fr
a'
ij
f
(CHAP. 3 )
weakly s t a r i n L m ( a , in
-. aij e t
c"(M, I(C'II~~,-S c gi -. gi i n c0(M,ar
9
2
B 1.
We have
Now
sup i(fr
x € 0 Thus
- ad,'-
'A
- +')+
(Eu'
+r(X)
1
independent of r .
independent o f r and E.
E
I f we p u t
we t h e n have
so t h a t it f o l l o w s , from MIRANDA 111, Theorem 3 7 . 1 , p. 169 (k = 0 ) t h a t C
(independent o f r , E ) .
W2%)
( * ) We i m p l i c i t l y assume t h a t we a r e working with a r e p r e s e n t a t i v e of f , measura b l e and bounded o n 8 ; l i k e w i s e t h e second p a r t i a l d e r i v a t i v e s o f $ a r e assumed measurable and bounded.
(SEC. 3 )
373
OPTIMAL STOPPING : STATIONARY CASE
Proceeding t o t h e l i m i t i n
we o b t a i n
E,
We e x t r a c t a sub-sequence such t h a t ur + w weakly i n w ~ ~ P ( o ) S i n c e t h e i n j e c t i o n map from particular ur Thus
w
*
5
into
dyp(@) i s compact,
we have i n
~“(8).
w in
-+
W2”(&)
vx
6,
E
WIT
= 0.
Furthermore
A’
+
ur
a u r = Aur
+
aur
+
(Ar
- A)ur
and
where
Thus
(nr
- ~
aur
+
) + uo i n~ LP(s).
Since ~u~ + aw
+
AW
weakly i n LP(c+)
we s e e t h a t
Aw
+
aw 5 f .
Also,
+
aur
A’
ur - f r
-+
aw
+ Aw -
f weakly i n L p ( S ) .
Since ur - ~l~ + w
so
-
i in
~‘(8)
it t h e n f o l l o w s t h a t (A’ u’+
aur- fr)(ur- $’)dx
from which it can r e a d i l y b e deduced t h a t
(Aw
+
aw - f ) ( w - $ ) = 0 a . e .
- so
(Aw
+
aw
- f)(w-+)&
374
OPTIMAL STOPPING PROBLEMS & V . I . ' s
(CHAP. 3 )
and t h e r e f o r e w = u, which t o g e t h e r with t h e uniqueness o f t h e l i m i t proves t h e m desired r e s u l t . We continue t h e study of Example 1, without assuming t h e c o n t i n u i t y of g , o r t h a t JI E W2y"(0). I n o t h e r words, we r e p l a c e (3.129) by
(3.1 29)'
assumptions (3.129) with
g
g E Co(6);Rn)
instead of
measurable and bounded,
,
and we r e t a i n (3.130) (but not ( 3 . 1 3 9 ) ) .
w
I n t h i s case, from Theorem 3.13, t h e s e t o f functions
(3.1 42)
I
w ~ c o ( u ) , w l c v x E b ,w Q(t)f
w l Jo
possesses a m a x i m u m element
dt
+
I,=o
satisfying
,
Q(h)w V h 2 0
u which i s i n f a c t
Further @ ( ti)s t h e semi-group corresponding t o t h e Markov process ( y ( t ) = Y(tAz),$)
.
I n t h i s c a s e , it is u n f o r t u n a t e l y impossible t o express u as a s o l u t i o n of a problem of t h e t y p e (3.140), s i n c e u cannot belong t o W2yp(@). The following i s then a n a t u r a l question: Suppose t h a t a? is a sequence such t h a t lj
and suppose t h a t ur i s defined from a y j , i n t h e same way a s u from a . . i t h e can we conclude from t h i s t h a t l J u + u ? o t h e r d a t a gi, f , JI being unchanged); Since t h e v a r i a t i o n a l theory i s a p p l i c a b l e t o u r , ur i s a s o l u t i o n of t h e v a r i a t i o n a l problem
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
s o t h a t ur may be obtained much more simply t h a n theorem: THEOREM 3.16.
U.
375
We t h e n have t h e following
Under t h e assumptions (3.129)', (3.130) and ( 3 . 1 4 4 ) , then
Proof. The method c o n s i s t s of reducing t o t h e conditions o f a p p l i c a t i o n of To do t h i s we consider a sequence J, g . such t h a t Theorem 3.15. j' J
l g j l Ic gj
,
E C0(U)
gj -+g
.
(*I
We denote by ~'(+~,g~),ur(+,g)
in
L'+'(o)
, u(Gj,gj) , U($,g)
...t h e functions
corresponding t o t h e various d a t a items, using an obvious n o t a t i o n . 3.15, we have
u
From Theorem
Furthermore, from (3.1k3) we have
We denote by QZ t h e measure corresponding t o ( a r , O ) corresponding t o (ar,gj ).
("IBy
We have
s u i t a b l e extension we can r e p l a c e O b y Rn.
and by Px. t h e measure
rJ
376
OPTIMAL STOPPING PROBLEMS & V . I . ’ s
(CHAP. 3)
Now, we have
where
and
We t h u s have
We w r i t e ( * )
where CT does n o t depend on 8 .
Y .=expZ . rJ
Y,
rJ
exp 2
and
and s i n c e we o b t a i n
Ia-’l
,[ g . I J
a r e bounded by c o n s t a n t s which a r e independent of r and j
I-
I-
( * ) The t e c h n i q u e which f o l l o w s is due t o STROOCK-VARADHAN C11 p . 498.
,
(SEC.
3)
377
OPTIMAL STOPPING : STATIONARY CASE
where Ch i s independent o f then
E
x
and r.
We choose y i n such a way t h a t 2y’ = 2+&;
Q:
IZrj-zrI2 S c1T[c1hllgj*llL2+6 2
2 Noting t h a t E Y ., E rJ and t h a t
n
(R
Y2 a r e bounded by a c o n s t a n t r
we r e a d i l y o b t a i n
+
hI
C2T which depends o n l y on T ,
1
thus
Therefore
Now we have
I n view of t h e p r e c e d i n g r e s u l t s , t h e d e s i r e d r e s u l t can now b e deduced e a s i l y .
m
Remark 3.9. Within t h e s e t t i n g o f Example 1, we a g a i n have t h e p r o p e r t y t h a t The proof i s i d e n t i c a l t o t h a t g i v e n f o r t h e e x i t t i m e T i s Px a . s . f i n i t e . Theorem 7 . 1 i n Chapter 2 (we a g a i n c o n s i d e r t h e t e s t f u n c t i o n w
= 2
- exp - y / x - x o /
and we n o t e t h a t Aw 2 C > 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
<
if 8
8
T~
otherwise
+ a
,
which i s an a d m i s s i b l e s t o p p i n g time.
We n o t e t h a t
Furthermore 1
Let
0 C O 6 C6'
5 E O6 * We denote by T~ = :T 7 S
x
Let
z
such t h a t
d(E.,6)
< 6
.
t h e e x i t time from
6,;
thus
6
T S T l x X
.
be t h e s o l u t i o n of
(A, +A,)" = 1
,
elBG= 0
.
From t h e theorem o f de GIORGI C11, NASH [ll, extended t o t h e boundary by
STAMPACCHIA [ 2 1 , we have lz(5)
- z(c')/
cl
J
>
.
(SEC. 3 )
381
OPTIMAL STOPPING : STATIONARY CASE
Now, from I t o ' s formula we have
- Z(YX(TX)))
E"(z(Yx(zp
Since yX(rx)
E
-E P-
=
.
3G we see t h a t we have 6
i(z
-
7)
=
- s" 5(yx(7 6 ) )
.
Furthermore y x (6~) E a06 and t h e r e e x i s t s 5, iYx(7
thus
7)
6
- 5,1
such t h a t
= 6
- ~ ( 5 , )1 = C 6'
Iz(Y~(~ )
E 3 0
;
s i n c e z ( c ) = 0 , we t h e r e f o r e have
(3.149)
.
E"(z6- T) 5 C 6'
Thus
From (3.149) and s i n c e
If1 is bounded on
.
X, IC, 6'
e',we have
Suppose a l s o
-6 6 z = i d { t 2 z Iyx(t) We note t h a t ? &
-6 7
5 T'
<
p
.
G]
and moreover
7' s y ( z d )
.
E a0
Since t h e n
{? < we have
7'
et
T6-
z6
3
6 ] c
,
6
sup
6
+
7 ~ & ~ 6 7
3
IY
.
3)
-Y
OPTIMAL STOPPING PROBLEMS
382
&
V.I.'s
(CHAP. 3 )
Now
Consequently
We now demonstrate an estimate in the opposite direction.
and since JI 2 0 on
&I,
it then follows that we have
It is clear that
and from what we have seen above
2 2
- C[6 i+ exp - -163 .
Thus, finally, in view of (3.150) we have
(3.1 51)
IuE(x)
-
U(X)
I 5+'6[C
exp
- b3 v ] ,
6 2 0 (+)
For a l l 9 we have
(SEC. 3 )
O P T I W S T O P P I N G : STATIONARY CASE
which proves t h e d e s i r e d r e s u l t .
3.9
383
8
Unbounded open domain
Synopsis I n t h i s s e c t i o n , we s h a l l endeavour t o i n t e r p r e t t h e V . I . problems which were s t u d i e d i n S e c t i o n s 1.11, 1 . 1 2 , 1.13. We s h a l l t h u s consider t h e c a s e o f an unbounded open domain and, i n o r d e r t o s i m p l i f y somewhat, we s h a l l l i m i t o u r s e l v e s t o t h e c a s e 6 =R n ( * ) . 3.9.1
Bounded coefficients
We assume t h a t
(3.1 52)
I:” a
E C1 (Rn)
,
aij bounded,
a . measurable and bounded, a
3
bounded
k
measurable and bounded on
R”
,
...
As i n ( 3 . 2 6 ) , ( 3 . 2 7 ) we c o n s t r u c t a s e t ( n , U , @ , & , i ? ( t ) ) t o g e t h e r with a p r o c e s s y ( t ) = y x ( t ) which i s a r.olution o f
(3.1 53)
dY ( t ) = g ( y ( t r ) d t
+
&(t))dF(t)
, ~ ( 0 )= x
We r e c a l l t h e r e l a t i o n a l formulas
(3.1 54)
(*)
f
E LpsP(Rn)
+
bounded
,+
E W1””(Rn)
, A+
E LppP(Rn)
p
.
A s w a s done i n Chapter 2 i n t h e c a s e o f e q u a t i o n s . The r e s u l t s adapt t o t h e c a s e o f unbounded open domains, by u t i l i s i n g where necessary c e r t a i n techniques r e l a t i n g t o bounded open domains i n o r d e r t o d e a l w i t h e x i t t i m e s .
384
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.’s
From Theorem 1 . 1 5 , t h e r e e x i s t s one and o n l y one f u n c t i o n u ( * ) s a t i s f y i n g t h e properties
(3.155)
u E V
n
W2,p,p
8
AUS
f a.e.
, U S (i, ,
(Au-f)(u-+) = 0 a . e .
Our o b j e c t i v e i s t o i n t e r p r e t u ( x ) . For any stopping t i m e 0 with r e s p e c t t o
st
we put
(3.1 56)
Since J, i s bounded, t h e q u a n t i t y EJ, i s w e l l d e f i n e d . We s h a l l now show t h a t t h e same i s a l s o t r u e of t h e i n t e g r a l , at any r a t e i f
is s u f f i c i e n t l y large.
p
This r e s u l t s from t h e f o l l o w i n g lemma, which i s complementary t o Lemma 3.1.
LEMMA 3 . 8 .
Suppose we have 0 E L’” is sufficiently small, #e have
and p
and B > 0 .
If p is sufficiently large
(3.157)
Proof.
and
u
It w i l l s u f f i c e t o prove independent o f Q .
(3.157) w i t h Q
E
a (R”)
and Q b 0 and with p
Let t h e r e be h > 0 and T > 0 , T > h (we s h a l l subsequently be l e t t i n g t o 0 and T t e n d t o + - ) ,
h
tend
We have
where ( s e e ( 3 . 2 8 ) )
Suppose we have a > 1 and l e t a ’ be t h e conjugate o f a . we have
(*)
Independent o f
(u
s u f f i c i e n t l y small).
From HSlder’s formula
OPTIMAL STOPPING : STATIONARY CASE
how s i n c e
lo-lgj
5
K we have
E 5:
5 exF
a(a-1)
2
kt
Therefore if
K( a-1 )
(3.1 58)
81
>2 C: e4'ltdt
it i s c l e a r t h a t t h e i n t e g r a l E
is w e l l d e f i n e d .
Furthermore
( s e e Lemma 3.1) we have
We put
P2=P3+P4 and l e t b > 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 <1 2
T tend t o +
m,
so that
t
(SEC. 3 )
387
OPTIMAL STOPPING : STATIONARY CASE
It only remains to verify the compatibility of the parameters.
rn
We then have THEOREM 3.19. Under t h e asswnptions (3.152), (3.154), f o r p s u f f i c i e n t l y large and y s u f f i c i e n t l y small, the solution u of (3.155) iS given by U(X)
= Inf
e
Jx(o)
.
Moreover there e x i s t s an optimal stopping time given by
gX
= id I s 2 0IuGTx(s)) =
Proof. Let % +. u in $ypau and let uN we have for any stopping time 8
E
Q(Y~(~))I .8(Rn).
.
Applying Ito's formula to
We note that
In addition to which,
where
E
is chosen sufficiently small and > 0 ;
where C is independent of N, 8, T.
Thus
applying Cauchy-Schwarz we have
3,
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
bU
N
-
au
0
( uniformly
in
T,B)
.
Consequently
BAT buN
. u dff exp -
iz
-.
.a
u n i f o r m l y w i t h r e s p e c t t o 8,T. A s Aun
+.
Au i n L p y u , it follows from Lemma 3.8 t h a t
We can t h e n p r o c e e d t o t h e l i m i t i n (3.161), from which it f o l l o w s t h a t
Since u
5
JI we can t h u s deduce
( * ) From Lemma 3.8 if
p2
88
B
IGc4p 1
,
p
>
(n+2) (K+48,
48~
(SEC. 3 )
389
OPTIMAL STOPPING : STATIONARY CASE
We can then l e t T tend t o + -; using an argument similar t o t h a t given e a r l i e r , we can show t h a t t h e i n t e g r a l s ( o r d i n a r y and s t o c h a s t i c ) converge. Moreover t h e We t h u s o b t a i n term r e l a t i n g t o I$ 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
jr
-
0 f o r a l l T and t h u s s i n c e
-
fE
5 s B"
j'
/fI(eq
ao)dt
< +=
,
we o b t a i n
lim sup R++= Thus
5
-
,
0
- ji ao)dt
(f/(exp
T
,
V T
.
uniformly i n 8 .
Furthermore
Hence again
IIR-. 0
,
uniformly i n 8.
Consequently
-
- Infe JXb)1 R +0+ =*
lu.&x) Since we know t h a t
v
x
.
-
q")u i x ) we have proved once again t h e f i r s t p a r t of Theorem 3.19. We s h a l l now give t h e analogue of Theorem 3.7. a (u,v) defined i n (l,iOO), and we assume t h a t
We consider t h e b i l i n e a r form
IJ
(3.1 65)
JI
i s uniformly continuous and bounded on f
bounded
.
Rn ; f E Co(Rn)
We have proved i n Section 1 t h e e x i s t e n c e and uniqueness of t h e s o l u t i o n
,
u of
392
OPTIMAL STOPPING PROBLEMS
We know that u
E
&
V.I.’s
(CHAP. 3)
Co(Rn).
THEOREM 3.20. Under the assumptions (3.152) and (3.165), we obtain t h e same c o n c h i o n s as i n Theorem 3.19.
Proof. We consider the penalised problem. For uniqueness of the solution u of
L! 5
po,
we have existence and
By the usual techniques we then have (noting that here f is bounded, which is a simplification by comparison with the situation in Theorem 3.19).
where
We can show as in Theorem 3.7 that (due in particular to the uniform continuity of $I), we have
u
4
O
~nf ~ e
,(e) ,
uniformly in
x
.
Here the situation is even simpler because the exit time considered. The proof is concluded as in Theorem 3.7.
T
no longer has to be
As indicated in Remark 1.19, we shall now give a result concerning the support
of the solutions of the V.I. (3.155) in the case where
THEOREM 3.21. Under the asswnptions (3.1521, (3.165), (3.167) the solution o f (3.166) has compact support.
u
Proof, Let there be r > 0 such that f(x) t y for 1x1 2 ro and suppose that R > r , We take x such t8at 1x1 2 R and we denote by T = T the exit time from Furthermore, as J, = O,x we have the :pen domain 151 161 > ro}.
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
S i n c e f i s bounded, we have f 2 - c .
J,(@) 2
E”
1:
Moreover s i n c e a
393 C M , we have
- M t dt .
f exp
We p u t
We have
We have
c
+-E
M
e
-Mt
(e-
1 -(t-QAt) E
-(Mt
- x t s e1
+ -1( t - e A t ) ) E
dt 2
-
YE
.
OPTIMAL STOPPING PROBLEMS & V.I.’s
394
( C W . 3)
I
1
Since T - AT i s also t h e e x i t time from t h e open domain(c 15 > r )corresponding t o t h e i n i t i a l s t a t e BAT), it follows from t h e s t r o n g Markov pro$erty t h a t we have
<(el
=
:1
I?
y e
-Mt
., -Men?
Y
d t + E e
We put
thus
If we l e t T be a f i x e d value ( t o be determined l a t e r ) , it can e a s i l y be seen
that
-MBAZAT $(e)rE”(Yl
We put
thus
m
e
M
++ Y
- (---++I
e
-m
xeA-c 2 T
OPTIMAL STOPPING
(SEC. 3 )
Let
h
be a parameter t o be chosen.
c . ,
2 hp
E [(eAzAT)
( s e e Chapter 2 , Theorem 2 . 3 ) .
P/2
:
STATIONARY CASE
395
We w r i t e
+ ATA AT)^]
We t h e n p u t
thus
<(e)zx+r+z
.
F i r s t o f a l l we n o t e t h a t we can choose T, which is a c o n s t a n t independent o f
396
x,
1-
e i n such a way t h a t X
E, E~
(with 1 -
E~
M > 0);
2 0.
We t h e r e b y f i x we n o t e t h a t
T.
Now i f we c o n s i d e r t h e f u n c t i o n of z ,
f o r z t 0 and z 5 T , we observe t h a t p ( 0 ) = 0 and t h a t
if
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ’ s
h
i s chosen equal t o 1
We t h e r e b y f i x h. We t h e n put
we have:
It t h e n f o l l o w s t h a t Y t 0 .
