Asymptotic Methods in

Asymptotic Method s i n the Theory o f Stochasti c Differential Equation s

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Translations o f

MATHEMATICAL MONOGRAPHS

Volume 7 8

Asymptotic Method s i n the Theor y o f Stochasti c Differential Equation s A. V. Skorokhod

^//i^^^Svjo America n Mathematica l Societ y }h Providence , Rhod e Islan d

10.1090/mmono/078

A. B . CKOPOXOi l

ACHMIITOTHHECKHE METOilbl TEOPM M

CTOXACTH^ECKHX IIH$$EPEHIIHAJIbHbIX

yPABHEHMM

« H A Y K A » , M O C K B A , 198 7

Transla ted fro m th e Russia n b y H . H . McFade n Transla t ion edite d b y Be n Silve r

2000 Mathematics Subject Classification. P r imar y 60-XX ; Secondar y 34 -XX , 4 7 - X X .

ABSTRACT. Th e topic s in this monograph ar e ergodic theory fo r Markov processe s and fo r solution s of stochastic differentia l equations , stochasti c differentia l equation s containin g a smal l parameter , and stabilit y theor y fo r solution s o f systems o f stochasti c differentia l equations . Th e mai n par t o f the materia l i s presente d fo r th e firs t time . Th e boo k i s intende d fo r specialist s i n th e theor y o f random processe s an d it s applications .

Bibliography: 6 6 titles .

Library o f Congres s Cataloging-in-Publicat io n Dat a

Skorokhod, A . V . (Anatoli i Vladimirovich) , 1930 -Asymptotic method s i n th e theor y o f stochasti c differentia l equations . (Translations o f mathematica l monographs , v . 78 ) Translation of : Asimptoticheski e metod y teori i stokhasticheskik h differentsial'nyk h uravnenii . Includes bibliographica l references . 1. Stochasti c differentia l equations . 2 . Asymptoti c expansions . I . Title . II . Series .

QA274.23.S5313 198 9 519. 2 89-1769 8 ISBN 0-8281-4531- 4 (hardcover) ; ISB N 978-0-8218-4686- 5 (softcover )

© 198 9 b y th e America n Mathematica l Society . Al l right s reserved . Reprinted b y th e America n Mathematica l Society , 2008 .

Translation authorize d b y th e Ail-Union Agenc y fo r Authors ' Rights , Mosco w

The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government .

Printed i n th e Unite d State s o f America .

Information o n copyin g an d reprintin g ca n b e foun d i n th e bac k o f thi s volume . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s

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10 9 8 7 6 5 4 3 2 1 1 3 1 2 1 1 1 0 09 0 8

Contents

Foreword i x

List of Notation x i

Introduction xii i

CHAPTER I . Ergodic theorems 1 § 1. General ergodic theorems 1

1.1. Ergodic theorems for semigroups of measure-preservin g transformations 1

1.2. Homogeneous Markov processes. Invariant measures and ergodic theorems 6

1.3. Harris recurrence 1 5 §2. Densities for transition probabilities and resolvents for

Markov solutions of stochasti c differential equation s 2 3 2.1. Nondegenerate diffusio n processe s 2 4 2.2. Diffusion processe s with degenerate diffusion 2 7 2.3. Processes with jumps 3 5

§3. Ergodic theorems for one-dimensional stochasti c equations 4 1 3.1. Diffusion processe s on the line 4 2 3.2. Diffusion processe s on an interval 5 3 3.3. Processes with reflection a t the boundary 5 5

§4. Ergodic theorems for solutions of stochastic equations in Rd 5 7 4.1. Invariant measures for processes on compact spaces 5 8 4.2. Locally compact spaces 6 3 4.3. Solutions of stochasti c equations in Rd 6 6

CHAPTER II . Asymptotic behavior of systems of stochastic equations containing a small parameter 7 7 §1. Equations with a small right-hand side 7 7

1.1. A general theorem on convergence to a diffusion proces s 7 7

vi CONTENT S

1.2. Ordinary differentia l equation s with a rando m right-hand sid e 8 0

1.3. A theorem o n integra l continuit y wit h respec t t o a parameter fo r diffusio n processe s 9 7

1.4. Stochasti c equations with smal l diffusio n 9 9 §2. Processes with rapi d switchin g 10 2

2.1. Processes with a discrete component 10 3 2.2. An ergodic theorem fo r jump processe s 10 6 2.3. An estimate fo r a process with a discrete component 11 0 2.4. A limit theore m fo r processe s with rapidl y varyin g

discrete component 11 4 2.5. Dynamical system s with rapi d switchin g 11 7

§3. Averaging over variables fo r system s of stochasti c differentia l equations 13 4 3.1. A general theorem o n averagin g 13 4 3.2. A diffusion proces s under the influence o f a rapid

dynamical syste m i n the presence of feedback 14 4 3.3. A dynamical syste m unde r the influence o f a rapid

diffusion process . Neutral cas e 15 6 3.4. A dynamical syste m unde r th e influenc e o f a rapid

diffusion process . Neutral case , large times 16 3

CHAPTER III . Stability . Linea r systems 18 3 §1. Stability of sample paths of homogeneou s Marko v processe s 18 3

