The Theory of Stochastic Processes III

392 Pages·1979·6.874 MB·English

by  Iosif Il’ich Gihman, Anatolii Vladimirovich Skorohod (auth.)

Grundlehren der
mathematischen Wissenschaften 232
ASeries 0/ Comprehensive Studies in Mathematics
Editors
S. S. ehern J. L. Doob J. Douglas, jr.
A. Grothendieck E. Heinz F. Hirzebruch E. Hopf
S. Mac Lane W. Magnus M. M. Postnikov
W. Schmidt D. S. Scott
K. Stein J. Tits B. L. van der Waerden
Managing Editors
B. Eckmann J. K. Moser
I. I. Gihman A. V Skorohod
The Theory of
Stochastic Processes 111
Translated from the Russian
by S. Kotz
Springer-Verlag
Berlin Heidelberg N ew Y ork
Iosif lI'ich Gihman
Academy of Sciences of the Ukranian SSR
Institute of Applied Mathematics and Mechanics
Donetsk
USSR
Anatolii Vladimirovich Skorohod
Academy of Sciences of the Ukranian SSR
Institute of Mathematics
Kiev
USSR
Translator:
Samuel Kotz
Department of Mathematics
Temple University
Philadelphia, PA 19122
USA
AMS Subject Classification: 34F05, 60Hxx
Library of Congreu, Cataloging in Publication Data
Gikhman, losif lI'ich.
The theory of stochastic processes.
(Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer
Berücksichtigung der Anwendungsgebiete; Bd. 210, 218, 232)
Translation of Teoriya sluchainyh protsessov.
Includes bibliographies and indexes.
1. Stochastic processes. I. Skorokhod, Anatolli Vladimirovich, joint author. 11. Title. III. Series:
Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen; Bd. 210, [etc.].
QA274.G5513 519.2 74-2552
Title of the Russian Original Edition: Teoriya sluchainyh protsessov, Tom IH. Publisher: Nauka,
Moscow, 1975.
All rights reserved.
No part of this book may be translated or reproduced in any form without written permission from
Springer-Verlag.
© 1979 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1979
9 8 7 6 5 4 3 2 1
ISBN-13: 978-1-4615-8067-6 e-ISBN-13: 978-1-4615-8065-2
DOI: 10.1007/978-1-4615-8065-2
Preface
It was originally planned that the Theory of Stochastic Processes would consist of
two volumes: the first to be devoted to general problems and the second to specific
cJasses of random processes. It became apparent, however, that the amount of
material related to specific problems of the theory could not possibly be incJuded
in one volume. This is how the present third volume came into being.
This voJume contains the theory of martingales, stochastic integrals, stochastic
differential equations, diffusion, and continuous Markov processes.
The theory of stochastic processes is an actively developing branch of mathe
matics, and it would be an unreasonable and impossible task to attempt to
encompass it in a single treatise (even a multivolume one). Therefore, the authors,
guided by their own considerations concerning the relative importance of various
results, naturally had to be selective in their choice of material. The authors are
fully aware that such a selective process is not perfecL Even a number of topics
that are, in the authors' opinion, of great importance could not be incJuded, for
example, limit theorems for particular cJasses of random processes, the theory of
random fields, conditional Markov processes, and information and statistics of
random processes.
With the publication of this last volume, we recall with gratitude oUf associates
who assisted us in this endeavor, and express our sincere thanks to G. N. Sytaya,
L. V. Lobanova, P. V. Boiko, N. F. Ryabova, N. A. Skorohod, V. V. Skorohod,
N. I. Portenko, and L. I. Gab.
I. I. Gihman and A. V. Skorohod
Table of Contents
Chapter I. Martingales and Stochastic Integrals 1
§ 1. Martingales and Their Generalizations 1
§ 2. Stochastic Integrals 46
§ 3. Itö's Formula . . . . . . . . . . . 67
Chapter 11. Stochastic Differential Equations 113
§ 1. General Problems of the Theory of Stochastic Differential Equations 113
§ 2. Stochastic Differential Equations without an After-Effect 155
§ 3. Limit Theorems for Sequences of Random Variables and Stochastic
Differential Equations . . . . . . . . . . . . . . . . . . . . 184
Chapter III. Stochastic Differential Equations for Continuous Processes
and Continuous Markov Processes in fYlm . . . . . . . 220
§ 1. Itö Processes 220
§ 2. Stochastic Differential Equations for Processes of Diffusion Type 257
§ 3. Diffusion Processes in m'" .......... . 279
§ 4. Continuous Homogeneous Markov Processes in fYl'" 317
Remarks .. 374
Bibliography 377
Appendix: Corrections to Volumes land II 380
Subject Index 385
Chapter I
Martingales and Stochastic Integrals
§ 1. Martingales and Their Generalizations
Survey of preceding results. We start by recalling and making more precise the
definitions and previously obtained results pertaining to martingales and semi
martingales (cf. Volume I, Chapter II, Section 2 and Chapter III, Section 4).