We t a k e
E
5 E~
(with
OPTIMAL STOPPING : STATIONARY CASE
(SEC. 3 )
where
thus
It follows from this that
Since d(x,S)
2
R - r o , we see that if /y(OATAT)-xls h then
z R
d(y(BAzAT),S)
- ro - h
9
thus
But in this case,
There exists cl
For
E S
5 E
such that for
E ~ we , therefore have
We can always assume
E
2 1.
Then
E S
E~
we have
397
O P T I W STOPPING PROBLEMS & V.I.'s
398
2 2
Y -2
+
( Y + c
M+
2(ME1+ 1 )
(CHAP. 3 )
CIE
(R-r0-h)4
1
and t h u s i f R i s such t h a t
4Cl(
iq Y +$M CE , +
(R-r -h) 4 2 we s e e t h a t Z Z 0.
1)
Y
We t h e r e f o r e choose R s u f f i c i e n t l y l a r g e , a s i n d i c a t e d above.
Then f o r 1x1 t R , we have K"(8) 2 0 , V
E 2 E~
and 8.
I t t h e n f o l l o w s from
( 3 . 1 6 8 ) t h a t we have
J (8) L
ov e
f o r 1x1 2 R
and t h u s u ( x ) Z 0 which i m p l i e s u = 0 ( s i n c e u 2 0) a n d c o m p l e t e s t h e proof o f t h e rn theorem. 3.9.2
Unbounded Coefficients.
We now wish t o c o n s i d e r t h e c a s e o f unbounded c o e f f i c i e n t s . We s h a l l suppose t h a t t h e c o e f f i c i e n t s a r e s u f f i c i e n t l y r e g u l a r t o be a b l e t o u s e a strong formulation o f I t o ' s e q u a t i o n , because G i r s a n o v ' s t r a n s f o r m a t i o n n e c e s s i t a t e s , i n t h e c a s e when t h e c o e f f i c i e n t s a r e unbounded, assumptions which We make t h e a r e d i f f i c u l t t o v e r i f y ( s e e Chapter 2 , S e c t i o n 4, Theorem 4 . 2 ) . f o l l o w i n g assumptions ( s e e Chapter 2 , S e c t i o n 7 . 2 and S e c t i o n 1 . 1 3 i n t h i s chapter).
(3.169)
a,. = a IJ ji
aij
E LYR")
,
E c ~ ( R ~ ) ai E
, z a i j ci
C. 2
cI (Rn , ac
J
a 2
c12 , a > o
E c'(R")
. a -
bounded, axk bounded
and i n c l u d i n g o r d e r 3.
(SEC. 3 )
(3.1 74)
OPTIMAL STOPPING
:
- dx') I
- g(x') 1 S
[Oh)
101
s KO
+
Igb)
Igb)
9
STATIONARY CASE
Is
$(I+
Klxa'
1x1)
399
I
.
and s i n c e f - A9 s a t i s f i e s t h e same growth assumptions a t i n f i n i t y as f , t h e right-hand s i d e of (3.175) i s w e l l d e f i n e d , a s we have shown i n Theorem 7 . 2 , Chapter 2. We t h e n have
THEOREM 3.22.
Under the assumptions (3.169), (3.1701, (3.171)we then have u E ( x ) = I n f J:(v) V
and there exists an optimal control 1 if
+ ( t )=
UE(Y(t)) 2
i(Y(t))
0 otherwise.
Proof.
We approximate f by a sequence fN d e f i n e d a s f o l l o w s
f
if
- N S f s N
,
N
if
f > N , - N
if
fN =
and l e t
sE= GN be t h e s o l u t i o n o f 1 A iiNE+ - UNE = f N - A+
and UNE
= ifNE +
$
.
f < - N
,
OPTIMAL STOPPING PROBLEMS & V.I.'s
400
(CHAP. 3 )
We c a n show t h a t we have
u
NE
(XI
= Inf J
p
V
where J N E ( v ) i s d e f i n e d by ( 3 . 1 7 5 ) w i t h f
N r e p l a c i n g f.
We s h a l l e s t a b l i s h a p r i o r i e s t i m a t e s f o r u
NE'
1,
S i n c e I f N \ 5 If
from t h e growth p r o p e r t i e s o f f , A+ ( * ) and a l s o t h e e s t -
i m a t e s o f t h e moments o f t h e s o l u t i o n s o f t h e I t o e q u a t i o n s ( s e e Theorem 7 . 2 , Chapter 21, we have
where C1 does n o t depend on N ,
E,
x.
We t h e n have
Making u s e o f t h e f a c t t h a t
- fN(d1 I
jfN(C)
If(S)
- f(dI
and p r o c e e d i n g a s f o r t h e proof o f Theorem 7 . 2 , Chapter 2 , we o b t a i n
(3.1 77) It is c l e a r t h a t f o r we have
u
NE
Therefore u
+
E
f i x e d , by v i r t u e of t h e e s t i m a t e s ( 3 . 1 7 6 ) and ( 3 . 1 7 7 )
1 u weakly i n HT. E
also s a t i s f i e s t h e e s t i m a t e s ( 3 . 1 7 6 ) and ( 3 . 1 7 7 ) . g E = f - A + - A
1
u
But i f we p u t
E'
we have
and
*
O
Consequently g
6
E
+ l z + -
E E - g E '
belongs t o
Lz and a l s o t o a s p a c e o f t y p e LP"(with
p o n e n t i a l weight ; u n c t i o n ) . Since t h e coefficients of A apply t h e techniques o f Se c tio n 3.9.1; th u s
ex-
a r e bounded, we can
OPTIMAL STOPPING
(SEC. 3 )
ri
CASE
: STATIONARY
401
,
E W21P,P
From t h i s r e g u l a r i t y r e s u l t , we can e a s i l y o b t a i n t h e s t a t e m e n t s c o n t a i n e d i n t h e t h e o r e m , by a p p l y i n g t e c h n i q u e s a l r e a d y e n c o u n t e r e d . We now c o n s i d e r t h e m i l a t e r a l problem : f i n d
(3.178)
We have t h e f o l l o i r i n g theorem:
Under the asswnptions of Theorem 3 . 2 2 , there e x i s t s one and THEOREM 3.23. only one solution of ( 3 . 1 7 8 ) with the usual interpre t a t i o n :
where, f o r m y stopping time 8,
We have shown i n Theorem 3.22 t h a t t h e s o l u t i o n uE o f t h e p e n a l i s e d Proof. problem ( 3 . 1 7 2 ) s a t i s f i e s
\
/U,(X)
+ \&up
I
C L I + Ix
.
I"]
F u r t h e r m o r e , from t h e i n t e r p r e t a t i o n o f u , and u s i n g a n argument a l r e a d y E employed s e v e r a l t i m e s , we have uE
s: j+"
- +S E S
If-.4+Iexp
C[1+
- p t exp --dtt
(clt+ 1x1
2m
3
EClt
) exp q]exp
0
t - pt exp --dt
and i f 6 i s s u f f i c i e n t l y l a r g e
Ic a b + IXI")E so t h a t
(%- +)+ 5 c2(1+
Ix
e
I")
.
But t h e n i f we w r i t e ( 3 . 1 7 2 ) i n t h e form
A
O
u
E
+ a
O
u =hE=f-AluE-
(Uc-
E
+I+
E
we have lh,l
s
Cg(l+
Ixlm+')
and t h u s , s i n c e t h e c o e f f i c i e n t s o f A
f :L
n LptP
a r e bounded, u
remains w i t h i n a bounded
402
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP. 3)
We can t h e n r e a d i l y p r o c e e d t o t h e l i m i t , and t h i s e s t a b l i s h e s s u b s e t o f W2yp". t h e existence of a s o l u t i o n u of (3.178). We c o n c l u d e t h e p r o o f ( u n i q u e n e s s and 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 , which go t o g e t h e r ) a s i n Theorem 3.19. m Remark 3.11. The s o l u t i o n u o f ( 3 . 1 7 8 ) n a t u r a l l y c o i n c i d e s w i t h t h e Theorem 3 . 2 3 may t h u s b e i n t e r p r e t e d a s a r e g u l a r i t y s o l u t i o n u of ( 1 . 1 4 5 ) . result. We s h a l l now g i v e t h e e q u i v a l e n t o f Theorem 3.20.
THEOREM 3.24. Under t h e a s s w n p t i o n s ( 3 . 1 6 9 ) a n d (3.165) t h e s o l u t i o n ( 1 . 1 4 5 ) c o i n c i d e s w i t h t h e lower bound of
u of
Moreover t h e r e e x i s t s an o p t i m a l s t o p p i n g time.
Proof. a sequence
We i n t e r p r e t t h e p e n a l i s e d problem. . ) I
J'
i s associated u
f . c'B(Rn),
J
E:
ij +
J, i n C o ( R n ) ,
f. J
To do t h i s we approximate J, by +.
f i n Co(Rn);
which i s i n t e r p r e t e d by Theorem 3 . 2 .
w i t h $Jj;f j t h e r e
Now, w i t h
E
fixed
J
so t h a t
(3.1 79) and
u
El J
+.
u
E
i n c'(R").
S i n c e JI and f a r e bounded, u t h e formula
u
(X) e
€3
i s bounded
(E
remaining f i x e d ) .
Moreover from
B
we can deduce, by v i r t u e o f t h e uniform convergence, t h a t a n a n a l o g o u s formula i s t r u e f o r u , w i t h f , $I i n p l a c e o f f . , J J From t h e formula t h u s o b t a i n e d and from Lemma 3 . 3 on m a r t i n g a l e s , we can a g a i n deduce ( 3 . 1 7 9 ) t o g e t h e r w i t h t h e e x i s t e n c e o f a n o p t i m a l v . The p r o o f is t h e n concluded a s i n Theorem 3 . 7 . m
$I..
I n v e s t i g a t i o n of a p a r t i c u l a r i n e q u a l i t y .
3.10.
We now c o n s i d e r t h e i n e q u a l i t y d e s c r i b e d b r i e f l y i n Remark 1 . 2 2 ( * ) Let O b e a bounded open domain o f R", domain,
E
n
0, a l s o o f c l a s s C 2 .
For J,
o f c l a s s C 2 , and l e t E b e a n open 2 E L (O),we i n t r o d u c e d i n Remark 1 . 2 2
( * ) The i n e q u a l i t i e s a r i s i n g o u t o f t h e games p r e s e n t e d i n S e c t i o n 1 . 1 4 w i l l b e i n t e r p r e t e d l a t e r on.
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
a(u,v-u) 2 (f,v-u)
(3.180)
v
x=
v E
{v E
VI
v
s
403
+
on
EI,
u E K . To s i m p l i f y somewhat we t a k e J, = 0 and a = 0(*). We have a l s o seen t h a t t h e V . I . (3.180) may be reduced t o t h e g e n e r a l gase, although t h e o b s t a c l e needs t o be modified. We consider 9 , a s o l u t i o n of
where F = 8 -
E,
1
t h e n we s e t
Y
=
Q
on
F
0 on E .
Since t h e open domain F i s r e g u l a r and 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 ( f o r ssiim mpplliicciittyy)),, tthhee ffuunncct ti o i onn ww EE CCoo( (FF) ) and and t thhuuss YY VV.I. .I.
EE
CCo(S). o(S).
a(B,v-a) 2 (f,v-a)
(3.181)
v
ii=
v E
{v E v l v s Y ]
W ee cconsider onsider th W t heenn t thhee
.
, ff E P
We have seen i n Remark 1 . 2 2 t h a t we have
(3.182)
.
u = i l
We s h a l l now provide a p r o b a b i l i s t i c proof of ( 3 . 1 8 2 ) and i n t e r p r e t t h e problem solved. Since t h e c o e f f i c i e n t s of t h e o p e r a t o r A a r e r e g u l a r , it has corresponding Let t o it an I t o e q u a t i o n , which can be i n t e r p r e t e d in t h e strong sense. t h e e x i t time from @, ( r e s p . y x ( t ) be t h e p r o c e s s thus d e f i n e d and T ( r e s p . ):T from
F)
(T:
I-
Tx),
For any stopping time
8, we put
J o and l e t
(3.183)
w(x)
I
Inf
ielYx(e) P
F if
Jx(e) e < 7)
.
We s h a l l show
u = w E Co(6)
and t h e r e e x i s t s an optimal s t o p p i n g t i m e
t o t h e e x i t t i m e from t h e open domain C = { u < 0) u
F
e*
equal
.
( * ) Our aim h e r e not being t o o b t a i n t h e most g e n e r a l theorem p o s s i b l e , but r a t h e r t o show t h a t t h i s V . I . s o l v e s a new problem (which a f t e r t r a n s f o r m a t i o n , however, reduces t o t h e g e n e r a l c a s e ) .
404
OPTIMAL STOPPING PROBLEMS & V.I.'s
(CHAP. 3 )
We n o t e t h a t
Moreover (from t h e g e n e r a l t h e o r y )
where
F i r s t we show t h a t
(3.186)
w ( x ) = E(x).
I n f a c t , i f 8 i s a d m i s s i b l e f o r (3.183) t h e n it i s c l e a r l y a d m i s s i b l e f o r Since
(3.185).
yi(Yx(en7x))
= y(Yx(e))
xeCTx
and yx(e)
d
F if 8 < T~ ( s i n c e 8 i s a d m i s s i b l e ) ,
it t h e n follows t h a t Y ( y x ( 8 ) ) x
~ ,= 0~ and
thus
X
Jx(e) =
jx(e).
Consequently w 2 12.
In o r d e r t o demonstrate t h e o p p o s i t e i n e q u a l i t y , we c o n s i d e r a n a r b i t r a r y s t o p p i n g time 8. From t h e s t r o n g Markov p r o p e r t y , we have
We t h u s s e e t h a t
Observing t h a t T; ( e A T )
( t h e f i r s t i n s t a n t o f e x i t from F subsequent t o 8 A T )
is an a d m i s s i b l e stopping t i m e f o r (3.183), it t h e n f o l l o w s t h a t
i (el
=
J~(TO
Y(eAT)
anu t h u s ti L w, g i v i n g (3.186).
)
(SEC. 3 )
OPTIMAL STOPPING : STATIONARY CASE
405
We next consider t h e penaZised probZem
(3.187)
Bu +-u1
+ XE'f
9
u
n o .
€4
Naturally we have
u (x) = Inf J~(v)
v
where
50'
Jx(v) E = 1
-T 1
f(exp
j:
v
xE
ds)dt
.
We now show t h a t
(3.188)
u
2
w.
I n f a c t suppose we have
2
={u
< O } u F
which i s an open domain s i n c e u CE.
It i s c l e a r t h a t 8
i s continuous.
Let
e be t h e e x i t time from
i s admissible for t h e problem (3.183).
Now, applying
I t o ' s formula, we o b t a i n
*..
Jo
hence (3.185). We now t r y t o e s t a b l i s h an i n e q u a l i t y i n t h e o t h e r d i r e c t i o n . Let 8 be a stopping time and l e t
v e ( s ) = 0 i f s < 8 , o r 1 i f s t 8. We t h u s have
f dt
JZ(v) = E
+E
f(exp
-4f
sne
xE
ds)dt
.
We introduce
E6 =
ISIS
E
9
d(S,bE)
<
6)
and we put
F = E 6
6
uF.
With any admissible stopping times 8 , we a s s o c i a t e an admissible
e 6 defined by
406
OPTIMAL STOPPING PROBLEMS & V . I . ' s
e
i f y(8)
1 F6 and 8
inf { t 2 e l y ( t ) We s h a l l n o t e t h a t i f y ( 8 )
E
(CHAP. 3 )
< T , or 8 t T ,
1 F6}
i f y ( 8 ) € F g and 8 <
Fg and if
e
T.
i s admissible, then 8
<
'E
* y(6)
E
( s i n c e y ( 8 ) cannot belong t o F i f 8 < T). We a l s o remark t h a t i f 8 <
B6
<
7 and
y(B6)
T
and
e6
E dE6
- 36
de)
E E~
> 8 , then e i t h e r
,
or else
e6
I 'c
Thus i f
(3.190)
e <
The6
we have
y(e )
and
6
E
as6 -
b~
. .
Now we have
We n o t e t h a t i f
B6 <
T, then y ( B g ) E
-
E6.
In actual fact either
( * ) We can t a k e 8 = 1 h e r e , but we could g e n e r a l i s e t h e assumptions concerning t h e c o e f f i c i e n t s and apply de Giorgi-Nash. We t h u s t a k e 8 5 1.
E
6
(SEC. 3 )
8
6
=
e
<
OPTIMAL STOPPING : STATIONARY CASE
T
i n which c a s e y ( e )
y(e6) E
E
- E6 ,
6
4
and
e6
< T,
or a l t e r n a t i v e l y
407
so t h a t
e < e6 <
'5
and y(e6)
C[6
+ 6B +
- dE
€ dE6
Now suppose
e6E
P
= infIt 2 e,ly(t)
then
e6 <
T
o
Ef
.
E E d~
y(e6)
But we t h e n have
Now
Consequently, we may w r i t e
It t h e n follows t h a t
UE(x) S v(:J
) g Jx(e6) + C[6
+
B
+ exp
63 S Jx(e)
+
E
+ exp
- -$ 3
'6 and s i n c e 8 i s a r b i t r a r y , we have
-
We have t h u s proved t h e f o l l o w i n g e s t i m a t i o n e r r o r
It n a t u r a l l y r e s u l t s from t h i s t h a t uc converges uniformly t o w.
u
+
u , say i n L2, we c l e a r l y have u = w and u
6
C
6,
E Cot&.
t h e optimal stopping t i m e , a s i n Theorem 3.7.
Since
We t h e n show t h a t
.
408
OPTIMAL STOPPING PROBLEMS & V . I . ' s
Thus we have indeed proved (3.184) and ( 3 . 1 8 2 ) .
3.11.
An
(CHAP. 3 )
8
extension of r e g u l a r i t y .
We give here a p r o b a b i l i s t i c proof of t h e r e g u l a r i t y r e s u l t demonstrated i n Theorem 1.11. We consider t h e V . I .
where
f E L2(a) , JI
(3.196)
i s Lipschitz continuous on
and
6,
+Ir,
2 0 (*)
,
"
(3.1 97) We immediately note t h a t we can t a k e for a ( u , v ) a more general form, with
constant coefficients.
We know (from Theorem 1.11) t h a t we have t h e following r e g u l a r i t y r e s u l t :
(3.1 98)
u i s Lipschitz continuous on
0.
We s h a l l provide a probabiZistic proof of ( 3 . 1 9 8 ) . We may always assume, without l o s s of g e n e r a l i t y , t h a t we have i n a d d i t i o n f = 0,
(3.199)
$Ir
= 0.