1.1. Definition 18 3 1.2. A Feller process on a compact metri c space 18 7 1.3. Stabilit y and instabilit y o f one-dimensional continuou s

processes 19 6 1.4. Stabilit y and instabilit y o f Felle r processe s in a locally

compact spac e 19 8 §2. Linear equations i n R d an d th e stochasti c semigroup s

connected wit h them. Stability 20 5 2.1. Linea r equation s 20 5 2.2. Operator equations . Representation o f solution s 21 1 2.3. Commutative cas e 22 2 2.4. Homogeneous case . Invariant subspace s 22 5 2.5. Mean squar e stability 23 1 2.6. Stability with probability 1 23 6 2.7. p-Stability 24 6

CONTENTS vi i

§3. Stability of solutions of stochastic differential equation s 25 1 3.1. Stability an d instabilit y i n first approximation 25 1 3.2. Diffusion equation s with homogeneous coefficient s 25 9

CHAPTER IV . Linear stochastic equations i n Hilbert space . Stochastic semigroups . Stability 27 1 §1. Linear equations with bounded coefficient s 27 1

1.1. General equations i n Hilber t spac e 27 1 1.2. Linea r equation s 27 9 1.3. Linear stochasti c equations i n Hilber t spac e 28 5 1.4. Stochasti c Hilbert-Schmid t semigroup s 29 1

§2. Strong stochastic semigroup s with second moments 29 6 2.1. Stron g and weak random operator s 29 6 2.2. Processes with independen t increment s that ar e

continuous i n | | • || 5 30 0 2.3. A stochastic differentia l equatio n 30 5 2.4. Second-order stochasti c semigroups o f bounde d

variation 30 9 2.5. Stochasti c equations o f diffusion typ e with constan t

coefficients 31 8 §3. Stability 32 2

3.1. Examples of stable and unstabl e infinite-dimensiona l systems 32 2

3.2. Stability i n the mean squar e 32 7

Bibliography 33 3


Foreword

The 198 2 boo k o n stochasti c differentia l equation s writte n jointly b y the autho r an d Iosi f Il'ic h Gikhma n di d no t includ e a number o f area s in this theory that are important for applications . Therefore , w e decided to write a book that would bring together materia l relating to applied ar-eas in the theory of stochasti c equations. W e intended to treat equations in infinite-dimensional spaces , in particular, infinite system s of stochasti c equations; the theory of linear equations in infinite-dimensional spaces and the semigroups connected with them, in particular , stochasti c partial dif-ferential equation s of evolution type; equations for conditionally Markov processes an d the equations o f nonlinea r filtration connected wit h them; and the asymptotic behavior of solutions of stochastic equations, including ergodic theory, the method o f averaging, an d the theory of stability . Th e plan of th e book wa s discussed fo r a fairly lon g time, an d we convince d ourselves at last that it was impossible to present all these topics in a single book. W e then decided to treat the last topic. Thi s choice was made under the influenc e o f th e interests of Iosi f Il'ich , who, a s a student o f Nikola i Nikolaevich Bogolyubov, had directed much attention to the study of the asymptotic behavior of systems undergoing random perturbations.

A serious illness did not permit Iosif Il'ic h to work on this book. No w he i s no longer, but the book i s published. I t would certainly have been different i f h e had take n part in it s writing—h e ha d a better feeling fo r the "physical" aspects of mathematica l theorie s and could convey thi s in his expositions, thus giving them more substance. Moreover , he knew far more than was written in his (and others') works.

While recognizing how far this book was from what we had envisioned, I wrote it nevertheless, hoping at least by the choice of topic to pay homage to the shining memory of my teacher and friend.

A. V. Skorokhod

ix


List of Notation

R—the rea l line.

i?+—the se t of nonnegative numbers .

a A b and a V b—the smalle r and larger o f the respective number s

a,beR.

Rd—the rf-dimensional Euclidea n space .

\x\—the absolut e value of a number x € R o r the norm o f

a vector x e X, wher e X i s a Euclidean space .

(x,y)—the inne r produc t i n a Euclidean space .

X x Y —the Cartesion produc t o f set s X an d Y.

(x9y)—an elemen t of X x Y\x e X, y e 7 .