Let {n, ~, P} be a prob ability space, let T be an arbitrary ordered set (in what
follows only those cases where T is a subset of the extended real line [-00, +00]
will be discussed) and let {iSt, tE T} be tI current of a-algebras (iSt C ~): if tl < t2
then iStl C iSt2' The symbol {g(t), iSt, tE T} or simply {g(t), iSt} denotes an object
consisting of a current of a-algebras {iSt, tE T} on the measurable space {n, ~}
and a random process g(t), tE T, adopted to {i1"t, tE T} (i.e., ger) if i1"t-measurable
for each tE T). This object will also be referred to in what follows as a random
process.
A random process {g(t), iSt, tE T} is called an I5t-martingale (or martingale if
there is no ambiguity concerning the current of a-algebras iSt under considera
tion) provided
(1) Elg(t)1 < 00 \itE T
and
E{g(t) I iSs} = g(s) for s < t, s, tE T;
it is called a supermartingale (submartingale ) if it satisfies condition (1) and
moreover
E{g(t)ll5s} ~ g(s), s < t, s, tE T
(2)
(E(g(t)ll5s}?g(s), s < t).
Observe that the above definition differs from that presented in Volume I since
we now require finiteness of the mathematical expectation of the quantity g(t) in
all cases. Previously, in the case of the supermartingale, for example, only the
finiteness of the expectation EC(t) was assumed.
2 I. Martingales and Stochastic Integrals
The definition presented herein is equivalent to the following: U(t), 15" tE T} is
a martingale (supermartingale ) if for any set Bs E ~ sand for any sand t belonging
to T such that s < t,
JB, JB,
g(t)dP = g(s) dP
Supermartingales and submartingales are also called semimartingales.
In this section we shall consider mainly semimartingales of a continuous
argument.
The space of all real-valued functions on the interval [0, Tl which possess the
left-hand limit for each tE (0, Tl and which are continuous from the right on
[0, T) will be denoted by gg or by gg[O, Tl
Analogous meaning is attached to the notation gg [0, T), gg [0,(0), and gg [0, 00 l.
A number of inequalities and theorems concerning the existence of limits plays
an important role in the martingale theory. The following relationships were
established in Volume I, Chapter 11, Section 2:
If g(t), tE T, is a separable submartingale, then
(4) q=~ p>l,
p-l'
E(g(t)-bf
E [b)
(5) v a, ~sup b ;
-a
leT
he re a + = a for a ;;,: 0 and a + = 0 for a < 0 and v[a, b) denotes the number of
crossings down ward of the half-interval [a, b) by the sampie function of the
process g(t) (a more precise definition is given in Volume I, Chapter 11, Section 2).
We now recall the definition of a closure of a semimartingale.
Let {g(t), 151, tE T} be a semimartingale and let the set T possess no largest
(smallest) element. The random variable TI is called a closure from the right (left) of
the semimartingalet(t) if one can extend the set T by adding one new element b
(a) which satisfies
t<b (t> a) VtE T
and complete the current of u-algebras {6" tE T} by adding the corresponding
u-algebra 6b (6a) so that the extended family of the random variables g(t), tE T',
T' = T u{b} (T' = Tu {a}) also forms an 61-semimartingale.
Theorem 1. Let g(t), tE T, be a separable submartingale, Tc (a, b), and the points
a and b be the limit points for the set T (-00 ~ a < b ~ (0). Then a set A of probability
o exists such that for we A :
1. Martingales and Their Generalizations 3
a) in every interior point t of the set T the limits ~(t-) and ~(t+) exist;
b) if sup {E~+(t), tE T} < 00, then the limit ~(b-) exists; moreover, iffor some to
the family of random variables {~(t), tE [to, b)} is uniformly integrable, then the
limit ~(b-) exists in L, as weil and ~(b-) is a closure fram the right of the
subm artingale ;
c) if limt-+a E~(t» -00 then the family of random variables {g(t), tE (a, ton is
uniformly integrable, the limit ~(a +) exists for every CI) E A also in the sense of
convergence in LI. and ~(a +) is a closure fram the left of the submartingale.
Proo/. The existence with probability 1 of the one-sided limits ~(t-) and ~(t+) for
each tE [a, b] under the condition that sup {E~+(t), tE (a, b)} < 00 was established
in Volume I, Chapter III, Section 4.
Furthermore, if the family {Ht), tE [to, b)} is uniformly integrable then the
convergence of ~(t) to ~(b -) with probability 1 as ti b implies that this con
vergence is also valid in L, and that ~(b -) is a c10sure from the right of the
submartingale ~(t), tE (a, b). This is proved in the same manner as in the case of
submartingales of a discrete argument (Volume I, Chapter II, Section 2).
We need now to verify assertion c).
Let 1= limrla E~(t). Since E~(t) is a monotonically nondecreasing function the
existence of the limit is assured. Moreover, the equality
implies that
sup EI~(t)l,;;;; 2E~+Cto)-1 = C < 00.