The s t a t e of t h e system i s given h e r e by YX(t) = x + w(t)
(3.200)
where w ( t ) i s a standard Wiener p r o c e s s , with values i n Rn. p r e t a t i o n of t h e s o l u t i o n
u
Thus t h e i n t e r -
i s (by v i r t u e of (3.199)
(3.201) with t h e u s u a l n o t a t i o n ( T
X
= e x i t time from t h e open domain
).
A s i n t h e d i r e c t proof of t h e c o n t i n u i t y of t h e right-hand s i d e of ( 3 . 2 0 1 ) , we introduce f o r 6 > 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 <
9 <
T
~
yx(9) ,
E
0-0, and
there exists a point 5
- PI s 6 .
lyx(e)
E
r
such that
As $ ( E ) = 0, we see from the Lipschitz property of $ that we have I+(yX))I
and thus we have
IJx(e)
6
T=S
e < zx
IS
C6
.
I
Ix--xI
1v
s C6 if
- Jx(e:)
We note that we have ( 3.202)
(YX(t)
- Y,,
Consequently if e 6 < + imp1ies d(yx,(t)
m,
(t)
=:
then 9 4 <
, 30) S
6
-
~f and yx(t)
Ix-x'/
We deduce from this the following result: x x*j 6 if 6 2 then ex < ( 3.203) 2 Assuming that 6 2
v,
,
.
t
,v
.
t E [O,T:]
+=r
we then consider
E Qfor ~ t
zxI 2
ex6
.
E
C O , T6 ~ I , which
(CHAP. 3)
OPTIMAL STOPPING PROBLEMS & V . I . ' s
410
so t h a t w e have
(3.204)
aij
E C'(Rn)
a i , a. a
2
, a.,
1J
bounded,
bounded, dxk
L i p s c h i t z continuous on R
Z aij E . 5 . 2 I J
a
Z Ei
n'
5 > 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 <
-
1 (ao+ F v ) d~
.
Since 0 i s r e g u l a r , we can reduce t o c o n s i d e r i n g t h e c a s e 0 = 0. be such t h a t
: N E
8(0) , CN
Ir
=
0
Then, l e t
,
uN -,u i n ~ ~ ( 6 ) . We now c o n s i d e r t h e problem
which possesses a s o l u t i o n
E
tP
EN
.
(Q)
By t e c h n i q u e s a l r e a d y employed, we can i n t e r p r e t u E N by
u
EN
(x,t)
P
Inf J;:(v) V
N where f x t ( v ) i s o b t a i n e d by u s i n g t h e formula ( 4 . 3 4 ) with
GN
replacing
We now e a s i l y s e e t h a t we have
thus
IUEN(X,t)
-u
EM (x,t)
I
Likewise we have
IUEN(X,t)
-
Ct(v)I
-
o
-o
uniformly on
Using i n a d d i t i o n t h e f a c t t h a t
uniformly on
D x [o,to], v t o
.
ux
.
[o,to],
v
to
<.
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.’s
420
uE
~~
+
us
f o r example i n L
2
(8.10,TC )
we t h e n deduce t h a t uE(X,t)
= Inf J:t(~)
,V
t
< T
V
Since(4.33)is cases.
.
c l e a r l y t r u e f o r t = T , we have c l e a r l y proved ( 4 . 3 3 ) i n a l l
I t t h e n f o l l o w s from ( 4 . 3 3 ) and ( 4 . 3 4 ) , and by v i r t u e o f t h e f a c t t h a t ; ( x ) 6 $ ( x , T ) L C , t h a t we have u (x,t) 6 C
(4.35)
(E
) ( independent of x , t ) .
We t h e n i n t r o d u c e t h e f u n c t i o n 6 which i s a s o l u t i o n o f
(4.36)
(4.37)
Next we p u t (4.38)
so t h a t w
WE(X,t)
f
U&(X,t)
- B(x,t)
is a s o l u t i o n of t h e problem
(4.39)
Naturally w addition:
and o f 6 , and i n
s a t i s f i e s t h e r e g u l a r i t y p r o p e r t i e s of u
(4.40)
1
I n f a c t , i n (4.39) t h e nonlinear term i s equal t o - ( u E
E
- $1’
which i s bounded
with respect t o x , t , according t o ( 4 . 3 5 ) . It t h e n s u f f i c e s t o apply t h e r e g u l a r i t y r e s u l t s f o r l i n e a r p a r a b o l i c P.D.E.’s. We now p u t
(4.41
1
w(x,t) = Inf
e
We p r o v e n e x t t h a t we have
~ ~ , ( e- )B ( x , t ) .
(SEC. 4 )
OPTIMAL STOPPING : EVOLUTIONARY CASE
421
(4.42)
E
(4.43)
.
bounded
I t w i l l s u f f i c e t o prove ( 4 . 4 2 ) w i t h h = 0.
LN s u c h
We c a n t h e n f i n d a s e q u e n c e
{
N
iiN pN
We d e n o t e by
-u
E &6)
, lENl
that
bounded ( i n d e p e n d e n t l y o f N) ,
~~l~
~ ~ (, 0 ) = o
in
lpNl
C
BN
E
GN.
We n o t e t h a t
i n d e p e n d e n t l y o f N, x , t ,
p N ( x , t ) -. p ( x , t ) u n i f o r m l y for x E Moreover
.
t h e s o l u t i o n of ( 4 . 3 6 ) corresponding t o
-
8, t E Colto], v
B (Y ( 0)) Xe<,AT
T
We n o t e t h a t
Co(G), t o t h a t ( 4 . 4 2 ) i s t r u e f o r
B &Y ( 8 11x e
to <
a.s.
a n d t h e r e f o r e s i n c e BN i s oounded, we h a v e
ii pN(Y(e))xe<,AT
exp
-
1:
a.
dh
-.
P
-
( y ( ~ ) ) x ~ exp < , ~ ~
1;
.a
d~
In addition
We c a n t h e n p r o c e e d t o t h e l i m i t i n ( 4 . 4 2 ) , w r i t t e n i n t e r m s of
-
3,8,.
Thus w e h a v e c l e a r l y proved ( 4 . 4 2 ) u n d e r t h e a s s u m p t i o n ( 4 . 1 3 ) .
u
Even i n so d o i n g , we know t h a t We now d i s p e n s e w i t h t h e a s s u m p t i o n ( 4 . 4 3 ) . i s bounded above ( b y J l ( x , T ) which i s b o u n d e d ) . We t h e n p u t
E =Ev(-!J) ?J
so t h a t t h i s o n c e more r e d u c e s t o c a s e ( 4 . 4 3 ) .
-
We may t h u s w r i t e ( 4 . 4 2 ) u s i n g Now, when !J
+
+
; +;
P ’ *lJ.
-
L e t B correspond t o u LJ P
422
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
and from t h e maximum p r i n c i p l e
and from t h e monotone convergence theorem, we have t h e same r e s u l t w i t h 5- and 6-. !J
We may proceed t o t h e l i m i t i n ( 4 . 4 2 ) , which i s t h u s proved i n a l l c a s e s .
( 4 . 4 1 ) , we o b t a i n
Using ( 4 . 4 2 ) i n
Uniform convergence of we to
W.
We s h a l l now prove t h e following r e s u l t :
(4.46)
w -. w
~ ' ( 5 .)
in
We f i r s t Once a g a i n , we cannot apply Theorem 4.4 d i r e c t l y , because 5 { C o ( z ) . n o t e t h a t by v i r t u e o f t h e i n t e r p r e t a t i o n o f ue and a l s o o f (4.421, we have
(4.47)
w ( x , t ) = Inf Lxt(') 0.
where we have put
and we observe t h a t
( a l t h o u g h 5 i s not continuous, 5 i s a c t u a l l y continuous i n
8
if
6
<
T).
Based on t h e r e g u l a r i t y p r o p e r t i e s ( 4 . 4 0 ) , we can apply I t o ' s formula, so t h a t we deduce
(SEC. 4)
423
OPTIMAL STOPPING : EVOLUTIONARY CASE
and consequently W'ZW
.
In order to obtain an estimate in the opposite direction, it is more convenient to work with u and Jxt(9), instead of w and Lxt(8), this being permissible since E
J;,(V)
- Jxt(e)
= L:~(v)
- Lxt(e)
, v V,
e
.
Without loss of generality, we may assme f = 0, Q = 0 (by subtracting some convenient functions). We then have
We proceed in a similar manner as for Theorem 4.4, with certain variations. Let 9 be an arbitrary stopping time and let ve( s )
o
if s
< e ,I
if s 2
e
;
we have
ds
+ I1 + 111 + IV + v + VI
.
424
OPTIMAL STOPPING PROBLEMS & V.I.'s
Finally, making
use
of the continuity of $I, as in Theorem 3.7, we have
Finally we obtain the estimate
Iwc(x,t)- w ( x , t ) hence (4.46). It then follows that
Ig
C[6+
3+ p ( 6f ) + exp -
6
3
(CHAP. 3 )
(SEC.
4)
OPTIMAL STOPPING : EVOLUTIONARY CASE
uE(x,t)
inf Jxt(@) V x , t 9
u , f o r example weakly i n L2 (Q), we have c l e a r l y proved ( 4 . 3 0 ) .
and a s u As w
-
425
E
p E C o ( b x [o,T[)
C o ( z ) and
t h e second p a r t o f ( 4 . 3 0 ) c e r t a i n l y h o l d s .
Optimal stopping time. It remains t o show t h a t (4.28) i s a n o p t i m a l s t o p p i n g t i m e . we have
(4.50)
-
9t:
i t
We show f i r s t t h a t
*
I n f a c t , we n o t e t h a t
9xt
I
= i n f (s E [ t , T ]
w(y(a))
=
($-P)(Y(s))~
Thus
(4.51) If i n f a c t
e
< T , we have ( s i n c e i n t h i s c a s e @
W(Y(6))
0
(J,
is c o n t i n u o u s i n 8 )
- B)(Y(b
so t h a t
- P)(Y(e)) A
WE(Y(b 2
(4J
hence ( 4 . 5 1 ) h o l d s good i f 6 < T , and If that
ext
( 4 . 5 1 ) i s c l e a r l y t r u e i f 0 = T.
= t , ( 4 . 5 0 ) i s c l e a r l y t r u e , from
- 65 T - 6
*ex,
t
-.
(4.51).
If
ext
> 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)
<
($-p)(X,t)
b e c a u s e o t h e r w i s e t h e r e s u l t is o b v i o u s . Suppose we have 6 > 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
<
0 and 0'
-
dh
a '/:o
i,thus e 6
to w we obtain
.
4 A.
If 8 < T, we have A < T and therefore since 6 is continuous in w(y(e6)) = + ( y ( e 6 ) )
- p(y(e6)) - T
8. [o,81
we have
6
and proceeding to the limit, we obtain
thus A L e .
Therefore if e < T, we have
eo
4
e.
If 8 = T, we can show that A = T. Otherwise A < T and the preceding-line of argument again applies,-hence ( 4 . 5 3 ) , thus contradicting the fact that 8 = T. Therefore we have e6 4 8 in all cases.
-
w(x,t) = Now if
i<
TAT, we have
Since w = 0 for t =
T
E w(y(Lz))
exp
-
a.
w(y(6)) = ( $ - B ) ( y ( i ) )
dh
.
.
or t = T, we see that
which clearly proves that 8 is optimal.
4.6
sr
is continuous on Q, it follows from (4.52) that we have
Since the function w
m
Extension IV. Further weakening of the assumptions concerning JI. 0 and
We replace (4.29) by
(SEC.
4)
OPTIMAL STOPPING : EVBLUTIONLBY CASE
427
(4.54) measurable and bounded,
J,
-
L2 (0). There exists
E
a sequence J,. u. such that J' J and
+.J 2
6
, +j
,
IQjI S C
S
J,(x,T), then the sequence
We shall note that if ;(x)
4
4J
at all points of 9 ,
i. = ;is
acceptable.
J The maximum weak solution of the V.I., u(x,t), is still defined, in the same way
as the functional J ( e ) , defined in (4.18). xt Theorem 3.8, namely THEOREM
have
4.6.
We then have the equivalent of
Under the asswnptions (4.14), (4.23) and (4.54). we once more
(4.55)
u(x,t) = Inf Jxt(9) 9
where u is the maximwn weak solution of the V . I . (4.24) and where Jxt(9) is defined i n (4.18). Proof.
Let u. be the maximum weak solution of V.I., corresponding to the -J
data values Qj, uj, and let JJ (a) be the corresponding functional. xt Since Theorem 4.5 is applicable to u. Jxt(9), we have J' u.(x,t) = Inf J
9
Arguing as in Theorem 3.8, we see that for t < T, we have
so
that
Furthermore, for all 9 Jit(9) +.
Jxt(e)
-y
and thus, as for Theorem 3.8, we have
uj(x,t)
~ ~ ~ v( t e < )T
the convergence being evident for t = T. We conclude the proof as for Theorem 3.8, making use of the stability theorem, (see Theorem 2.28,Section 2.22).
8
428
OPTIMAL STOPPING PROBLEMS & V . I . ' s
(CHAP. 3)
Remark 4.2. A s i n t h e s t a t i o n a r y case ( s e e Remark 3 . 3 ) , we can no longer a s s e r t , merely under t h e assumptions of Theorem 4 . 6 , t h a t an optimal stopping rn time e x i s t s . 4.7.
Extension V .
Ogerators which a r e not i n diveraence form.
We now consider t h e nonstationary analogue of what was c a r r i e d out i n Section 3.7. However, s i n c e t h e study of two-parameter semi-groups i s r e l a t i v e l y l i t t l e developed, we s h a l l not adopt a s general a framework as t h a t considered i n We s h a l l only i n v e s t i g a t e t h e case of d i f f u s i o n s ( s e e Section Section 3.7. 3 . 7 . 4 ) , which w i l l allow us t o g i v e a theorem f o r e x i s t e n c e and uniqueness i n t h e case of p a r a b o l i c V . I . ' s , with o p e r a t o r not i n divergence form. We consider a family A ( t ) , t E [o,T], of second-order d i f f e r e n t i a l o p e r a t o r s ,
(4.56) where
(4.57)
(4.58)
d
a bounded open c?omain of Rn o f c l a s s C 2 , s a t i s f y i n g ( 3 . 7 8 )
Next suppose we have f , $ ,
satisfying (*)
(4.59)
We define a i n such a way t h a t
aa* -
= a (where a = ( a ) ) , ij
We t a k e Q = Co([o,
: 3
E
-I
; R")
,
a-algebra generated by
w(h),h E [t,s], V t 5
From t h e theory of STROOCK-VARADHAN [ll, f o r any x
E
Rn, t
E
[o,-[,
B
there
e x i s t s a p r o b a b i l i t y measure PXt on (Q,3+.)anda process w ( s ) ( s L t ) which i s a t xt s t a n d a r d i s e d Wiener process and a as martingale f o r t h e measure P , such t h a t i f t we put ( * ) To simplify somewhat we s h a l l r e s t r i c t our a t t e n t i o n t o t h e homogeneous case ( d a t a values zero on C = r x Io,T[).
(SEC. 4)
OPTIMAL STOPPING
; w) = w(s)
?(a t h e n we have f o r s
E
429
EVOLUTIONARY CASE
:
Ct,
- st" -[
i:
g(Y(h),h) dh +
x +
?(s)
o ( Y ( A ) J h ) dwt(h)
.
We d e s c r i b e as a d i f f u s i o n stopped on e x i t from 8, t h e p r o c e s s
T
s E
Y(sAz)
y(s) =
where
[tt
-[
i s t h e e x i t t i m e from @ o f y ( s ) .
Fo r any s t o p p i n g time 6
(4.60)
E
C t , T I , we d e f i n e t h e f u n c t i o n a l
st
Jxt(e) =
;1
- a(A-t)
f(Y(h),h) exp
+ sIt Q (g(e),e) xe
dh
- a(8-t)
exp
where a t o (we could t a k e a more g e n e r a l a c t u a l i s a t i o n f a c t o r a ) .
u
Furthermore we c o n s i d e r t h e following problem: f i n d
E
(4.61 )
us
JP (Q) Q
bU
--+B t
v
(u-Q)(.-
9
x , t E rj ; uIz = 0
A(t)u
%+
+
satisfying
a USf
A(t)u
+
au
u(x,T) =
a.e.
- f) =
in 0
Q
;(XI
J
, a.e.
in
9
.
We s h a l l now prove
THEOREM 4.7. Under t h e asswnptions (4.57), (4.58), (4.59),there e x i s t s one and only one solution u of problem (4.61). Moreover we have u ( x , t ) = Inf Jxt(e)
e
where J x t ( e ) is defined by (4.60). Lastly
et
Lnf
(8
E Ct,T]
/ ~ ( 9 ( 8 ) J a=)
+ (Y(s),s)]
i s an optimal stopping time. Proof. (4.62)
(4.63)
We f i r s t s t u d y t h e p e n a l i s e d problem: f i n d u
such t h a t
OPTIMAL STOPPING PROBLEMS & V.I.'s
430
- E +A ( t ) bWO
(4.64)
Wol c
=
0
w
0
+a
Wo(X,'P)
J
wo i
I
(CHAP. 3 )
f
c(X)
lD2'lJP(Q)
Wo
.
(4.65) I n o r d e r t o i n t e r p r e t wo, we f i r s t of a l l suppose t h a t we have
f 6 &(Q)
(4.66)
zn E
Let
J
c
Fn(xJt) = Suppose we have h > 0.
do) GI,
, zn
c2s1
We p u t
6
-
J
wo
bz
-dt n + A(t)
= 0
in
~P~J"P(Q)
zn + a zn
P
gtzn
(Y(t+h), t+h) exp
[
- ah -
(zAT)V(t+h)
- EXt ,..
*-
.
From I t o ' s formula, we have
Ext z n ( i ( ( r A T ) v ( t + h ) ) ,(TAgV(t+h)) exp
(4.66)'
.
a(s-t) Fn(Y(s)Js) ds
.
-
a ( ( r A T ) V ( t + h )- t )
-
But y ( s ) p o s s e s s e s f o r almost a l l s > t , a p r o b a b i l i t y d e n s i t y p ( x , t ; S , s ) we have such t h a t ( c f . STROOCK-VARADHAN 111, p.490) f o r a l l B > 1, B <
t h u s also
However t h e n
OPTIMAL STOPPING
(SEC. 4 )
+ 0 because
Fn + f
:
.
i n Lp(Q)
n + m
+
But f o r p > z
-f
1, we haveb2'1yP(Q) c C 0 ( G ) with continuous i n j e c t i o n , t h u s
Proceeding t o t h e l i m i t i n
wo i n C o ( G ) .
stw'(f'((zAT)V(t+h)), I
But wo theorem.
