Foundations of Stochastic Analysis

Preface

Stochastic analysis consists of a study of different types of stochastic processes and of their transformations, arising from diverse applications. A basic problem in such studies is the existence of probability spaces supporting these processes when only their finite-dimensional distributions can be specified by the experimenter. The first solution to this problem is provided by the fundamental existence theorem of Kolmogorov (1933), according to which such a process, or equivalently a probability space, exists if and only if the set of all finite-dimensional distributions forms a compatible family. This result has been analyzed and abstracted by Bochner (1955), who showed it to be a problem on projective systems of probability spaces and who then presented sufficient conditions for such a system to admit a limit. The latter becomes the desired probability space, and this abstraction has greatly enlarged the scope of Kolmogorov’s idea. One of the purposes of this book is to present the foundations of this theory of Kolmogorov and Bochner and to indicate its impact on the growth of the subject.

An elementary but important observation is that a projective system uniquely associates with itself a set martingale. In many cases the latter can be represented by a (point) martingale. On the other hand, a (point) martingale trivially defines a projective system of (signed) measure spaces. Thus the Kolmogorov-Bochner theory naturally leads to the study of martingales in terms of the basic (and independent) work due to Doob and Andersen–Jessen. However, to analyze and study the latter subject in detail, it is necessary to turn to the theory of conditional expectations and probabilities, which also appears in the desired generality in Kolmogorov’s Foundations (1933) for the first time. This concept seems simple on the surface, but it is actually a functional operation and is nontrivial. To facilitate dealing with conditional expectations, which are immensely important in stochastic analyses, a detailed structural study of these operators is desirable. But such a general and comprehensive treatment has not yet appeared in book form. Consequently, after presenting the basic Kolmogorov-Bochner theorem in Chapter I, I devote Chapter II to this subject. The rest of the book treats aspects of martingales, certain extensions of projective limits, and applications to ergodic theory, to harmonic analysis, as well as to (Gaussian) likelihood ratios. The topics considered here are well suited for showing the natural interplay between real and abstract methods in stochastic analysis. I have tried to make this explicit. In so doing, I attempted to motivate the ideas at each turn so that one can see the appropriateness of a given method.

As the above description implies, a prerequisite for this book is a standard measure theory course such as that given in the Hewitt-Stromberg or Royden textbooks. No prior knowledge of probability (other than that it is a normed measure) is assumed. Therefore most of the results are proved in detail (at the risk of some repetitions), and certain elementary facts from probability are included. Actually, the present account may be regarded as an updating of Kolmogorov’s Foundations (English translation, Chelsea, 1950, 74 pp.) referred to above, and thus a perusal of its first 56 pages will be useful. The treatment and the point of view of the present book are better explained by the brief outline that follows. A more detailed summary appears at the beginning of each chapter.

After introducing the subject, the main result proved in Chapter I is the basic Kolmogorov-Bochner existence theorem referred to above. To facilitate later work and to fix some notation and terminology, a résumé of real and abstract analysis is included here. Occasionally, some needed results that are not readily found in textbooks are presented in full detail. Most of these (particularly Section 4) can be omitted, and the reader may refer to them only when they are invoked. Chapter II is devoted entirely to conditional expectations and probabilities containing several characterizations of these operators and measures. The general viewpoint emphasizes that the Kolmogorov foundations are adequate for all the known applications. This is contrasted with (and is shown to include) the new foundations proposed by Rényi (1955). Then the integral representation of Reynolds operators is given as an application of these ideas, to be used later for a unified study of ergodic-martingale theories. Chapter III contains extensions of the Kolmogorov-Bochner theorem. The existence theorem of Prokhorov and certain other results of Choksi are also proved here. A treatment of direct limits of measures is necessary. This topic and infinite product conditional probabilities (Tulcea’s theorem) are discussed. The work in this chapter is somewhat technical, and the reader might postpone the study of it until later. Chapters IV and V contain several aspects of (discrete) martingale theory. These include both scalar- and vector-valued martingales, their basic convergence, and many applications. The latter deal with ergodic theory, likelihood ratios, the Gaussian dichotomy theorem, and some results on the convergence of partial sums in harmonic analysis on a locally compact group. At the end of each chapter there is a problem section containing several facts, including important results in information theory, and many additions to the text. Most of these are provided with copious hints.