'E(a,'ol
In view of Chebyshev's inequality P(B,),;;;; CI N, where B, = {1~(t)1 > N}, i.e.,
P(B,)"'" 0 as N...,. 00 uniformly in t. Let e > 0 be arbitrary and t, be such that
E~(t)-l < e/2 for all t< t,. Then for tE (a, td
JB, 1~(t)1 dP = JW'»N) ~(t) dP+ JW'»-N} ~(t) dP-E~(t)
,;;;; JW'»N~(t0 dP + JWtl>-N} ~(tl) dP - E~(t)
,;;;; JB, 1~(t,)1 dP+~,
so that JB, 1~(t)1 dP < e for all tE (a, td and N sufficiently large. Thus the family
{~(t), tE (a, toJ} is uniformly integrable. The limit !im,"a ~(t) exists with probability
1; therefore it also exists in the sense of convergence in LI,
Now ~(a +) is indeed a c10sure from the left of the submartingale {~(t), tE T}.
This follows from the fact that it is permissible to approach the limit as s ! a under
the integral sign in the inequality
s < t, BEn 15,. 0
'ET
4 1. Martingales and Stochastic Integrals
Remark. It is evident that assertion c) of the theorem is valid for sequences also. In
this case the assertion can be stated as folIows:
If {. .. ~(-n), ~(-n + 1), ... , ~(O)} is a submartingale and lim,. E~( -n) > -00,
then the sequence {g(-n)} is uniformly integrable, the limit ~oo = !im ~(-n) exists
with probability 1 in LIas weil and is a dosure from the left of the submartingale
{~(n), n = ... -k, -k + 1, ... , O}.
In what follows we shall call a semimartingale uniformly integrable if a
corresponding family of random variables ~(t), tE T, is uniformly integrable. We
shall refer to a martingale as integrable if sup {EI~(t)l, tE T} < 00.
Theorem 2. Let T c (a, b) and let a and b be limit points of the set T (-00 ~ a < b ~
(0). In order that martingale {~(t), 6" tE T} be uniformly integrable, it is necessary
and sufficient that a random variable 1] exist such that
(6) Ehl < 00, tE T.
If this condition is satisfied, we can set 1] = limtib ~(t) and the variable 1] will be
uniquely determined (mod P) in the dass o[ all (T{6t, tE T}-measurable random
variables.
Proof. By Theorem 1, if the martingale {~(t), 6t, tE T} is uniformly integrable,
then it possesses a closure from the right and therefore admits representation (6).
Now let martingale ~(t) admit the representation given by formula (6). Then
which implies that
(7) \;fBE6t.
In particular, EI~(t)1 ~ EI1] I. Therefore Chebyshev's inequality implies that
P{I~(t)1 > N} ~ 0 as N ~ 00 uniformly in t. Applying inequa!ity (7) to the set
B = Bt = {1~(t)1 > N} we verify that the family {g(t), tE T} is uniformly integrable.
We need now to prove the uniqueness of representation (6) in the class of all
(T{ts" tE T}-measurable random variables. Suppose that two such representations
exist in terms of the random variables 1];, i = 1, 2. Then
E{?ltst}=o \;ftE T,
where ? = 1] 1 - 1]2·
Thus
for all A belonging to 6t and all tE T, and consequently for all A belonging to
1. Martingales and Their Generalizations 5
um"
tE T}. Since the variable? is u{Ö\, tE T}-measurable, it follows that ? = 0
(mod P). 0
Remark. If U(t), tE T} is a martingale and T contains a maximal element then the
family of random variables ~(t), tE T, is uniformly integrable.
A u{I5" tE T}-measurable random variable TI appearing in representation (6)
is called the boundary value 0/ the martingale ~(t), tE T.
It was shown in Volume I, Chapter III, Section 4 that under very general
assumptions there exists-for a given semimartingale-a stochastically equivalent
process {(t), 15" t;;;.:O} with sampie functions belonging to .@[O,oo), and,
moreover, the current of u-algebras ~, is continuous from the right, i.e.,
'<It>O.
In this section we shall assurne, unless stated otherwise, that the semimartingales
under consideration possess these properties.
Quasi-martingales. Let {15" t;;;.: O} be a current of u-algebras continuous from the
right (1St = ISt+).
Definition. A process {~(t), t;;;.: O} adopted to 1St is called a quasi -martingale
(lSt-quasi-martingale) if
EI~(t)1 < 00 '<It;;;.:O
and
n-I
sup L EI~(td-E{~(tk+l)II5,.}I=v<oo,
k~O
where the supremum is taken over arbitrary values of n and tJ, t2, ... , tn;
O,,;;;;to< tl < t2<· .. < tn <00.
It will be shown below that a study of quasi-martingales can be reduced to a
study ofsemimartingales.
As examples of quasi-martingales, one may note martingales, super
(sub )martingales for which inf E~(t» -00 (sup E~(t)< 00) and also processes
which are differences of two supermartingales. It turns out that these examples
exhaust all the possible quasi-martingales.
Set
8(s, t)=~(s)-E{~(t)ll5s} (s<t), a(t)=E~(t).
Then
n-l n-l
L la(td-a(tk+I)1 = L IE8(tb tk+I)I,,;;;; V,
k~O k~O