E
n
(zAT)V(t+h))
- ah -
EXt wo (g(t+h),t+h)exp
Co(G)
4 31
EVOLUTIONARY CASE
and f E ~ ( Q ) .
s"'
i n ( 4 . 6 6 ) ' we o b t a i n
exp
- a((TAT)V(t+h)-t)
(zAT)V(t+h) eva(s-t) f EXt !t+h
We can t h u s l e t h +
0,
(r(s) ,s)ds
.
by v i r t u e o f Lebesgue's
We deduce from t h i s t h e r e p r e s e n t a t i o n formula
(4.67)
wO(x,t)
EXt
P
+ Ext i
t
f(g(s),s)exp
(g(T))exp
which i s t h e r e f o r e v a l i d i f f ,
- a(s-t)ds
- a(T-t)xTcT-
u satisfy
(4.66).
But ( 4 . 6 5 ) i m p l i e s , s i n c e
I f now f 6 L P ( Q ) , ii E ~ ~ " ( n0 is) ' P ( ~ ) ,we can f i n d a sequence f , s a t i s f y i n g (4.66) such t h a t
fN-f
in
LP(P)
cN-ii
in
w2?P(@
Suppose we have
.
w i corresponding t o f N ,
$.
We w r i t e (4.67) f o r f N , 7$,
wi
and we proceed t o t h e l i m i t by v i r t u e of ( 4 . 6 7 ) ' . We again o b t a i n ( 4 . 6 7 ) , s i n c e t h e right-hand s i d e of ( 4 . 6 7 ) i s independent of t h e r e p r e s e n t a t i v e f chosen. We next consider an i t e r a t i v e scheme; we d e f i n e wn+'
n+l
(4.68)
--d Wd t w nal . wn+l
+ A(t) wn+l+ a wn+' + +(wn-
I z = 0 , w"+'(x,T) b2,1,~(Q)
.
= :(x)
,
+I+
from w" by
= f
,
OPTIMAL STOPPING PROBLEMS
432
&
V.I.'s
(CHAP. 3 )
From the probabilistic interpretation, we deduce that wn+l ( x , t )
- wn(x,t)
E
and thus considering wn(t)
so
[(wn-'-
EXt I
o)+(s(s))-
wn(x,t) as an element from Co(6), V t , we obtain
that we deduce, as with Theorem 3.10, that the series wo + w1 - w0 +
converges in Co(Co,TI ; wn
+
Co(&)),
in
(2-d + ( s ( s ) ) l d s
- -...+
wn+l - w"
to a function w.
+
- -
.
Thus
~'(6).
In particular, wn remains within a bounded subset of Co(G). Utilising the a priori estimates from the linear parabolic problem, we then deduce that wn remains within a bounded subset of IP~'~'~(Q). Thus wn
+
w weakly in b2""(Q)
and strongly in C o ( ? L ) .
We c a n then proceed to the limit in (4.68),from which we deduce that w is a solution of (4.62). Uniqueness is demonstrated using the same type of estimates. In fact if w1 and w2 are solutions, then arguing as above we see that we have
and thus, from Gronwall's lemma we necessarily have w l = w2. Then we give an interpretation of u
.
Employing arguments which we have
used several times already, we have (4.69)
Due to the regularity of the functions $I We have, in fact,
, we can transform expression (4.70).
(SEC. 4 )
OPTIMAL STOPPING : EVOLUTIONARY CASE
433
thus
But t h e n we have
st
and from t h e l a s t assumption of ( 4 . 5 9 ) , it follows t h a t (ue- +)(x,t)
s C EXt
T (exp
- (a+$(s-t))ds 1
=
C,E
But then we o b t a i n
Using (4.71) i n (4.62), we deduce from t h e estimates r e l a t i n g t o t h e l i n e a r parabolic problems t h a t we have
(4.72) We then e x t r a c t a subsequence again denoted by u
u We have u
E
Co(G).
4
u
such t h a t
weakly i n l@2'1ap(Q). Moreover s i n c e t h e i n j e c t i o n mapping from lf2a1'p(Q)
into
L p ( Q ) is compact, we have, a p a r t from e x t r a c t i n g an a d d i t i o n a l sub-sequence,
u -. u s t r o n g l y and a . e . i n Lp(Q). From t h e c o n t i n u i t y of t h e t r a c e s , we have u I x = 0 and u(x,T) = i ( x ) . Arguing as i n t h e proof of Theorem 3.14, we r e a d i l y deduce t h a t u i s a solWe 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 u t i o n of problem ( 4 . 6 1 ) . problem ( 4 . 6 1 ) . We now show t h a t every s o l u t i o n u of ( 4 . 6 1 ) i s such t h a t
434
OPTIMAL STOPPING PROBLEMS & V . I . ' s
u ( x , t ) = Inf
(4.73)
e
(CHAP. 3 )
Jxt(e)
which, o f c o u r s e , a l s o p r o v e s u n i q u e n e s s . Now t h e proof o f ( 4 . 7 3 ) i s i d e n t i c a l t o t h a t o f Theorem 4 . 1 ( w i t h t h e s t r o n g d i f f u s i o n s r e p l a c e d by weak d i f f u s i o n s ) , hence t h e r e s u l t . The method f o r Theorem 4 . 1 a l s o shows t h a t Bt
i s an optimal stopping time.
I
Remark 4.3. When Ji i s no l o n g e r r e g u l a r , it i s no l o n g e r o b v i o u s t h a t problem ( 4 . 6 1 ) p o s s e s s e s a s o l u t i o n . I n t h i s c a s e we can u t i l i s e t h e nonhomogeneous semi-group O ( s , t ) , s 2 t , a s s o c i a t e d w i t h t h e weak d i f f u s i o n s t o p p e d on e x i t from @and we can o b t a i n t h e a n a l o g u e o f ( 3 . 1 4 2 ) . We c o n s i d e r t h e s e t of w(x,t) satisfying
(4.74)
t h i s s e t t h e n p o s s e s s e s a maximum element u = I n f J x t ( B ) .
I
e
Remark
4.4.
Estimation e r r o r .
We can o b t a i n t h e analogue o f Remark 3.2, n m e l y
where
a
i s chosen i n such a way t h a t
Remark 4 . 5 .
l + a
>
a
"+1.
I
2
( N o n - c y l i n d r i c a l open d o m a i n s ) .
We a d o p t t h e s e t t i n g o f Remark 2.18. We now assume t h a t Q i s a nonc y l i n d r i c a l open domain i n Rn x R t , c o n t a i n e d i n 0 < t < T . We d e n o t e by
T
xt
t h e e x i t t i m e from Q o f t h e p r o c e s s ( y ( s ) , s ) , xt
We n o t e t h a t t S T 2.18 i s a g a i n g i v e n $?
where
S
T.
The m a x i m u m weak s o l u t i o n
u(x,t) = Inf Jxt(e) t5BST
u
i.e,
d e f i n e d i n Remark
4)
(SEC.
4.8.
OPTIMAL STOPPING
:
EVOLUTIONARY CASE
435
I n f i n i t e horizon
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 ( 4 . 1 4 ) w i t h Q = Bx l o , -[, ( @ b e i n g assumed bounded i n o r d e r t o c l a r i f y t h e c o n c e p t s ) . We d e f i n e t h e t r a j e c t o r y y x t ( s ) a s t h e s o l u t i o n of
dy = o(Y(s),s) dw(s)
s
> 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(<).
en
1;
(exp
( x , t ) E Co(O x [o,n])
.
-
1;
(ao
and t h u s z=(x,t) E
Co(<).
dA) d s
+
z E ( x , t ) when n -+ +
Furthermore we have
,
P u t t i n g zn = 0 f o r t 2 n, we have
Now
exp
z
+%)
-, uniformly
in x,t
- y(s-t) E
<.
ds
Therefore
440
OPTIMAL STOPPING PROBLEMS & V.I.'s
z 2 z since
e
E
UI
v
2 zn
n
.
8 , and 0 i f s 2 8 , it Also, t a k i n g f o r any stopping times 8 . v e ( s ) = 1 i f s can e a s i l y be seen, using arguments a l r e a d y given (based on t h e f a c t t h a t f i s bounded) t h a t
z s z c c c and t h u s
Thus we c l e a r l y have z general c a s e l e t
FN Let z
N
Z.
-+ z uniformly on
z
E
Co(z),
, FN
E B(Q)
i f f c D ( Q ) . To demonstrate t h i s i n t h e
-.?
e-yt
e+
LP(Q)
in
be t h e f u n c t i o n ( 4 . 8 5 ) corresponding t o fN.
.
For a l l 8 we have
and t h u s
from which it follows t h a t
z N ( x , t ) e-yt uniformly on
2.
Since zN
E
-
z(x,t) e+
Co(z),
we c l e a r l y have z
E
Co(z).
We s h a l l now show t h a t we have
In fact if t
S
z n ( y x t ( s ) , s ) < 0.
s < 8 , then f o r
-
consider t h e compact s e t
z
n
n
s u f f i c i e n t l y l a r g e we have s < 8
Since zn t z, it follows t h a t z ( y x t ( s ) , s ) < 0 .
e
a x
[t,el.
uniformly on
From D i n i ' s lemma,
6 x [t,S]
.
NOW
and thus
We furthermore
(SEC.4)
OPTIMAL STOPPING : EVOLUTIONARY CASE
441
hence ( 4 . 8 9 ) . I n order t o prove t h e theorem, it is c l e a r l y s u f f i c i e n t t o prove t h a t we have
(4.91)
-
dZ
(=
C-z)
+ a(t;z,c-z)
2
(H,c-z)
V C ~ V l C s O l z s OI e-yt z E ~ ~ ( ,v e-yt )
E E dt
L~(H)
.
Since u-J) i s obviously a s o l u t i o n of ( 4 . 9 1 ) , we can c l e a r l y see t h a t (4.90) m holds.
Remark 4.6. When 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 a r e r e g u l a r (Lipschitz continuous) and when we can use a strong formulation, it i s s u f f i c i e n t i n (4.78) t o have yo > 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
<
T
,x
€ Rn
Here t h e t r a j e c t o r y y X t ( s ) i s simply
(4.99)
Yxt(S)
= x
+ v2(w(s)
-w(t))
and t h e f u n c t i o n J x t ( e ) can b e w r i t t e n
(4.1 00)
we assine (4.97) and
THEOREM 4 . 9 .
E
(4.101)
f
(4.1 02)
uE
L Y Rx~3L"(R~)
-,T[)
, ~2 - M
(*)
, f 2 fo > o
;
then (4.1 03)
u 1 0
for
t S T o = T - -
M fO
Proof.
From ( 4 . 1 0 0 ) we have Jxt(e) 2 fo E ( 0 - t )
= f o E[(8-t)Xe=T
2 (fo(T-t) if t 2 T
.
- M)
+
- M P(&T) (e-t)
P(&T)
Xe
.
- M P (&T)
2 0
But t h e n u ( x , t ) B 0 f o r t 2 To;
hence ( 4 . 1 0 3 ) s i n c e u 2 0.
We now g i v e t h e analogue o f Theorem 2.19.
( * ) We c o u l d t a k e (--,TI
i n s t e a d o f (0,T) ( s e e Theorem 2 . 1 8 ) .
I
444
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
We adopt t h e conditions of Theorem 4.9.
THEOREM 4.10.
that
We asswne i n addition
We then have, with u again denoting t h e solution o f t h e V . I . :
Proof.
Let S =
(XI
1x1
< ro]
process
A
-. C;
+ y2(w(A)
-
.
We d e n o t e by
~ ( 8 ) )A ~2
where f o r s i m p l i c i t y we have w r i t t e n T = T xt.
xe=T and hence t h e r e s u l t .
thus
J(e)
Iyxt(T)
I 2 ro
6s
t h e e n t r y time i n t o S of t h e
.
I n f a c t we have
'
We n e x t c o n s i d e r
- JE(e) 2 - f o
.
E
S i n c e we a l s o have T = 7 t h a t if we p u t
se
'7
s
'I
y(eAr), enz 1
aE(~,s) = E e - F ( T E ~
it follows from t h e s t r o n g Markov p r o p e r t y
-
8)
then
J E ( e ) = f o E (BAT-t)
+ fo
E
- (foe + M)E OE
(y(BhT),
BAT)
.
(SEC.
4)
OPTIMAL STOPPING
EVOLUTIONARY CASE
:
We t h e n w r i t e , a s i n Theorem 3.21,
and we have
Furthermore, we a l s o have - 1
If
7
Es
$
s +
Y E , then
15 +
there exists h
.\y2 (w(h)
t h u s t h e r e e x i s t s h E [s,s
Consequently, we have
- 4s)) I + YE]
E [sts
< ro
such t h a t
+ yE]such
that
445
446
(CHAP. 3 )
C o l l e c t i n g t h e s e r e s u l t s t o g e t h e r , we o b t a i n
o r , depending on t h e choice o f h ,
JE(e) 2 E
(eAT-t)[f,-
(fo
E
+ M)
$- 1
C L(T-t) hp
1
-
But t h e n , s i n c e J(8) 2 JE(e) f E , we can r e a d i l y deduce, by l e t t i n g c t e n d a t o 0 , t h a t u ( x , t ) L 0 , and hence th% r e s u l t .
Remark 4.8.
h= 2
If we t a k e p = 2 , we o b t a i n
gave us t h e b e t t e r e s t i m a t e
h = 0
.
I
, whereas
Theorem 2 . 1 9
8
Remark 4.9. Using t h e same t y p e o f methods, Theorems 4.9 and 4.10 may be extended t o t h e c a s e o f d i f f e r e n t i a l o p e r a t o r s A ( t ) more g e n e r a l t h a n - A . We now g i v e a n o t h e r t y p e o f theorem c o n c e p i n g t h e s u p p o r t s . assume (4.104) and we c a l l S t h e support of u .
We no l o n g e r
We have
THEOREM 4.11.
Under t h e assumptions of Theorem 4.9. we consider for a > 0
8
(SEC. 4) and
OPTIMAL STOPPING
:
EVOLUTIONARY CASE
P 2 2 the points (x,t) such that
Let Q
aP
be the set o f points s a t i s f y i r g (4.106). Then we have
u = 0
(4.1 07)
Proof.
( x , t ) E Qap
for
,
for
T-t
s u f f i c i e n t l y small.
Let x,(x) be the characteristic function of S .
Jxt(e)
2 fo E(8-t)
- M 1 xS(Yxt(T)
xbT
We have for all 8
.
Next we put
We once more have J(e)
We then define
8'
- J'(e)
2
- fo P .
by
s-e
ds =
(T-el)
It is clear that 8 ' is ' 3 measurable.
exp
- 8'-' .
Furthermore we have
447
L48
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
and as i n t h e preceding theorem we s e e t h a t t h i s is bounded above by
We t a k e
h
as i n t h e preceding theorem.
S(e)2 E
(4.1 09)
It then follows t h a t
- -Ifo (T-8) - M z] 8'E
exp
e
where we have put 2 = P(Y'(e)18;S,T) xly(B)-xl
(4.108)we have
Now from
P(y(e),e;s,T) =
s
de)+
<
h
lyu E s
But when ly(e)-xl < h , we have
~ ~ ( = w )(A E
y(e)
thus
I f we put
B =
h
2(T-8)
'12
+ lyu
E
s]
exp
-
1Al2 2(T-8)
ah ( & l n / 2 (T-8)'/2
.
(SEC.
4)
449
OPTIMAL STOPPING : EVOLUTIONARY CASE
we obtain
(where d = d(x,S)). F o r rl
E
lO,l[,
we can define C(q) such that
(exp
- r2
n-1 r dr
'Fnus it follows that
s C, c ( d
I(W)
exp
~ ( q )exp
- ~~(1-r))
v PBo
.
(d-h)* -(14.
4(T-e)
Since, for T - t sufficiently small, the function z + z' lLog z I 3 is increasing on CO,T-tl, it follows from the choice of x (see (4.106))that we have
d2 h
+
(2+ a ) ( T - e )
f
/Log(T-8) I 4r
and consequently we have
We now choose
rl
such that
It follows from this that ~ ( w )s
c1
c(d
exp
- ( I + allI ~ o g(T-e)[
Consequently, we have
fo(T-9)
-I
2 2 fo(T-8) P
=
C,
c(q)
- M C1 C(q)(T-9)'+
if T-t satisfies the condition
>
fo- M C1 C(q) (T-t)'?
Consequently, for (x,t) E Q
CrP
<,(el
2 o
1'
(T-e)[fo- M C, C(q) (T-9)'1]
2 (T-e)rfo- M C1 C(q) (T-t)'l]
(4.1 10)
(T-e)I+ 'I
0
a
0
.
and T-t sufficiently small, we have
, v e,
g
,
450
OPTIMAL STOPPING PROBLEkB & V.I.'s
so t h a t u
(4.1 1 1
2
(CHAP.
0 , which c o n c l u d e s t h e p r o o f o f t h e theorem.
a.
lj
= a j i : Zaij
,
ai E C ' ( Q a. 2 b a.. dai
2
'
We now t a k e
ba
dXk
"k
aij
bounded
a.
da.
,
,
+0
da.
A,ZJ,& bt
'
bounded
,
C2"(q)
c'(P) ,
ba.
:
I
da Bt
bounded
.
satisfying
f
If
(4.1 13)
E
2
> 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
(e-t)
we c o n s i d e r t h e sequence
t
+ nk
t
if
if
8 = t
t
+ (n-l)k < 8s nk + t
.
,nl
1
,
(SEC.
4)
OPTIMAL STOPPING : EVOLUTIONARY CASE
459
We observe t h a t f o r k s u f f i c i e n t l y small we have
and t h u s s i n c e t h e f u n c t i o n $ Now 0 prove (4.156) for 'ak.
t, t + k, and f o r
k
is continuous and bounded, it w i l l s u f f i c e t o only t a k e s a f i n i t e number o f v a l u e s , namely
.., t
+ Nk
n? 1 (0, = t + nk)
I
It
+
(n-1)k
<
n k
Furthermore
Now we have
Consequently, combining t h e s e r e s u l t s we o b t a i n
i.e.,
(4.156) with
'a
= 'ak.
We s h a l l now prove t h a t we have
+
t)
t+nk
c Zt
.
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
460
In f a c t , we have
hence (4.157). Using t h e r e l a t i o n s ( 4 . 1 5 6 ) , (4.155) and (4.157) i n ( 4 . 1 5 4 ) , we o b t a i n , from t h e d e f i n i t i o n of J x t ( e )
(4.1 58)
u(x,t)
.
s Jxt(e)
We now t a k e 8 = 8 , a s defined by ( 4 . 1 5 2 ) . This time we have t h e r e l a t i o n (4.154), with e q u a l i t y in2tead of i n e q u a l i t y . By a s i m i l a r argument t o t h a t used previously we show t h a t (observing t h a t 8 is c l e a r l y a stopping time with r e s p e c t t o Z!),
-
We have t h e r e f o r e proved t h e theorem i n t h e case where f , u a r e bounded. We now consider t h e general case,
(4.113), (4.114)on f , iI we have (4.1 60)
f E Lp'@(Q)
and
,
F i r s t we n o t e t h a t , i n view o f t h e assumptions
E L"'(0)
with
p
arbitrary,
p
E
[l, m
[
p > 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
Au
+ aul f
if
u =
AU
+
if
u = GZ
auL f
$1
on
.