References to the literature are interspersed in the text with (I hope) due credits to various authors, backed up by an extensive bibliography. However, I have not always given the earliest reference of a given result. For instance, all the early work by Doob is referenced to his well-known treatise, and similarly, certain others with references to the monumental work of Dunford-Schwartz, from which an interested reader can trace the original source.

The arrangement of the material is such that this book can be used as a textbook for study following a standard real variable course. For this purpose, the following selections, based on my experience, are suggested: A solid semester’s course can be given using Sections 1-3 of Chapter I, Chapter II (minus Section 6), Sections 1 and 2 of Chapter III, and most of Chapter IV. Then one can use any of the omitted sections with a view to covering Chapter V for the second semester. (This may be appropriately divided for a quarter system.) There is a sufficient amount of material for a year’s treatment, and several possible extensions and open problems are pointed out, both in the text and in the Complements sections of the book. For ease of reference, theorems, lemmas, definitions, and the like are all consecutively numbered. Thus II.4.2 refers to the second item in Section 4 of Chapter II. In a given chapter (or section) the corresponding chapter (and section) number is omitted.

Several colleagues and students made helpful suggestions while the book was in progress. For reading parts of an earlier draft and giving me their comments and corrections, I am grateful to George Chi, Nicolae Dinculeanu, Jerome Goldstein, William Hudson, Tom S. Pitcher, J. Jerry Uhi, Jr., and Grant V. Welland. This work is part of a project that was started in 1968 with a sabbatical leave from Carnegie-Mellon University, continued at the Institute for Advanced Study during 1970-1972, and completed at the University of California at Riverside. This research was in part supported by the Grants AFOSR-69-1647, ARO-D-31-124-70-G100, and by the National Science Foundation. I wish to express my gratitude to these institutions and agencies as well as to the UCR research fund toward the preparation of the final version. I should like to thank Mrs. Joyce Kepler for typing the final and earlier drafts of the manuscript with diligence and speed. Also D. M. Rao assisted me in checking the proofs and preparing the Index. Finally, I appreciate the cooperation of the staff of Academic Press in the publication of this volume.

CHAPTER

I

Introduction and Generalities

This chapter is devoted to a motivational introduction and to preliminaries on real and abstract analysis to be used in the rest of the book. The main probabilistic result is the Kolmogorov–Bochner theorem on the existence of general, not necessarily scalar valued stochastic processes. Also included is a result on the existence of suprema for sets of measurable functions. Several useful complements are included as problems.

1.1 INTRODUCING A STOCHASTIC PROCESS

Stochastic analysis, in a general sense, is a study of the structural and inferential properties of stochastic processes. The latter object may be described as an indexed family of random variables on a probability space. This brief statement implies much more and contains certain hidden conditions on the family. To explain this point clearly and precisely, we use the axiomatic theory of probability, due to Kolmogorov, and show how the basic probability space may be constructed, with the available initial information, in order that a stochastic process may be defined on it. Other axiomatic approaches, notably Rényi’s, are also available, but the methods developed for the Kolmogorov model are adequate for all our purposes. This will become more evident in Chapter II, which elaborates on conditional probabilities, where Rényi’s model is discussed and compared.