&x,<^{X)—the a-algebr a o f Bore l subsets of a metric space X.

(Ra)ms—Lebesgue measur e on a set S.

sf ®>38 —the product o f cr-algebra s srf and SS.

\lstfn—the smallest cr-algebr a containing srfn.

a(£Q, a e A) —the cr-algebr a generated by the variables {£ a,ae A}.

L(X, Y) —the linea r space of linear operator s fro m a linear spac e

A" to a linear space F .

\\A\\—the norm o f a linear operator A e L{X, Y).

A*— th e operato r adjoin t t o A {A* e L(Y,X)).

{e^}—an orthonorma l basis in a Euclidean spac e X.

irA = 2^^(Ae k,ek).

x o y e L{X, X) —defined b y (x o y)z = (x, z)y, wher e X i s a Euclidean

space.

(p'{x)—the function i n L(X, Y) define d fo r tp : X — • Y b y the

equality

<p'(x)y = j-9(x + ty) I /=o, x,yeX, teR.

\\<p\\= sup|«?(x)| .

Cx—the spac e of continuous functions o n X.


Introduction

Asymptotic problem s fo r stochasti c differentia l equation s aros e an d were solved simultaneously with the very beginnings of the theory of such equations, because the founder of this theory, I . I . Gikhman, was consid-ering first and foremos t problem s o n asymptoti c behavior , an d h e con -structed the equation s themselve s partly i n orde r to be abl e to pose an d solve thes e problem s rigorously . I n thi s he , a s a studen t o f N . N . Bo -golyubov, wa s continuin g th e tradition s o f th e ne w directio n develope d in the 1930' s by N. M . Krylov and Bogolyubov i n investigations o n non-linear mechanics—th e stud y of system s subjec t t o th e actio n o f rando m perturbations. A cycle of papers by Krylov and Bogolyubov [l]-[5 ] were devoted t o thes e investigations . The y established , i n particular , ergodi c theorems for Markov processes with a phase space of a very general form. Special mention should be made of [1] , in which a study was made of the behavior o f a system subjec t t o th e actio n o f a rapidly variabl e rando m force that becomes a "white noise" in the limit. I t is this paper that served as an impetus for the creation by Gikhman of the theory of stochastic dif-ferential equations . I n [l]-[5] various approaches were considered to the rigorous definition o f a dynamical syste m subjec t to the actio n o f a ran-dom force o f "whit e noise" type, as well a s the definition o f a stochastic differential equatio n in a random field of forces with independent values, and results were obtained on the asymptotic behavior of the system when the field varies (fo r example , whe n impuls e action s becom e continuou s actions). (It o used the convenient concep t of a stochastic integral to con-struct a stochasti c equatio n i n [1 ] and [2] ; this for m o f th e equatio n i s more accepted at present.)

We indicate tw o direction s i n the asymptoti c investigatio n o f system s with random actions: 1 ) investigation of the behavior of systems as t —> oo , and 2) investigation of systems depending on a small parameter as this pa-rameter tends to zero. Th e mixed problem also relates here—investigation

xiii

XIV INTRODUCTION

of a syste m a s a parameter tend s t o zer o an d t tend s t o infinit y simul -taneously.

The mai n system s considere d ar e thos e describabl e b y Marko v proc -esses that are, in turn, solutions of stochastic differential equations . How -ever, many of the results are simpler to formulate an d prove for Markov processes, and even for processes of a more general form. I t is often con-siderations of convenience that dictate the choice of the form of a system. We remark als o that , i n additio n t o problem s o n th e behavio r o f a sys-tem, ne w problems connected with the study of the asymptoti c behavio r of distributions (transitio n probabilities) aris e for stochastic systems.

In considering the asymptotic behavior of a system as t —• oo we are pri-marily interested in a definite "stabilization" of the system. Thi s term can be used t o characteriz e an y regularity tha t manifest s itsel f i n th e behav-ior of th e system . Th e crudest typ e o f suc h stabilization i s boundedness in probability . Unde r fairl y natura l assumption s abou t th e probabilisti c properties of the system, boundedness in probability implies ergodicity— this property characterize s mor e precisely th e behavior o f th e syste m o n the whole unbounded interva l o f variation. Eve n when the system i s not bounded in probability, it can fail to diverge to infinity but instead return to a neighborhood of the original state with probability 1 . The n it has an infinite invarian t measure , an d we ca n judge th e qualitativ e behavio r o f the system on the basis of exact quantitative laws.

Although ergodi c theor y (includin g ergodi c theor y fo r Marko v proc -esses) is very well developed, some questions connected with this theory, as well as some results relating specifically to solutions of stochastic equations, are appearing here for the first time i n a monograph. Shurenkov' s book [1] contains the most complete reflection of the state of ergodic theory for Markov processes, along with a detailed bibliography.