C 2 0
,
'
,
a l l these i n e q u a l i t i e s holding a.e. of
( 5 . 6 ) and
give an interpretation of u as the value of a stochastic differential game where the decisions of the players are stopping times.
5.1.2
Penalised problem
Suppose we h a v e € > 0. we have
We i n t r o d u c e t h e f o l l o w i n g problem.
Find u
-E
such t h a t
Before i n v e s t i g a t i n g t h e e x i s t e n c e and u n i q u e n e s s o f ( 5 . 7 1 , we c o n s i d e r a s t o c h a s t i c d i f f e r e n t i a l game problem a s s o c i a t e d w i t h ( 5 . 7 ) . F i r s t o f a l l t h e r e c o r r e s p o n d s t o t h e p a i r o , g i n a unique manner a weak s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n . Suppose Q
=
,
C ( c 0 , m[;Rn)
t
3 = o - a l g e b r a g e n e r a t e d by w(a) y(t;o)
= w(t)
.
,
01 ag t
,
and
There e x i s t s one and o n l y one p r o b a b i l i t y measure Px on (n,$) f o r which we can c o n s t r u c t a s t a n d a r d i s e d Wiener p r o c e s s which i s a Zt m a r t i n g a l e and such t h a t we have
Suppose we t h e n have two s c a l a r p r o c e s s e s v l ( t ) , v 2 ( t ) adapted t o that
at
and such
464
OPTIMAL STOPPING PROBLEMS & V.I.'s
(5.9)
a.s.
v i ( t ) E [0,1]
Let T be t h e time a t which
a&
,v
t
i s reached.
(CHAP. 3 )
. We put
We now prove THEOREM 5 . 1 Under t h e asswnptions (5.1), ( 5 . 2 ) and ( 5 . 5 ) , t h e r e e x i s t s one and only one solution u of ( 5 . 7 ) and furthermore we have
u ( x ) = Kin Max
(5.1 1 )
v2
Proof. For
9
.
< ( v 1 , v 2 ) = Max Min < ( v 1 , v 2 )
v1
v1
v2
F i r s t , we assume t h a t a > 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
<
+,(Y(t))
Consequently, from (5.27) we have
f (y(t))exp
- at
dt
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.’s
472
C
and we have
hence
I
(5.31).
We can assemble t h e r e s u l t s o b t a i n e d i n t o
Under t h e assumptions (5.lj, (5.2) and (5.5) there e x i s t s one THEOREM 5.2. and only one solution of problem (5.6). The s o l u t i o n u(x) i s t h e value of t h e stochastic game with stopping time (5.26), where t h e functional Jx(S1,S2) i s defined i n (5.25). There e x i s t s a saddle point of t h e game, S1, S2 f o r which t h e values are given by (5.28). ,
.
A
Remark 5.2. The e s t i m a t e s (5.22) and (5.23) have c l e a r l y played a fundamental However if t h e o p e r a t o r A can be c a s t i n r o l e i n obtaining the desired r e s u l t . divergence form, which supposes t h e a . . a r e Lipschitz continuous t h e n we can 1 J obtain t h e estimate
(5.32) without assuming i n
(5.5) f
t h a t we have
- AJ12 - aJ12S C
Let u s , i n f a c t , w r i t e
dx, 8
A = - Z
ij
,
f
- Ajil - a$, 1 - 2 .
b + dx .
aij
1
Z
J
i
a
5, 2
ai
b axi
0 V
Si,So
We assume t h a t we have ( c f . ( 1 . 7 8 ) ) .
(5.33)
61-11Z aij Sicj + X
We deduce from
(a
+
(5.7)t h a t
A) (us- +2)
We m u l t i p l y by [ ( u E - $I*) Observing t h a t
aiSi5,+
+
1 (uE-
+ 3 p-1
+,I
+
2
- E1 ( ~ c -
E
4~~)-= f
(which i s zero on l’).
R
.
- AJ12 .
473
STOCHASTIC DIFFERENTIAL GAMES
5)
(SEC.
thus
Similarly
These e s t i m a t e s a r e c l e a r l y l e s s p r e c i s e t h a n ( 5 . 2 2 ) and ( 5 . 2 3 ) but they
1
s u f f i c e t o prove ( 5 . 3 2 ) , s i n c e f - ;(uE subset o f Lp(@)
.
for
- q2)
Remark 5.3. U(X).
- EX We t n e n put
(5.34) We t h u s have
(5.36)
(5.37)
1 - q1)+ -(u E E
I t w i l l be u s e f u l i n what fol:ows We n o t e t h a t we have
EX $i(~(S1AS2Az)) exp
(5.35)
+
S1 A S 2 A ~ 0
(Qi
+
remains i n a bounded
t o g i v e t h e e q u i v a l e n t of ( 5 . 3 2 )
- aSIAS2AT
= +i(x)
a Q i ) ( y ( t ) ) exp
- at
dt
.
474
OPTIMAL STOPPING PROBLEMS & V . I . ' s
Vie(t)
(5.40)
q i p =
for t <
1
for S.
1E
siE s t
f o r t t S.
fiit(t)
(CHAP. 3 )
1E
< S.
1E
+ h
+ h.
But we have ( c f , proof of Theorem 5.1), (5.41)
Jf(Vl,,v2) 2 u,(x)
t
-a$ !
(exp
:ASl
- at)(exp -
1:
v 2 ds)
-- ( VIE d s ) [ V , , ( t ) ( U , b ( t ) ) - +,(y(t))) TAsl + (u,(Y(t)) - +,(Y(t)))-] d t = u,(x) - EX X . x (exp
E
slE
We note t h a t X We have
2 0.
We examine t h e values of X according t o t h e values of
X = 0, i f S
1E
' T*
If SIE 5 T , we d i s t i n g u i s h two c a s e s : a)
Case a ) We have
thus
S1,
<
7-h
,
b)
S l e5 2-h
.
475
STOCHASTIC DIFFERENTIAL GAMES
xs
S
be- JI,
sup
.
I(Y(t))
12
Case b ) We have n
thus
We can combine c a s e s a ) and b ) i n t o a s i n g l e e s t i m a t e , namely
Ex X S EX {
(5.42)
+
sup
s1 E-
h
Iue-
Iu,- G1 I ( Y ( t ) )
sup
s ctsr
JIl l ( y ( t ) ) x
x
1 c-
W2 , P (a)
inequalities,
(5.43) where a, C
C
IluEII
But from t h e e s t i m a t e
a r e independent o f
E,
p1 (6)
=
sup x,x'E
u,
Uc(Y(slE))
SUP
v
x,xI
t h e function
I q X )
-q
- JI1(Y(SlE)) = 0
Moreover i f we have S1 cSt S S 1 E+
] =
Ex{X1+ X2]
c
u
x 9
1
-
0 with 6
,
- y ( S I E )I S
lY(t) h
t h e n it f o l l o w s from t h i s t h a t
X1 S CoAa + p1 (A)
,
I n a d d i t i o n we have
PX(
sup
S l i t
(*I
The c o n s t a n t s
c
q,+ h
pY(t)
.
( * ) , we deduce, u s i n g t h e Sobolev
(x-x' 1 26
Now we have
h
a > 0.
Furthermore, from t h e c o n t i n u i t y o f JI,,
15.44)
EC
1' E+h $z>S1
- uc(xI) 1s c o I x a ~ I a,
IUJX)
1'
X 2 S Coha
- Y(S1,)
+
p , (A)
2 A j S C
. h -
A*
denoting v a r i o u s c o n s t a n t s , independent o f
E .
*
476
OPTIMPL STOPPING PROBLEMS & V.I.'s
(CHAP. 3)
and t h u s , having r e g a r d t o t h e f a c t t h a t Iu ( x ) I 5 C , we o b t a i n
ExXS We choose
X = hG.
C
h
[-
A*
+
ha
+
pl(h)]
.
Consequently, we have demonstrated t h e e s t i m a t e
I n ( 5 . 4 5 ) , t h e constant C is independent of
E,
x, h , v2.
Moreover h and
,-.
a r e a r b i t r a r y and Bls is independent of
it would be p r e f e r a b l e t o w r i t e v" Is = vleh)'
A s i m i l a r argument allows u s t o demonstrate t h e e s t i m a t e
where p i s d e f i n e d i n t h e same vay a s p1 w i t h $, i n p l a c e of 2
I ) ~ .
We s h a l l now e s t a b l i s h some new e s t i m a t e s . With t h e t r i p l e t ( S o2 , Let So be an arbitrary stopping time. 2 s h a l l a s s o c i a t e a c o n t r o l V 2 ( t ) d e f i n e d a s follows
slE,
(5.47)
t
We note t h a t V o ( t ) i s a p r o c e s s adapted t o 3 s i n c e 2
la;(t) = i j = ~ since
slE, SO2
a r e stopping times.
Y
2
v2 but depends on the choice of h ( t h u s
s > , 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;<
-r ,a s1
exp
- -: I
+ 2 (exp
- as:}
exp
,I
TAS,
J
Vi)ds)(exp
AS; (O1 E+
dt
- at)dt
s1 CAT
To abbreviate this expression, we put
g(Blc’B;)
f
+ E” I + E” I1 + E” I11
f(Slc,S.,,
.
We shall now estimate the quantities I, 11, 111. We have
We consider first the quantity I. I = O We now assume S
1E
Case Ia:
SIE A $ 2
if
.
7
A So < r ; we then have two possibilities. 2
e S1c-
s l E<
a d
SO
2
.
z
Then
f exp
(5.49)
- at exp -
n
l
J
t
V I E ds
.
s1
We divide this into two sub-cases.
Sub-case 1 : &1
S I E + h < T .
We have
I=
+
PE+
f exp -at (exp
s1 E
h
f(exp
- )t-S1 dE
- at)(exp
LIE+
f1Eds)exp
h
Thus
If I(exp
(5.50)
- at)(exp If \(exp
Sub-case I : a2 Then
/I/
+h)
S,E
s4,
E
t
z
/f/exp
- -t-S1)dt
E
- at)dt .
. - at
exp
- +;-slE)dt
.
- Thd t .
478
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
Thus combining the two sub-cases into a single formula we can write:
III
(5.51) Case Ib:
If
s1 E
> S;
and S;
I
(exp
<
z
- -t-S1 )dt
E
+ exp
If/dt
.
.
We have
We once more consider two sub-cases. Sub-case
I
.
s1
b, '
f exp
L
> s;
- at exp
t-s;
, so2 <
Is T
Sle$
J
--
dt +
.
T
t
7
f(exp
- $ /s,(VIL+
6;)ds)exp
1E
from which, as for ( 5 . 5 1 ) , we easily deduce
Sub-case Ib2 :
f(exp
- at)
t-s;
exp
,
s;
SIE > s ;
--
dt
S I E > 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
.
<
We then have I11 = 0 and thus
I1 + I11 =
We note that
1 exp
LIE .
blE(s)ds) exp
- at dt
-
- as1
We then consider two sub-cases: Sub-case IIa Then I1
+ I11
:
1
SIE + h
<
.
T
- -!$t-SIE))exp
I
-
at dt
- JI1 (y(SIE))(exP - as,E) ds)exp
-", 1
4~~ (exp
thus
Sub-case 11 : &2
Then
s1 E
+ h > T > SIE
.
-7
- at dt)exp - -h
V1 Eds) (exp
E
- at)dt
480
OPTIMAL STOPPING PROBLEMS & V.I.'s
But we have
We can summarise the two sub-cases in a single formula
We now consider
and since $I
S
q 2 , we see that
(CHAP. 3 )
(SEC. 5 )
STOCHASTIC DIFFERENTIAL GAMES
We t h e n c o n s i d e r two sub-cases.
SLtb-case (11 + I I I )
[Si
,
Si + h[
<
+h
S;
V I E + ?: = 1
We n o t e t h a t containing
:
7
for
481
. ,
S I E + h[
t E S[,:
t h i s interval
,thus
t -s;
--)exp--
at dt
(4, E+ vi)+2(eXp
- +,(y(S;))exp - a
So 2
(VIE+ V i ) d s ) e x p
-
--h . E
We a r e t h e n i n a s i t u a t i o n a l r e a d y e n c o u n t e r e d i n t h e a n a l y s i s o f sub-case 11,.
1
+
, with
Sub-case (11 + I I I )
:
b2
We have
so
+
2
h i z
4;
.
We t h u s have
.
I n view o f t h e f a c t t h a t J12(y(')) 2 0 , we s e e t h a t we can w r i t e
W e can combine f o r m u l a s ( 5 . 5 5 ) and ( 5 . 5 6 ) i n t o one :
(5.57) 11 + 111 I
C[E
+ exp
- -h+
Thus, u s i n g arguments a l r e a d y p u t
fact that
/u,(x)
SUP
S, EAz Sz St S ( S 1 E+h)Az
I+2(Y(t))
- +2(Y(s;))ll
482
OPTIMAL STOPPING PROBLEMS AND V.I.'s
we o b t a i n
(5.58)
E (I1 + 111)
s C[E +
a
-p+hT + h4 + i
exp
-
We have t h u s demonstrated t h e e s t i m a t e
-41
pl(h
(CHAP. 3)
+
-14
P2(h
)I
a
and s i n c e Si i s a r b i t r a r y , we o b t a i n a
(5.60)
We now seek t o o b t a i n an e s t i m a t e i n t h e o t h e r d i r e c t i o n . a r b i t r a r y stopping time.
We have
Let S;
be a n
With t h e t r i p l e t ( S y , S 2 E , J-2E) we a s s o c i a t e
(SEC. 5 )
We a g a i n have
STOCHASTIC DIFFERENTIAL GAMES
(5.54).
483
We are n e x t g o i n g t o bound t h e q u a n t i t y I1 from below.
We c o n s i d e r two c a s e s Case 11,:
s2c
SP
’
s2E
< T
.
We have 1
V2E(s)ds)(exp
b e c a u s e i n t h e i n t e r m e d i a t e c a s e So
-
S2E
, we
- at)dt
have
We t h e n s e e , as for ( 5 . 5 5 ) t h a t we have
Case I I ~ :
s2= >
S;
,
S:
<
.
Then
(?:+
V2E)ds) (exp
We a r e a g a i n d e a l i n g w i t h a n e s t i m a t e a l r e a d y e n c o u n t e r e d . summarising t h e p r e c e d i n g e s t i m a t e s , we o b t a i n
which, t o g e t h e r w i t h
(5.46) gives
- at)dt
Finally,
484
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
and s i n c e S? i s a r b i t r a r v . we o b t a i n
This t o g e t h e r with ( 5 . 6 0 ) allows us t o s t a t e t h e following approximation result : THEOREM 5.3. Under the assumptions ( 5 . 1 ) , ( 5 . 2 ) and ( 5 . 5 ) we h u e the following approximation error between the solution u ( x ) of ( 5 . 7 ) and the solution u(x) of (5.6)
(5.64)
- u(x) I I
Iu,(X)
C[e
+
a -
V X E
5.1.5.
exp
E'*+
6
- hr +h&
, where
Weakening of the assumptions on )I
+ h% +
1 1 + p 2 ( hT) 1
p, (h4)
h t 0 i s arbitrary. m 1' $2.
I n t h e case where t h e o p e r a t o r A can be put i n divergence form, t h e v a r i a t i o n a l techniques have allowed us t o prove t h e e x i s t e n c e of a s o l u t i o n of t h e V . I . ( 1 . 1 67) with assumptions on JI V which a r e c l e a r l y weaker t h a n ( 5 . 5 ) . Our o b j e c t i v e 1'
2
now i s t o give an 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 such a V . I . We w r i t e t h e o p e r a t o r A i n t h e form
(5.65)
d axi
I: - a
A = -
ij
ij
d~~
+ z a . 1
a
dxi
where we have put
(5.66) For u,v
E
1
H
(8)we put
We assume
(5.68)
a i j = aji a 2 0
,
t
aij
Lipschitz continuous, ai measurable and bounded,
V u
a(u,u) 2 pllull'
E H'(8)
,p >
0 ; furthermore ( 5 . 3 3 ) i s
satisfied;
(5.69)
f 6 Lp(o)
Q, s +2
, on
p
U
>%,
Qi f
, Q, lr"
co(b) , 0 1Q
~
;/
~
We know ( s e e Section 1 . 1 4 ) t h a t t h e r e e x i s t s one and only one s o l u t i o n the V.I.
(5.71)
IuEb.
a(u,v-u) 2 (f,v-u) V v E K
,
u
of
(SEC. 5 )
STOCHASTIC DIFFERENTIAL GAMES
485
F u r t h e r m o r e , t h e f u n c t i o n a l ( 5 . 2 5 ) r e t a i n s a meaning.
We s h a l l now p r o v e
THEOREM 5 . 4 . Under t h e asswnptions ( 5 . 6 8 ) , ( 5 . 6 9 ) , ( 5 . 7 0 ) , t h e solution u o f ( 5 . 7 1 ) belongs t o Co(B) and s a t i s f i e s u(x) = min m u
(5.72)
s2
$(s, ,s2) = m a r min $(s1 ,s2)
s1
Proof.
N
N
Suppose we have a sequence I),,
-
and
N
C,
s2
s1
Moreover t h e saddle point o f f ( S , ,S2) is
where Jx(S .S ) is defined i n ( 5 . 2 5 ) . d e f i n e d by'
N
-. C, and C 2
C2
I)2 o f f u n c t i o n s
in
E B(b) such t h a t
.
C0@)
We o b s e r v e t h a t t h e a s s u m p t i o n s ( 5 . 5 ) a r e t h e n s a t i s f i e d w i t h
a, N
N
$,,
with
However, as we have i n d i c a t e d i n t h e p o s s i b l e e x c e p t i o n o f t h e l a s t one. Remark 5.2, t h i s l a s t assumption i s r e d u n d a n t , when t h e o p e r a t o r A c a n b e p u t i n t o d i v e r g e n c e form and when we h a v e ( 5 . 3 3 )
.
t h e s o l u t i o n of t h e V . I .