Questions involving (asymptotic) stability of a system in a neighborhood of an equilibrium state or involving instability of the system arise naturally in the study of the behavior of systems on an infinite interval . Unde r very general assumptions , stabilit y implie s asymptoti c stabilit y fo r stochasti c systems, an d instabilit y wit h positiv e probabilit y implie s instabilit y wit h probability 1 . Linear systems for which the point 0 is the only equilibrium point are of special interest. Suc h systems are either stable or unstable. I n the latter case the system either diverges to infinity, or oscillates and hence has an invariant measure.

Gikhman founded the theory of stability for solutions of stochastic dif-ferential equations in [6] and [7], and then Khas'minskil developed it fur-ther i n [l]-[5] . W e not e tha t th e stud y o f stabilit y o f linea r system s i s

INTRODUCTION xv

closely connecte d wit h th e stud y o f product s o f independen t identicall y distributed matrices (about this see Bellman, Kesten, and Furstenberg (see Furstenberg [1]) , Tutubalin [1] , and Sazono v an d Tutubalin [1]) .

We mentio n als o result s o f Kulinich [1 ] tha t hav e no t appeare d i n a book: fo r recurren t processe s h e foun d condition s fo r th e existenc e o f a limit distributio n fo r a solution o f a stochastic equatio n unde r a suitabl e normalization.

Carrying result s relatin g t o stochasti c equation s i n finite-dimensiona l spaces ove r t o th e infinite-dimensiona l cas e is fa r fro m trivial . Althoug h the form o f stochasti c equatio n propose d b y Gikhma n i s insensitive t o a change in the dimension o f the space, the more natural form base d on the Ito integral needed a certain reinterpretation (Daletski i [1], [2]). The study of linear systems led to the concept of a stochastic semigroup (Skorokho d [1]> [2], [4], and Butsa n [1]) . Mean-squar e stabilit y o f solution s o f linea r equations involve s stability o f certai n no w nonrandom semigroup s i n th e Banach spac e o f linea r operator s actin g i n a Hilber t space . Ther e i s a fairly complet e exposition o f the theory of stability of such semigroups in Daletskii and Krein's book [1] . A small parameter i n the equation has the effect tha t som e term s i n th e equatio n becom e larg e i n compariso n wit h others, and sinc e a stochastic differentia l equatio n contain s fou r differen t terms (th e differentia l o f th e unknow n solution , th e drift , th e diffusion , and th e jumps), w e obtain differen t problem s wit h a smal l paramete r b y placing th e smal l paramete r a s a coefficien t o f differen t group s o f terms . Most natural is the problem when the system is determined by an ordinary differential equatio n wit h a smal l rando m perturbation . The n unde r a mixing conditio n fo r th e proces s o n th e right-han d sid e i t behave s like a solution o f a stochastic equation o f diffusion typ e on large time intervals . Another class of problems is connected with the presence of rapidly varying components i n th e system . I f thes e component s hav e ergodi c properties , then thei r effec t o n the remainin g component s i s "averaged", i.e. , fo r th e latter a closed equation i s obtained whos e coefficients ar e the coefficient s of the original equation , average d with respec t to an ergodi c distribution . These kinds o f theorem s generaliz e th e Bogolyubo v metho d o f averagin g to random systems. Gikhma n an d Khas'minskil occupied themselves with the justification o f the Bogolyubov method of averaging in various degrees of generality in the case of stochastic equations (se e also Stratonovich [1], [2], V. V. Sarafyan [1] , and Sarafya n an d Skorokho d [1]) .

We remark tha t fo r finit e Marko v chain s an d semi-Marko v processe s such a metho d o f averagin g wa s develope d b y Korolyu k an d Turbi n [1 ] (see also Turbin [1] ) as a method o f asymptoti c phase amalgamation .

XVI INTRODUCTION

A specia l plac e i s occupie d b y th e clas s o f problem s o n th e behavio r of a dynamica l syste m unde r th e influenc e o f a smal l diffusion . The y have bee n investigate d b y Venttsel ' an d Freidli n [1 ] (se e als o Venttsel ' [1], an d Sarafya n [1]) , and relat e t o th e determinatio n o f a n asymptoti c expression fo r th e probabilit y o f unlikel y events (larg e deviations ) suc h as, fo r example , th e syste m reachin g th e boundar y o f a domai n whos e interior contains a point o f stable equilibrium, du e to a small diffusion o r a transition o f the system from on e stable stat e to another .

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1. Fluctuation in dynamical systems under the influence of random per-turbations, "Nauka" , Moscow , 1979 ; English transl. , Random per-turbations of dynamical systems, Springer-Verlag , 1984 .


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