We d e n o t e by u N, uf
N
N
r e l a t i v e t o $J1, $J2i
N
c o r r e s p o n d i n g t o I)1, JI,,
N
$J1s
*,.N
and of t h e p e n a l i s e d problem
Likewise we d e n o t e by <(S,,S,)
JGc (vl,v,)
t h e functional (5.25)
t h e f u n c t i o n a l (5.10) corresponding t o
We t h e n have t h e f o l l o w i n g r e l a t i o n s ( c f . ( 5 . 2 5 ) )
(5.73)
I P , ( s , , s ~ ) - S ( mSa, ,(sup S ~I ) I ~ x
€6
- + y ( x ) l , x s u€p 0I+,(.) - +:(.)I)
,
and ( c f . ( 5 . 1 0 ) )
(5.74)
/ G E ( v 1,v2)- <(v,
,v2)
Is
m= (sup x EU
I+, (XI - +:(XI
I , sup x
IC~(X)
€5
Remembering t h a t we have
N
UE(.)
= sup Inf
= Inf SUP v2 v1
$NE(V1
= Inf sup
<(sl ,s2) =
,v2)
v1
sup Inf
s2 s1 t h e r e a l s o f o l l o w from ( 5 . 7 3 ) , ( 5 . 7 4 ) t h e r e l a t i o n s
(5.75)
$N&(V1
,
,v2)
v2 s2
<(s, ,s2)
,
- +:(x)
11
1
486
(CHAP. 3 )
OPTIMAL STOPPING PROBLZMS & V . I . ' s
(5.76)
But we have also
u (x) = Inf sup J p 1 , v 2 ) = sup Inf <(v1,v2)
(5.77)
v2
v1
v2
v1
This follows from a p p l i c a t i o n of Theorem 5.1 and from Remark 5.1. We a l s o note t h a t from ( 5 . 7 5 ) we n e c e s s a r i l y have
.
Inf sup Irx(S1,S2) = sup Inf Jx(S1,S2)
(5.78)
s2
s1
s2
Moreover, from ( 5 . 7 5 ) , ( 5 . 7 6 ) , ( 5 . 7 7 ) , we have
sup lU&(X) X
-
Jxq,s2) I s
Inf sup s2 s1
2
maz(ll+,4~ll,llc24~ll)+ ) . (SUP, . I
-
But, f o r N f i x e d , we have from Theorem 5.3,
- uN(x) 1
N sup Iu,(x) X
thus
lim sup(sup 1uE(x) E + O
N
- u"x)
I .
X
-
x
~nfsup
s2
from which, l e t t i n g N
+
-k
m,
thus SUP
1
x Eb Now, we know t h a t u have U(X)
when
f(s1,s2)1) s
2
0
- ( I I $ ~ - ~N, I I , I I + ~ ~ ~ I I )
- ~ n fsup f ( s , , s 2 )1) s2
- Inf sup
f(S,,S2)
s2
I
-
2
u f o r example i n L ( 0 ) .
I
-
s1
x
u p )
+
&
we deduce
lim sup(suplu,(x) & + O
0
0
=
c
.
Consequently we n e c e s s a r i l y
Inf sup f ( S l , S 2 ) y x E Ei s2
which moreover depends continuously on in accordance with t h e uniform converIt remains t o prove t h a t S1, 8, c o n s t i t u t e a saddle p o i n t . gence (5.75). We s h a l l f i r s t show t h a t we have
If u(x) = $,(x), t h e n S1 = C and ( c f . ( 6 . 2 5 ) )
But
STOCHASTIC DIFFERENTIAL GAMES
8
and t h u s
f(i,,S2) since $
11,
=q x )
x&)
Thus condition ( 5 . 7 9 ) c l e a r l y holds a t every point x such t h a t
6 0.
u(x) = $,(x).
= cL1(x)
We may t h u s assume t h a t u(x) > $,(x)
and t h e r e f o r e S1 > 0.
Suppose we have 6 > 0 such t h a t u(x)
>
$,(x)
+6
and 1
s 4 J 1 ( Y ( t ) ) ++J
S6 = i n f I t 2 0 I u ( y ( t ) ) thus
1
t E [ O , S6Az]
and t h e r e f o r e
(u,-
*I
cL1r( Y ( t ) )
y(t) E
= 0
8
and u ( y ( t ) )
- C,(Y(t))
6
2~
-
.
But we then have
s 8[
1
f exp
- at dt
We l e t E + 0 , and by v i r t u e o f t h e property o f uniform convergence o f u u, we o b t a i n
- a t at + u(y(zAS6AS2))exp 1 - a(zAS6AS2)] 1 .
U(X)
s Ex[
f(y(t))exp
to
458
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V . I . ' s
1
^
We t h e n l e t 6 +. 0 , SE 1. S1 ( s e e Theorem 3 . 7 ) , and s i n c e
u i s bounded, it
follows from Lebesgue's Theorem t h a t we have
which, i n view of ( 5 . 8 0 ) , c l e a r l y proves ( 5 . 7 9 ) . We can prove i n a similar manner t h a t we have
v s1
2 f(Slt5,)
U b )
and t h u s
which c l e a r l y shows t h a t S1, S
Remark 5.4.
2
i s a saddle p o i n t .
We do not know i f t h e stopping times SlE, S2€ converge r e s -
p e c t i v e l y t o S1, S ( a s i n t h e case of a s i n g l e p l a y e r ) . We can show however, 2 by techniques similar t o t h o s e of Theorem 3.7, t h a t we have
lim id s1 2
9, ,
lim id
sPEz i,
I
t h e d i f f i c u l t y being t h a t we no longer have t h e property u 5.2. 5.2.1.
u.
a
The nonstationary case.
Operators which are not i n divergence form.
We s p e c i f y A ( t ) , t
where (Q = o x ]O,T[, 2 = r x lO,T[)
i n t h e form
lO,TC,
E
A(t)
(5.81)
(5.82)
2
=
with
- ij E
Oa
a.
.(x,t)
1J
bounded open subset of Rn o f c l a s s C2;
4 = aji , aiJ E C O G ) Vt;l*.*Cn E R B > 0
a.
t
,
E aij
The f u n c t i o n s a i j , pi a r e extended t o Rn I)2,
2 ci ,
sicj 2
B
[O,Tl,
preserving t h e i r p r o p e r t i e s .
;
g . measurable and bounded on
Next we t a k e f , I),,
b q
d2
~ E gi(x,t) X.dX i j i7
satisfying
$. x
(SEC. 5 )
STOCHASTIC DIFFERENTIAL GAMES
489
We w r i t e
: 3 = u - a l g e b r a generated by w(X),X
Suppose we have Cl = Co([O,-[,Rn), and PXt a p r o b a b i l i t y measure on ( 0 ,
6 )which i s unique,
a s t a n d a r d i s e d Wiener process, which i s a
9t m a r t i n g a l e ,
E
[t,sl
such t h a t t h e r e e x i s t s f o r which t h e process
y(s;w) = w(s) i s a s o l u t i o n of t h e equation
(5.84)
Then if v ( s ) , v ( s ) a r e two s c a l a r processes adapted t o
1
(53 5 )
2
a.s.
v
E [0,1],
vi(s)
$
satisfying
s
and if S1, S2 a r e two stopping times such t h a t Si E [ t , T I , we put TAT 1 (5.86) Ct(v,,v,) = ( f + 7 V 1 o l + 7 V 2 Q2)(Y(s),S)
st[[ t
x {exp
-
(a(s-t)
+ xT
on
ft(sl,S2)
=
ft[
1
+7
- (a(T-t)
r A S AS
s” 1;
(v,+ v2)dh)jds
++
(vl+ v2)dh)]
- a(s-t)ds
f(y(s),s)exp
+
t Q1(y(sl)’sl)
X S 3 S 2 , S 1 < ~ A Texp
+
Q2(Y(S2)9S2)
XS2<S,ATAT
+
u ( y ( T ) ) xT=S AS
r
19
We then consider t h e following problem. such t h a t
exp
,
exp
- a(S1-t)
- a(s2-t)
- a(T-t)] .
Find u = u ( x , t ) , a f u n c t i o n defined
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.’s
490
L
xt
A
then Sl, S2 is a saddle point of the functional J Proof.
(S1,S2).
F i r s t of a l l we solve
(5.90)
by an i t e r a t i v e method, as i n Theorems 4.7 and 5.1.
We show t h a t
u , ( x , t ) = Min ax $Et(vl,v2) = ax Min C t ( v l , v 2 ) v2
as i n Theorem 5.1.
v2
v1
v1
Then we show t h a t (UE’
L
02)+
t
E
sc
as i n Theorem 5 . 2 , and we conclude t h e proof as i n Theorem 5.2.
Remark
5.5.
The analogue o f Remark 5.2. f o r t h e nonstationary c a s e .
We a l s o have THEOREM 5.6.
8
Under the asswnptions o f Theorem 5.5,
Proof.
S i m i l a r t o t h e proof of Theorem 5.3.
5.2.2.
Operators i n divergence form.
8
we have
8
(SEC. 5 )
STOCHASTIC DIFFERENTIAL GAMES
491
We now endeavour t o g i v e a n i n t e r p r e t a t i o n of t h e V.I. examined i n Theorem 2.22. We w r i t e
(5.92) with
+
. a =gi
b
E -a bx
j
ij
j
Lipschitz continuous,
a 2 0 , I:a
(5.98)
(5.101)
crn(6)
BU
V
Y
+I(X,~)
.
s ES
+ a(t;u,v-u)
-(,v-u)
with
measurable and bounded,
~ . ~ . s B x ~ ~ , v ~ , . . . ~ p, >E oR ,, 1 J
iJ
'i E H;(B) , u E
a.
2
1
v E H,(B)
G~(X,T)
; A(T)E
- (f,v-u) 2 o
, JI,
S V S 92
8
E L'W
.
492
(CHAP. 3 )
OPTIMAL STOPPING PROBLEMS & V.I.'s
THEOREM 5.7. Under t h e assumptions (5.94), (5.95), (5.96), (5.97), (5.981, t h e s o l u t i o n u of (5.99), , (5.102)s a t i s f i e s
.. . .
E Co(6
(5.1 03)
x [O,T]) * . .
S1,
and we again have (5.89) h e r e
S2
are defined as i n Theorem 5.5.
To s i m p l i f y t h e n o t a t i o n s l i g h t l y , we t a k e a = 0 . Proof. penalised problem
We consider t h e
(5.104)
We have (5.105)
u, E b2" "(0 x
V to
]O,to[)
<
,u
T
,
E Co(q)
If B i s a s o l u t i o n o f (5.106)
B E b2'1'p(0x
]O,to[)
V to
<
,
T
,
B E C"($
and i f we put
( 5.1 07)
w
C
m u
C
- B
t h e n wE i s a s o l u t i o n of t h e problem
- >+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 ,<) be t h e canonical space ( s e e Chapter 3, Section 3 . 7 . 1 ) and l e t y ( t ; w ) = w ( t ) . We know ( s e e g a p t e r 3 , Section 3.7.1, example) t h a t t h e r e e x i s t s one and only one measure P I P on ( p , J ) such t h a t we have
t
(SEC. 2 )
dy =
(2.1 6)
~ ( ~ 1dw(s) s )
where w ( s ) is a Wiener process and a We denote by
517
HAMILTON-JACOB1 EQUATION
T
=
T~~
s
3 't
> t
Y(t) =
X
J
martingale, w ( t ) = 0.
t h e exit time from @'of t h e process y ( s ) .
Now suppose t h e r e a r e two p l a y e r s 1 and 2; player 1 w i l l be c a l l e d t h e "muximiser", and player 2 t h e "minimiser". We d e s c r i b e as t h e strategy f o r player 2 a measurable mapping x , t +. v 2 ( x , t ) from 0 x [OIT] 4 V2 and as t h e strategy f o r player 1 a measurable mapping x , t , v 2 -t v l ( x , t , v 2 ) from 0 x [O,T] x V2 V1 ( * ) #
-
We define v , ( x , t ) , v l ( x , t , v 2 ) f i x e d element o f
V2'
for x
4 @ i n an
a r b i t r a r y manner ( f o r example a
V1 r e s p e c t i v e l y ) .
Having s p e c i f i e d t h e s t r a t e g i e s v 2 ( x , t ) , v l ( x , t ,v2) ( * * ) , we put
(2.1 7 )
We t h e n define on (Q,St)t h e measure (given by t h e Girsanov transformation)
where + ( s ) i s a standardised Wiener process and a 3' martingale.
t
Then, f o r every p a i r of s t r a t e g i e s v1(x,v2),
+ c(y(T))(exp We s h a l l denote by
Remark 2.7.
8t
v 2 ( x ) , we put
+
1;
cv
. xt t o t h e measure P . (h)dh)
1 2
t h e expectation with respect
Let v 2 ( x ) be a Y2-valued process adapted t o
3 : ,
xT -
~
3
and l e t v1(x,v2)
be a s t r a t e g y f o r player number 1. We can c l e a r l y render t h e right-hand s i d e o f
( * ) Players 1 and 2 a r e not i n a symmetrical s i t u a t i o n . (**)We s h a l l w r i t e v ( x ) , v1(x,v2) t o shorten t h e n o t a t i o n . 2
518
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
(CHAP
. 4)
(2.17) meaningful, with v2(s) i n p l a c e of y ( y ( s ) ) and v ( y ( s ) , v , ( s ) ) i n place of v , ( y ( s ) , v 2 ( y ( s ) ) ) . The right-hand s i d e o? (2.20) i s d i f i n e d l i k e w i s e and w i l l be denoted by JXt(v ( x , v 2 ( . ) ) . 1 xt t o J (v,(.),v2(x)).
Using analogous arguments, we can a s s i g n a meaning
The i n t r o d u c t i o n of such f u n c t i o n a l s i s , as we s h a l l s e e , p a r t i c u l a r l y u s e f u l when we wish t o o b t a i n s t o c h a s t i c c o n t r o l problems a s a s p e c i a l c a s e of games problems. rn We then have
We adopt the assumptions ( 2 . 1 ) , (2.2), ( 2 . 3 ) , ( 2 . 9 ) , ( 2 . 1 4 ) , ( 2 . 1 5 ) ; we then have uniqueness o f the solution o f (2.10)( i n b 2 ” ” ( Q ) ) . This solution i s given by THEOREM 2.1.
(2.21
1
u(x,t) = f i n ax St(vl(x,v2),v2(x)) v2(x) v1 (‘JV2)
Moreover there e x i s t s a pair of strate gie s v,(x),
.
v1(x,v2) such that we have
From Remark 2.5, L i s a f u n c t i o n which i s continuous with r e s p e c t t o Proof. From t h e theorem o f s e l e c t i o n of measurable s e c t i o n s des-a l l t h e variables. c r i b e d i n Remark 2.5, t h e r e e x i s t s a measurable f u n c t i o n v1( x , t , u , p , v 2 ) x , t E Q such t h a t
( * ) u, Du extension.
E
Co(6)
(for p sufficiently large).
Otherwise Du i s t h e Borelian
(SEC. 2)
519
HAMILTON-JACOB1 EQUATION
From I t o ‘ s formula we have
Likewise, we have
H ( x , s , u , ~ ) = Max L(X,s,u,Du,V1 , e , ( X , S ) ) V 1 1 L ~ x , s , u , D u , v l , ~ 2 ~ ~ ,v s ~v,~
(2.28)
-
Using an argument a l r e a d y g i v e n , t h e f i r s t i n e q u a l i t y i n (2.23) can be dedAgain applying (2.26) I n (2.28) we can t a k e vl = v1(x,c2). uced immediately. w i t h v1 = v (x,? ) and j2we o b t a i n
1
2
u(x,t) 2
St(v, (x,v2),e2(x))
hence (2.22), s i n c e v (x,v2) i s an a r b i t r a r y s t r a t e g y f o r p l a y e r 1.
1 v2 = v (x) i n (2.27) and a r g u i n g as b e f o r e , we a l s o have 2
But t a k i n g
so t h a t
(*)
v1,v2 a r e e i t h e r s t r a t e g i e s , or adapted c o n t r o l s , a s i n Remark 2 . 7
(CHAP. 4)
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
520
which t o g e t h e r with t h e second p a r t of (2.22), c l e a r l y proves (2.21).
Remark 2.8.
where
We a l s o have t h e r e l a t i o n
02(t) = V 2 ( y ( t ) , t )
.
This i s meaningful because t h e t r a j e c t o r y y does not depend on t h e c o n t r o l s ( i t
i s t h e measure
which depends on t h e c o n t r o l s ) .
Uniqueness follows from t h e f a c t t h a t any s o l u t i o n of (2.10) Remark 2.9. s a t i s f i e s (2.21) and (2.29). S I n t h e case where f , c , g do not depend on vl, Remark 2.10. (2.29) reduces t o t h e r e s u l t
2.3.
Solution of t h e H . J .
the relation
equation
We s h a l l now prove
Under t h e assumptions of Theorem 2.1, a solution u of (2.10)
THEOREM 2.2.
e x i s t s with (2.30)
Proof.
.
u E b””’P(Q) Let a:.
1J
be a sequence o f r e g u l a r c o e f f i c i e n t s such t h a t
The methods of Section 1 allow us t o prove t h a t t h e r e e x i s t s one and only one function ur such t h a t
We denote by yr t h e s o l u t i o n o f
dyr = ar(yr(s),s)dw(s), Yr(t) = x
,
s
>
t
,
(SEC. 2 )
and by
HAMILTON-JACOB1 EQUATION
<;::,
t h e measure c o n s t r u c t e d from (yr,v,,v2)
521
s t r u c t u r e d l i k e Q.
F i n a l l y , l e t Jr’Xt be t h e c o s t f u n c t i o n c o n s t r u c t e d i n a manner analogous t o Theorem 2 . 1 a p p l i e s , and t h e r e f o r e (2.20).
(2.33)
Now from t h e assumptions ( 2 . 1 4 ) , ( 2 . 1 5 ) t h e f u n c t i o n s f , c , g a r e 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 and s i n c e x V 1 x V2 i s compact, it f o l l o w s t h a t
a
f , c , g a r e bounded(*).
(2.34)
But t h e n ( 2 . 3 3 ) shows t h a t we have
lur(x,t)
Ic
.
c
But from equation ( 2 . 3 2 ) , we have
and from ( 2 . 7 ) ( * * ) , we a l s o o b t a i n
(2.35) But f o r a l l
E >
0
and t h e r e f o r e choosing t h a t we have
(2.36)
E
s u f f i c i e n t l y s m a l l , it f o l l o w s from ( 2 . 3 5 ) and ( 2 . 3 4 )
l l 4 l p ,p
5
c
*
But t h e i n j e c t i o n f r o m b 2 ’ l S p intoIbl’o’p a sub-sequence such t h a t
ur
+
u weakly i n ~
i s compact and t h u s we can e x t r a c t
2 and~ s t r o n 1gly ~ i n t 2 *~O > 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
by ( 2 . 1 8 ) and f t ( v 1 ( . ) , v 2 ( . ) ) by t h e right-hand s i d e of (2.20).
t h e f u n c t i o n a l J A ( v (.),v ( . ) ) possesses a saddze p o i z t , 1 2
jl(.), G2(.)
Then
given by
(2.47) where t h e f u n c t i o n s $ ( x , s ) , j 2 ( x , s ) c o n s t i t u t e a saddle point of L(x,s,u,Du,v~,v~). 1 We have t h e r e f o r e
This r e s u l t i s due t o t h e f a c t t h a t t h e t r a j e c t o r y y ( t ) does not depend on t h e controls. This i s a These only have an e f f e c t on t h e p r o b a b i l i t y measure. property which has no counterpart i n t h e d e t e r m i n i s t i c case.
3. 3.1.
THE HAMILTON-JACOB1 INEQUALITY. Penalised scheme.
OPERATOR NOT I N DIVERGENCE FORM.
Interpretation.
We now t a k e f u n c t i o n s $ , ( x , t ) ,
$,(x,t)
( t h e o b s t a c l e s ) such t h a t
(SEC. 3)
HAMILTON-JACOB1 INEQUALITY
525
(3.2) u
Under t h e assumptions of Theorem 2 . 1 and ( 3 . 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 ( 3 . 2 ) Player number 2 uses a c o n t r o l We d e f i n e t h e s t a t e o f t h e system by ( 2 . 1 6 ) . v (. ) , an adapted process with v d u e s i n V 2 ( * ) , and player number 1 uses a 2
strategy v1(x,v2). I n a manner analogous t o ( 2 . 1 7 ) and ( 2 . 4 6 ) , we put
fv
(3.3)
(9)
1 2
= f(y(s),s,vl (Y(s),v2(s)),v2(”))
( 9 ) = g(Y(s),s,vl(Y(s),v2(s)),v2(~)) 1 2 cv v ( 9 ) = C(Y(S),”,Vl (Y(S),V,(S)),V,(S))
Pv
1 2
9
9
.
We s h a l l now prove THEOREM 3.1.
The asswnptions a r e t h o s e o f Theorem 2.1 and (3.1).
j z ( x , t ) be d e f i n e d by
H(x,t,uE,he) =
L(x,t,uE,mE;vl V
and (*)
v p , t,v2) or a strategy v2(x,t)
1
d e f i n e d by
3.0;)
Let
726
STOPPING TIMES AND STOCHASTOC OPTIMAL CONTROL
(CHAP. 4 )
we then have
Proof. It i s s u f f i c i e n t t o prove (3.61, s i n c e t h e property (3.7) i s a s t r a i g h t f o r w a r d consequence of ( 3 . 6 ) . Now I t o ' s formula g i v e s
It t h e n s u f f i c e s t o argue a s i n t h e proof of Theorem 2 . 1 and a s i n t h e proof of Theorem 6.1, Chapter 3. Then (3.6) can e a s i l y be deduced, from which t h e r e s u l t t h e n follows. 3.2.
The Hamilton-Jacobi i n e q u a l i t y and t h e optimal s t o p p i n e t i m e problem.
We consider t h e following problem: f i n d
(*)
m
U E ( x , t ) if x
of O w i t h JI,
5
4
Oand t
0 and
2
E
0.
u
such t h a t
C0,Tl; $1, $ 2 a r e extended by continuation outside
527
HAMILTON-JACOB1 INEQUALITY
s
u c b2*l~P(c7) ;
on
US
- &at+
A(t)u
- H(x,t,u,Du)
= 0 a.e.
l - - +h at
A(t)u
- H(x,t,u,Du)
2 0
dU -+ A(t)u at
- H(x,t,u,Du) 5 0
G in
Q if Q1
<
u
if u
P
Q
I’
It
if u =
11
< 1
Q2
,
’
$2
Let S1, S2 be two stopping times with r e s p e c t t o 3; with
Si E [,,TI
.
We s h a l l now prove
Under the assumptions of Theorem 3.1 and (3.11), probtem ( 3 . 9 ) THEOREM 3.2. We then define C 2 ( x , t ) and $1(xpt,Tr2) possesses one and onZy one sotution. r e s p e c t i v e t y by
H(x,t,u,Du) =
1
L(x9t,uJh,vl,B2)
t
L(x,t,u,Du,V1 , V a l = M#x L(x,t,u,Du:vl ,v*) 1
v v2
*
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
528
(CHAP. 4 )
Thus in particular
(3.13)
u ( x , t ) = Min
w Min ax s1 v 2( . ) v 1 ( x , v 2 )
=MPX
E n Min lax s2 v2( . ) v 1 . ( x , v 2 )
s2
s1
=
Proof.
~ Min n ax ax s2 v & . ) s1 v1 (x,v2)
Suppose i = 1,2.
Pt Pt
Pt
.
Due t o t h e r e g u l a r i t y o f $ .
we have
Thus from ( 3 . 5 ) , t a k i n g i = 2 t o i l l u s t r a t e t h e c o n c e p t , w e have
Taking i n t o a c c o u n t t h a t
E(y(T))
t h a t f , c , g a r e bounded, t h a t $, we can w r i t e
so t h a t
s 4J2(y(T),T) , 4J2(y(?)) 3 0 , 4J1 C cL20n Q, D$, a r e a l s o bounded and t h a t ( 3 . 1 1 ) a l s o h o l d s ,
(SEC. 3 )
HAMILTON-JACOB1 INEQUALITY
529
Thus
.
s c
E
I n t h e same way we can prove t h a t we have
Thus u
where lg
satisfies (cf. (3.2))
1
5 constant.
From t h e r e l a t i o n s u
E
,
-4Jec.5 2-
u
,
- G I B C €
it a l s o follows t h a t lu,l< constant. 2.2, that
It then follows, arguing a s i n Theorem
We can t h e r e f o r e e x t r a c t a sub-sequence, again denoted by u E , such t h a t uE
+
u weakly i n L b 2 y 1 y p ( Q ) ,
s t r o n g l y i n ljllyoypand a . e .
It then follows t h a t
( u - + ~ ) +=
o
,
(u4,)- =
o
a.e.
and s i n c e u, $1, $2 a r e continuous, we c l e a r l y have
We a l s o have
H(x,t,uE,DUE)
-
H(x,t,u,Id
a.e.
We t h e n conclude t h e proof m i n g a l i n e of argument similar t o t h a t of Theorem The proof 6.2, Chapter 3. Thus t h e e x i s t e n c e of a s o l u t i o n o f ( 3 . 9 ) i s c l e a r . of ( 3 . 1 2 ) and (3.13) is then similar t o t h a t of ( 3 . 6 ) , ( 3 . 7 ) .
5 30
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
Remark 3.1. i s unnecessary. we have:
"(a)
remains t r u e ( s e e a l s o Remark 6 . 2 , Chapter 3 ) . COROLLARY 3.1. u in
~'(5).
Proof.
4)
I n t h e case where 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 , assumption(3.11) I n a c t u a l f a c t , as we have shown i n &Action 1 ( c f . ( 1 . 8 3 ) , (1.84)),
This i s s u f f i c i e n t t o imply t h a t 11UE11
to
(CHAP.
S C , and t h e r e f o r e Theorem 3.2 8
Under t h e assumptions of Theorem 3.2, t h e sequence uE converges
During t h e proof of Theorem 3.2, we saw t h a t
Thus
We d e f i n e zE t o be a s o l u t i o n of
From Chapter 3, Theorem 6.5, we can consider z a s t h e value 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 game, i n which t h e only d e c i s i o n varEables a r e t h e stopping times
S1'S2. The f u n c t i o n a l of t h e game i s given by (6.87) with H i n p l a c e of f and with u = 0. Furthermore u i s t h e value o f t h e same game with HE r e p l a c i n g H. From t h e convergence of H f H i n Lp, it follows t h a t
(3.1 7) From Theorem 6.6, Chapter 3, we a l s o have
(3.18) and (3.17), ( 3 . 1 8 ) c l e a r l y imply t h e d e s i r e d r e s u l t ( * ) .
(*)
8
We have not sought t o e s t a b l i s h an estimation e r r o r as i n Theorem 6.3, Chapter 3.
4)
(SEC.
4.
HAMILTON-JACOB1
v.1.’~
5 31
HAMILTON-JACOB1 VARIATIONAL INEQUALITIES
4.1
The games case
We a r e now concerned more p a r t i c u l a r l y with t h e case where t h e o p e r a t o r A ( t ) can be put i n t o divergence form, i n o t h e r words with t h e s i t u a t i o n discussed i n Section 1; we s h a l l a l s o o b t a i n s e v e r a l a d d i t i o n a l items o f information. We t h u s have
(4.1 with
A s u s u a l 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 assumption
We t a k e f , c , g s a t i s f y i n g ( 2 . 1 4 ) , ( 2 .1 5 ) and
(4.4)
From Theorem 1.1 ( * * ) t h e r e e x i s t s one and only one f u n c t i o n u such t h a t we
have
( * ) So t h a t A ( t ) coincides with ( 2 . 1 1 ) . and terms o f order 0.
(**)By v i r t u e of ( 4 . 3 ) ,
1dH ~ I1
We can o f course add f i r s t - o r d e r terms
(h + IuI + IPI)
-
( C W .
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
5 32
4)
(4.7)
(4.8)
(4.9) (4.10)
We then have
Moreover tre obtain the s m e concZusions as f o r Theorem 3 . 2 , n m e l y (3.12), (3.13) Proof.
-
We approximate $1, $ 2 , u by a sequence $,
+:
(4.1 2) '
N
E b2'1'p(Q) N
s +2
on
N
,
EN E J@)
G ,
+,lZ N
N
2
,$;
such t h a t
,
= 0 =
N
N c21
.z
,
(4.13)
We denote by uN, u !
t h e s o l u t i o n o f t h e V.I. and o f t h e p e n a l i s e d problem r e l a t -
F u r t h e r , from (3.10) we have
=
pN -,C
when
N -m
.
(SEC.
4)
533
HAMILTON-JACOB1 V . I . ' s
Likewise
Now, from t h e r e s u l t s o f Section 3, we have
We p u t w(x,t)
It follows from
Inf
id
sup
s1
s2
Pt
sup
V2(J
.
V1(X,V2)
(4.14) and (4.16) t h a t we have
Since t h e s o l u t i o n u ( x , t ) of t h e p e n a l i s e d problem i s r e g u l a r and Theorem 3.1 t h e r e f o r e remains valid: it follows from ( 4 . 1 5 ) t h a t we have (4.18)
But for a f i x e d value of N , we have from Corollary 3.1 u,N(x,t) - u N ( x , t ) I t then follows, by v i r t u e of
u
4
w
in
in
cO(Q)
.
(4.17) and (4.18) t h a t we have
c'(P)
.
5 34
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
Since we know t h a t u (4.1 9)
u
t
-+
,
w
uE -. u
4)
2
.u f o r example i n L (Q), we t h e r e f o r e have
u
-.
u
in
In p a r t i c u l a r we have proved (4.11). t h a t we have (4.20)
(CHAP.
strongly i n
c'(Q)
.
Furthermore, we have shown i n Theorem 1.1
L~(O,T:H;(O)).
We consider t h e i n s t a n t s S1, S2 and t h e s t r a t e g i e s C 2 ( x , t ) , Gl(x,t,v2)
defined
Y
i n Theorem 3.2. Altho h , i n c o n t r a s t t o Theorem 3.2, t h e function Du i s not continuous, but merely L (Q), t h e s t r a t e g i e s 0 ( x , t ) and Vl(x,t,v2) a r e w e l l defined as measurable functions of t h e argumengs ( s e e Remark 2 . 5 ) . We can always assume t h a t 0 and 0 a r e Bore1 functions obtained by extension. 2
1
We s h a l l now show t h a t (3.12) remains t r u e . Since (3.12) i s e v i d e n t l y t r u e i f t = T , we assume t < T.- F i r s t we prove t h e second i n e q u a l i t y i n ( 3 . 1 2 ) . I f u ( x , t ) = q 1 ( x , t ) , t h e n S1 = t and from (3.10) we have
s i n c e qlIz = 0 . Consequently, t h e second i n e q u a l i t y i n (3.12) i s s a t i s f i e d i f u(x,t) = ql(x,t).
(SEC.
4)
535
HAMILTON-JACOB1 V . I . ' s
and t h u s
.
(uE- 4Jl)-(Y(s)ys) = 0
xt be t h e measure d e f i n e d Now l e t v ( . ) be an a r b i t r a r y c o n t r o l , and l e t Qo 2 1 2 Finally by v 2 ( x ) and t h e s t r a t e g y V1(x,v2) o b t a i n e d by t h e formula ( 2 . 1 8 ) . t + h < T and suppose S i s an a r b i t r a r y s t o p p i n g suppose h > 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) = <(x), we have
z-
AS^)
~ ( Y ( T A AS,), ~, .cA~~
= u(y(*S11, i1
xglSs 1 ' 51
+ u'y's2)'s2)xS2<~1A~AT
+ C ( Y ( T ) ) X As<s ~~
1r
NOW i f
* . .
A
-
Zl<~hT,we have u(y(S,),Sl) = ( L 1 ( Y ( s l ) , s , ) . Moreover u
5
Ji2.
In
view of a l l t h i s , (4.27) implies
u(x,t)
s ft@, b,Vl
),v2(.):i1 ,s2)
which i s c l e a r l y t h e right-hand i n e q u a l i t y i n (3.12). The left-hand i n e q u a l i t y i s proved v i a s i m i l a r arguments;
4.2
hence t h e r e s u l t .
The c o n t r o l case
I n t h e c o n t r o l case, t h e assumptions concerning t h e o b s t a c l e can be weakened considerably f u r t h e r , and t h i s w i l l allow us t o o b t a i n t h e analogue of Theorem 4.1. We assume 9, = m 9 = Q with
-
,
2
We have proved i n Theorem 1 . 6 , t h a t t h e r e e x i s t s a rnaxzhwri weak solution of the
v.I.
s
(4.31)
T
6v
[-(= 1V-U)
+ a (t ;u,v-u)-(
+-$v(T)-;l22 u E L'(o,T;v)
v
0
, us
H(x, t ,u, Du) ,v-U) ]dt
v E y Q
,
a.e.
We s h a l l now prove
4.2 Under the asswnptions ( 4 . 2 8 ) , ( 4 . 2 9 ) , (4,30) and ( 4 . 2 ) , (2.141, ( * ) the rnuxiniuni weak solution of (4.31) satisfies
THEOREM (2.15)
(4.32)
u E co(6 x [O,T])
.
Moreover (*)
With a s i n g l e c o n t r o l , v2 = v ,
V2 = V
.
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
538
where, S being a stopping time in V , we put
E
(CHAP.
4)
Ct ,TI and v ( . ) being an adapted process with values
Proof. We follow t h e l i n e of argument used i n Theorem w i t h an i n t e r p r e t a t i o n of t h e p e n a l i s e d problem:
4 . 5 , Chapter 3.
u
We begin
For t h i s purpose we approximate by a r e g u l a r sequence, f o r which t h e i n t e r p r e t a t i o n i s s t r a i g h t f o r w a r d , due t o t h e r e g u l a r i s a t i o n o f t h e corresponding s o l u t i o n . We t h u s have We t h e n proceed t o t h e l i m i t , as i n Theorem 4 . 5 , Chapter 3.
Then we prove, a g a i n as i n Theorem
(4.36)
u
4
Inf
s,v( .I
4.5, Chapter 3 , t h a t
P(s,v(.)) =
w(x,t)
.
To t h i s end, we n o t e t h a t ue L w, t h e n t h a t f o r a l l S , t h e control
6,(~) =
i s such t h a t we have
0
if
s < S
1
if
S B S
(SEC.
4)
5 39
HAMILTON-JACOB1 v.1.'~
and where t h e constant C does not depend e i t h e r on S , o r on v ( . ) . This can e a s i l y be seen t o imply ( 4 . 3 6 ) . Since we know t h a t u + u from above (where u i s t h e m a x i m u m weak s o l u t i o n of t h e V . I . ( 4 . 3 1 ) ) , we 6ave c l e a r l y proved (4.32) and ( 4 . 3 3 ) . m
Remark 4.1. We do not know whether t h e lower bound i n (4.33) i s a t t a i n e d . i s reasonable t o conjecture t h a t t h e c o n t r o l s
I
= inf 1s E CtlT]
v(t) where
= +(Y(s),s))
u(Y(s),s)
It
,
,
= G(Y(t),t)
V ( x , t ) i s defined by
H(x,t,u,Du) = Kin
[f(x,t,v) + u c(x,t,v) + Du. g(x,t,v)]
V E V
r e a l i s e t h e lower bound. We do not know whether
u
-
u
,
The d i f f i c u l t y encountered i n t h e proof i s a s follows.
L~(o,T;H:(o))
strongly i n
In t h e absence of such a r e s u l t , we do not know whether
H(x, t,uE, D U E )
+
H(x, t,u,Du)
L2(Q)
in
.
We t h e r e f o r e cannot readopt t h e argument used i n Theorem 4.1. t h a t the controls
However we note
where O E ( x , t ) i s defined by
H(x,t,uE,DuE)
Min
I
[f(x,t,v)
+
uE c(x,t,v)
+
DUE. g(x,t,v)]
V E V
form a minimising sequence.
Remark 4.2.
&dtc
5.
Naturally i f $ s a t i s f i e s
L*(O,T;H'(O))
A(T)c E L2(6) , t h e lower bound.
m
,
62Q E L2(p)
7
and i f
then as i n Theorem 4.1, m
Q E L2(0,T;H1(0))
, $$€
L2(0,T;H'(S)) 1 ( 4 . 3 ) i s s a t i s f i e d along with E Ho(0),
&d Et
L2(Q)
and t h u s
s, B ( t )
u
realise
OPTIMAL CONTROL AND STOPPING TIMES WITH POLYNOMIAL GROWTH
o=
We s h a l l now consider t h e s i t u a t i o n where R n , b u t i n which t h e Hamiltonian H (See 2 . 5 ) ) no longer s a t i s f i e s t h e assumptions of Lipschitz c o n t i n u i t y and of l i n e a r
STOPPING TIMES
540
5.1
Assumptions
-
notation
-
AND STOCHASTIC OPTIMAL
t h e problem
CONTROL
(CHAP.
-
4)
Let V be a closed convex subset of R p , containing 0. We t a k e functions x [O,T] x V R , + ( x , t ) : R" x [O,T] R, g k , t , v ) : Rn x [O,T] x V f(xbt,v) : - R , c(x,t,v) : x [O,T] x V - R , d t ) : [O,T] -d(R ;Rm) such t h a t
where go, B(t)
I%J
Ic
a r e continuously d i f f e r e n t i a b l e w . r . t .
-
The function f s a t i s f i e s t h e p r o p e r t i e s
I
;:vrc,
I V l L C 6
,
p > 1
The function JI s a t i s f i e s
b'+ bXidX . b X
J
m e function
continuous and bounded
.
k
U satisfies
F i n a l l y t h e functions u and c s a t i s f y t h e p r o p e r t i e s
.
x,t.
Also,
(SEC. 5 )
POLYNOMIAL GROWTH
We s h a l l designate as an admissibte system a s e t
A
= (Q,O,P,$,
v(s),w(s), ~2 t )
where (Q,O,P) i s a p r o b a b i l i t y space, '3' s 2 t i s an i n c r e a s i n g sequence o f sub-oalgebras of 0 ,w ( s ) is a standardised h e n e r process and a Zt martingale, ( w ( t ) = O), and v ( s ) i s a process adapted t o z:, with values i n V and such t h a t t h e r e e x i s t s a constant Kv (dependent on t h e process v ) such t h a t
Iv(s,~)I
S Kv
t
V s2 t
, v
w
.
To such a system t h e r e corresponds t h e t r a j e c t o r y
For a stopping time 8 with r e s p e c t t o 3 : . such t h a t 8
E
C t , T I , we put
O u r o b j e c t i v e i s t o prove THEOREM 5.1. Under the asswnptions ( 5 . 1 ) , (5.21, ( 5 . 3 ) , (5.41, (5.51, ( 5 . 6 ) , (5. T), there then e x i s t s one and only one soZution of the V. I .
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
542
u E LP(O,T;W’*P,”)
(5.12)
- -d+t bU
L P ( o , T ; L P * ~ ,) p
(u+)(-=
++
I, p
JI
all
u(x,T)
+ A(t)u
u ( x , t ) = J~~(.$,$)
- H(x,t,u,Du))
= 0
a.e.
.
s
(2,641 such
Moreover t h e r e e x i s t s a pair
that
= i d ~,,&,e)
4,
5.2
>
A ( t ) u s H(x,t,u,Du)
us
(5.13)
dU ,E bt
.
Proof of Theorem 5,l
We put
and we consider admissible systems such that
(5.15)
V(9,W)
f.
Yk
We write
4, =
the set of admissit
We put
If we put ‘p
we have
= f
+ -aJI at
A(t)J,
?
systems satis “in@; (5.15)
,
(CHAP.
> o
4)
POLYNOMIAL GROWTH
(SEC. 5 )
(5.19)
u,(x,t)
= Inf sJ,e E A
543
Jxt(”l’e) k
then by an easy extension o f techniques a l r e a d y presented ( c f . Theorem 3.22 and Section 4.10, Chapter 3 ) , t h e function % i s t h e unique s o l u t i o n of t h e problem
UkS9 bU
(r,-+I(&+
-
Ukb,T)
#
- s(x,t,uk,Duk)) = <(I) . A(t)uk
= 0 a.e.
We now l e t k + + . From ( 5 . 1 9 ) t h e sequence uk is obviously decreasing. From t h e assumptions and t h e formula ( ? . l a ) , we a l s o see t h a t LL&x,t)
2
-
t
where M i s a constant independent Of k , x , t .
Thus
\(x,t)
I u(x,t)
. A
*
Furthermore, f o r any f i x e d k , problem ( 5 . 1 9 ) possesses a s o l u t i o n d , 8 , given by t h e usual formulas. We then have
where yxl ( s ) denotes t h e process corresponding t o t h e c o n t r o l V , but s t a r t i n g a t We note t h a t
XI.
544
STOPPING TIMES
= I + I1 + I11
AND STOCHASTIC OPTIMAL CONTROL
(CHAP.
4)
.
Now by v i r t u e of t h e f a c t t h a t a depends only on t i m e , we have
thus
We n o t e t h a t
and t h u s
I1
II'JT- O
lxl-x\
when
-
0
.
Since cp i s bounded from below, t h e r e e x i s t s a c o n s t a n t C such t h a t
lcpls cp + c
*
Taking account o f assumption ( 5 . 5 ) and d i v i d i n g t h e l e f t - h a n d s i d e of (5.21) by l x l - x l , we deduce from t h e preceding arguments and from assumptions ( 5 . 3 ) , ( 5 . 4 ) t h a t we have
Since
< and JI
a r e bounded f u n c t i o n s , we a l s o have
l i m sup X I - x
%(Xl,t)
IX'J
- %(X't) I
S
w(.,t) + C(u,(x,t)
+
1)
.
I n t e r c h a n g i n g t h e r o l e s o f x' and x, and t a k i n g i n t o account t h e f a c t t h a t i s bounded, we o b t a i n
\
(SEC. 5 )
POLYNOMIAL GROWTH
545
But from t h e r e g u l a r i t y r e s u l t s ( 5 . 2 0 ) , we have
We s h a l l prove t h a t we have
and t h a t t h e r e e x i s t s a f i x e d ko independent o f x , t such t h a t
It follows from (5.26) and ( 5 . 2 7 ) t h a t we have
(5.29)
On t h e b a s i s o f ( 5 . 2 9 ) , t o g e t h e r with t h e p r o p e r t i e s of (se from t h e f a c t t h a t is bounded, we can proceed t o t h e E m $ i n which we can d e d u c e k e e x i s t e n c e o f a solution. We s h a l l now PI W W ee ppuut t
L(X,t,V,A,p) = F.
B(t)V
+
c(xJtJv)h + f(x,t,v)
.
Let vk be a p o i n t such t h a t
Since (5.30)
vk
i s convex, we a l s o have
p. B(t)(v-vk) V V E V k
.
+
A c ' (x,t,vk).(v-vk)
+f'
(XJt,vk).(v-vk) 2 0
546
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
and s i n c e
B
.
lvkl s B(h,p) .+
+
We can e x t r a c t a subsequence, once more denoted by vk, such
m.
Vk
and
4)
> 1, it can be deduced t h a t vk i s bounded, t h u s
(5.32) We l e t k that
(CHAP.
-
v+
I + € 1.
Let v be f i x e d , i n V.
For k > I v / , v
L(X,t,Vk,h,P)
E
Vk, s o t h a t
s
Lb,t,V,h,P)
s
Lb,t,v,hyp)
and hence i n t h e l i m i t L(X,t,v*,h,P)
v
v
E v
.
Thus a t f i x e d x,t,X,p t h e function v + L a t t a i n s i t s minimum on Y. it can e a s i l y b e v e r i f i e d t h a t t h e f u n c t i o n
Moreover
i s continuous i n x , t , X , p and t h u s we can f i n d a measurable s e c t i o n j ( x , t , A , p ) such t h a t
Since
%
and D%
are bounded by fixed constants, (5.32) then clearly f o l l o w s .
Utilising the properties of
€$, (5.31) then
clearly follows.
0
Furthermore, using the Girsanov transformation, we construct a weak solution of the equation (5.34)
df = go(f,s)ds
.
f(t) = x
+ B(s)e(f,s)ds +
ds)dw(s)
From ( 5 . 3 4 ) , we o b t a i n an admissible s y s t e m d ; t a k i n g
6 = inf
It
I da(t),t) = +(s(t),t))
it follows from t h e r e g u l a r i t y p r o p e r t i e s o f t h e function
-
(*IBecause f',(v*).I+ 5 -p. B ( t ) V * hc'~ (I+).@ t h e preceding argument i n order t o o b t a i n ( 5 . 3 3 ) .
u
t h a t we have
and it s u f f i c e s t o use again
(SEC. 6 )
PRINCIPLE OF SEPARATION
547
Likewise, it can e a s i l y be shown, by techniques already encountered, t h a t
&,t)
s
Jxt(J!vJ) Vd,O
hence t h e r e s u l t .
Remark 5.1.
The r e s u l t s of Theorem 5 . 1 supplement what has been s a i d i n
Section 1.
8
(5.35)
f(X,t,V)
B
1x1
2ln
v ,Rp
-
= c
U = $ = O
+
IVl2
- fo
,
2.a
>
1
.
.
(5.36)
6.
THE PRINCIPLE OF SEPARATION.
We now r e t u r n t o t h e s e t t i n g of Section 4.12, Chapter 3 , i . e . corresponding O u r o b j e c t i v e i s t o consider t h e same problem with t o a p a r t i a l observation. simultaneous c o n t r o l v i a a "continuous" v a r i a b l e and v i a stopping times.
6.1.
Assumptions
-
Notation
-
The problem.
The n o t a t i o n and assumptions which w i l l not b e reproduced h e r e , a r e t h o s e of Let V be a compact conurn subset of Rp and l e t B ( t ; v ) Section 4.12, Chapter 3. be a mapping from C0,TI
(6.1)
X
V
B E Co([O,T]
-f
Rn such t h a t
x V)
dB lz lS C
.
F i r s t of a l l we give formally t h e evolution equations of t h e system and of t h e observation; t h e s e a r e :
STOPPING TIMS AND STOCHASTIC OPTIMAL CONTROL
548
(CHAP.
4)
(6.2)
I
(6.3)
1
Y(t) = dz
I
= +5 ,
H(s)y(s)ds
+
dC(s)
,
The expression (6.2) i s formal because we have not s p e c i f i e d t h e c l a s s of We s h a l l i n f a c t do t h i s below, but f i r s t it i s useadmissible c o n t r o l s v ( s ) . f u l t o make a number of comments. Since z ( s ) i s t h e observation, it i s n a t u r a l t o choose v ( s ) adapted t o z ( s ) . But we immediately perceive a d i f f i c u l t y , s i n c e z ( s ) i s not defined independently To overcome t h i s d i f f i c u l t y , it i s n a t u r a l t o proceed as follows. If of v(s). we choose f o r v ( s ) an arbitrary process with values i n V , t h e equation ( 6 . 2 ) can It i s i n f a c t s u f f i c i e n t t o s o l v e on t h e one hand be solved.
I
y l d s + G dw(s)
dy, = F
Y I W =
=
+5
and on t h e o t h e r hand
and it i s c l e a r t h a t y = y
+
y2.
Knowing y , (6.3) f u r n i s h e s t h e observation z.
I n g e n e r a l , of course, v ( S ) We thereby o b t a i n two processesses v(s) and z ( s ) . We can say t h a t v ( s ) i s admissible if v(s) i s w i l l not be adapted t o z ( s ) . adapted t o t h e process z ( s ) constructed as i n d i c a t e d above. Unfortunately t h i s c l a s s of admissible c o n t r o l s i s too wide ( f o r reasons which w i l l become c l e a r e r a l i t t l e l a t e r o n ) . I n order t o define more p r e c i s e l y what we s h a l l understand t o b e an admissible c o n t r o l , we introduce t h e following processes (which a r e independent o f t h e c o n t r o l ) , which have a l l been considered a l r e a d y i n Section 4.12, Chapter 3 :
(6.4)
(6.5)
(6.6)
The processes 5 and y correspond t o t h e s t a t e and t o t h e observation without c o n t r o l (B = 0 ) and have been introduced under t h e n o t a t i o n y , z defined i n s e c t i o n 4.12.
(SEC. 6 )
(6.11)
PRINCIPLE OF SEPARATION
d!
Ebds + PFR'
i(t)
=x
s i n c e R-2 i s d i f f e r e n t i a b l e .
1; (*)
dv(s)
,
But t h e n
=g
This was designated as
ZIt
i n S e c t i o n 4.12.
The l a t t e r n o t a t i o n w i l l be
r e s e r v e d f o r t h e o b s e r v a t i o n p r o c e s s (and i n t r o d u c e d l a t e r ) .
549
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
550
(CHAP.
4)
hence ( 6 . 1 3 ) . Let v(s) be a process with values i n V, adapted t o I:.
We can c o n s t r u c t t h e
system s t a t e y ( s ) and t h e observation z ( s ) , i n t h e manner i n d i c a t e d e a r l i e r . We can a l s o c o n s t r u c t t h e following p r o c e s s , termed a Kalman f i l t e r (which w i l l be j u s t i f i e d l a t e r ) .
I
(6.14)
df = @ds
+
B(s,v(s))ds
+
+
PH+R-l(Hy-Hf)ds
The s o l u t i o n of (6.14) i s c a r r i e d o u t i n two s t a g e s .
PH*R-ldE
,
We s o l v e successively
(6.15)
and we c l e a r l y have
9 = Q 1+ f ,
(6.17) We denote by
i:
*
not e x a c t l y t h e family generated by t h e observation process
( * ) , but t h a t defined by 1
(6.18)
2:
a
o-algebra generated by
z(X) and ?(A),
t
5
A
5 s
.
We then designate a s an admissible control, a V-valued process v(s) adapted t o
: I t h e f a m i l y of o-algebras , f a m i l y 2;
such t h a t , i n addition, v(s) is adapted t o t h e
Remark 6 . 1 . The above d e f i n i t i o n of a n admissible c o n t r o l i s c l o s e t o t h e "natural" d e f i n i t i o n o u t l i n e d e a r l i e r , a p a r t from two r e s t r i c t i o n s . The f i r s t i s t h a t v i s adapted t o t h e innovation i n s t e a d of being a r b i t r a r y . The second is t h a t , i n order t o j u s t i f y such a c o n t r o l i n p r a c t i c e , we would have t o assume t h a t we could observe not only t h e observation process z ( s ) i t s e l f , b u t a l s o t h e For a very wide c l a s s of c o n t r o l s , y ( s ) w i l l be adapted t o
Kalman f i l t e r $ ( s ) .
'2t (we s h a l l s t a t e p r e c i s e l y what t h i s i s ) . But, i n view of o u r a p r i o r i d e f i n i t i o n of t h e Kalman f i l t e r , we do not n e c e s s a r i l y have zs = is and t h e r e f o r e Z:
Z:
t
=
t'
We now t a k e functions f ( x , t , v ) , J l ( x , t ) , G(x) such t h a t
(*)
This i s denoted by Z:
(SEC. 6 )
551
PRINCIPLE OF SEF'AFATION
(6.21 1
.
bounded
;?:, 8
L e t t i n g 8 be a stopping time w . r . t .
(6.22)
JX,(v(.),e) = E
j:
C t , T l , we put ( " )
E
f(y,v,s)ds
xeCT + EE(Y(T))
+ E~J(Y(O),~)
%-T
.
O u r o b j e c t i v e i s t o c h a r a c t e r i s e t h e function
(6.23)
(6.26)
df
fl d s + B ( s , v ( s ) ) d s + PH*R4 dv(s)
P
9(t)
I
dv = R*
x
.
H(y-f)ds
+
R 7 dC
P
R*(d.z-@
,
ds)
thus (6.27)
( * ) A s i n Section 4 . 1 2 , Chapter 3, we could add an a c t u a l i s a t i o n term exp B(s-t), but not a general a c t u a l i s a t i o n term of t h e form
-
exp
1:
c(y,v,h)dh
.
STOPPING TIMES AND STOCHSTIC OPTIMAL CONTROL
552
Since $ ( s ) i s adapted t o , :I
(CHAP.
4)
r e l a t i o n (6.27) a l s o shows t h a t z ( s ) i s adapted
t o :1 and t h u s
it"c It" . Furthermore, w e c an e a s i l y v e r i f y t h e r e l a t i o n
where y* i s a s o l u t i o n of
(6.29)
Now (6.29) shows t h a t y * ( s ) i s adapted t o i s a l s o adapted t o
is, so t
at" I which proves.(6.24). Rf?mUk 6.2.
E(S)
It" c is
t
8
?,:
zst
fact that
and t h u s (6.28) shows t h a t y ( s )
It can e a s i l y be v e r i f i e d t h a t i f v ( s ) i s assumed t o be adapted
t o Z I (and not t o
Remark 6.3.
5:
that
b u t s t i l l of course t o 1 ' ) t I
then we have
IS =
t 3:. From (6.25), and t a k i n g i n t o account t h a t i s independent o f 9, we have
(6.30)
f(s)
= E[Y(s)
I
2: = I:,
and from t h e
t
which j u s t i f i e s t h e terminology "Kalman F i l t e r " f o r $ ( s ) , under t h e conditions where we consider t h a t we are a b l e t o observe
8
t' We now give an important example of an admissible control. Let v ( x , t ) be a mapping from Rn
continuous i n x.
x
C0,TI + V ,
measurable w . r . t .
t and
We put
K(s)
I
P(s)P(s)R-'
(5)
and we note t h a t K(s) i s d i f f e r e n t i a b l e w . r . t . s . (6.26) i n t h e following form ( i n view of (6.27))
Noting t h a t we may w r i t e
we a r e l e d t o introduce t h e following system o f integral equations.
(SEC. 6 )
PRINCIPLE OF SEPARATION
553
I n (6.31), i(s), % ( s ) , y ( s ) a r e unknowns and y ( s ) , v ( x , s ) , along with t h e o t h e r symbols, were defined e a r l i e r . We s h a l l prove i n Lemma 6.2 below t h a t (6.31) possesses a s o l u t i o n ( n o t n e c e s s a r i l y unique) adapted t o 3;. We then p u t
v(s) =
V ( K 4
+
K(s)P(s),s)
which d e f i n e s a process adapted t o 3 '.; We d e f i n e y,p,y*,z
a s before.
i(s)
Y*(s)
9
We now show t h a t D ( s ) i s admissible.
It i s c l e a r t h a t we have
P(s)
z(8)
#
r.(s) + 9(8)z(s)
E
f(s)
*
But then
v(s)
=
V(f(S),8)
which proves t h a t D ( S ) i s adapted t o remains t o prove
and t h e r e f o r e T ( s ) i s admissible.
We s h a l l condense t h e n o t a t i o n somewhat.
1:
We p u t
With t h e ahove n o t a t i o n ( 6 . 3 1 ) can be re-expressed i n t h e form (6.32)
It
The system (6.31) possesses a solution i ( s ) , 2 ( s ) , p(s) adapted
LEMMA 6.2.
t o 3;.
Proof.
5;
+)
= h(s) +
(r(e)(p(e) +
K((p(e),e)) de
.
I n ( 6 . 3 2 ) , h ( s ) i s a continuous process adapted t o 3 ' $ and
STOPPING TIMES AND STOCHASTIC OPTIMAL CONTROL
554
(6.33)
8
-
r(S)
9,s *
(CHAP.
,
i s measurable and I\r(s)l[1 C
~ ( ~ s si s) continuous i n (P
and measurable w . r . t . s a constant independent of 'p,s
I 1 c,
lK('?,s)
. and
In order t o solve (6.32) we adapt Carathkodory' s method ( s e e CODDINGTONWe define an approximation sequence 90 ( s ) by t h e r e l a t i o n s LEVINSON C11).
(6.34)
'pj(s)
h(s)
t5
on
Is-?
J
(r(e)cpj(e> +
cpj(s) = h ( d +
3
T-t ss t + -
K('pj(e),e))
T-t t+-SssT J
on
, de
.
The process cp,(s) i s continuous and (with w f i x e d )
T-t t c - 2 ~
if
J
> 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
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