Thus, if is a probability space, a mapping (real line) is a (real) random variable if Xt is a measurable function. To fix the notation and for precision, we shall present a résumé of the main results from real analysis in Section 2, which will then be freely used in the book. Let T be an index set and be a family of random variables on . If t1, …,tn are n points from T and x1, …,xn are in or are ± ∞, define the function Ft1, …, tn, called the n-dimensional (joint) distribution function of (Xt1, …, Xtn), by the equation

As n and the t points vary, we get a family of multidimensional distribution functions . Since , from (1) we get at once the following pair of relations:

where (i1, …, in) is any permutation of (1, …, n). The functions are monotone, nondecreasing, nonnegative, and left continuous. Moreover, and . The relations (2) and (3) are called the Kolmogorov compatibility conditions of the family . Thus any indexed family of (real) random variables on a probability space (or equivalently a stochastic process) determines a compatible collection of finite-dimensional distribution functions whose cardinality is that of D, the directed set (by inclusion) of all finite subsets of T.

The preceding description shows that even if the question of existence of a probability space is not settled, it is simple to exhibit compatible families of distribution functions. It will then be natural to inquire into their relation to some (or any) probability space. To see that such families exist, let f1, …, fn be positive, measurable functions on the line each of which has integral equal to 1. Define F1,2,…,n (= Fn, say):

It is clear that {Fn, n ≥ 1} is a family of distribution functions satisfying (2) and (3) with there. A less simple collection is the Gaussian family of distribution functions given by

where K = (kij) is a real symmetric positive definite matrix, is a point of , det(K) = determinant(K), and a prime denotes the transpose. An easy computation, which we omit, shows that the family {Gn, n ≥ 1} of (5) satisfies (2) and (3). Thus one can find many compatible families of distribution functions on . A fundamental theorem of Kolmogorov states that every such compatible family of distribution functions yields a probability space and a stochastic process on it such that the (joint) finite-dimensional distributions on the process are precisely the given distributions. We shall prove this (in a slightly more general form) in Section 3. Thus the existence of a probability space is equivalent to the selection of a compatible family of distributions. Depending on the type of this family (i.e., Gaussian, Poisson, etc.), the probability space , or the stochastic process, is referred to by the same name. Let us first recall some measure theoretical results for convenient reference.

1.2 RÉSUMÉ OF REAL ANALYSIS

In this section we present an account of certain results from measure theory, mostly without proofs. Our purpose is to fix some notation and to make certain concepts precise since the reader is expected to have this background. (The omitted proofs may be found in Halmos [1], Hewitt–Stromberg [1], Royden [1], Sion [1], or Zaanen [1].)

Most of the references to measure will be to the abstract theory set forth by Càrathéodory as follows. Let be a collection of subsets of a point set Ω for which and let be a function such that . We define the set function on Ω by

where . We say that is generated by the pair . Then is an outer measure. Let

for all . The following results holds.

1. Theorem (a) The restriction of to , denoted by , is -additive, and is a -algebra, containing the class of its -null sets (i.e., is complete);

(b) if is a semi-ring and is additive, then and is an -outer measure, i.e., for any

, where is the closure of under countable unions;

(c) under the hypothesis of (b), iff is -additive; and

(d) if , for each there exists a (the closure of under countable intersections), , such that .

[Hence each A has a measurable cover B if is finite and the hypothesis of (b) holds.]

Note that if is defined by for , then is an outer measure and even when is an arbitrary (not necessarily generated) outer measure on Ω. If , we say that is Carathéodory regular, so that admits no more extensions by this procedure. (It may be verified that = for all , although ≠ may occur.) Thus is Carathéodory regular iff it is an -outer measure. In this case, if An↑A, then . We present later the topological regularity of when Ω is a topological space. It should also be remarked that both (b) and (c) of the theorem may be false if is only a lattice.

On extension of measures from a smaller ring to a larger one, a positive answer is provided by part (b) of the above theorem. A more precise result is as follows:

2. Theorem (Hahn) Let Σ0 be an algebra of sets of Ω and let on Σ0 be a countably additive real, complex, or (more generally) -valued function, where is a reflexive Banach space (here means that the series converges unconditionally in the norm of ), then has a unique countably additive extension to , the -algebra generated by Σ0. The same result holds if is positive, and extended real valued if it is -finite in addition.

The vector case (i.e., is -valued) uses a theorem of Pettis stating that weak and strong -additivities are equivalent in a Banach space. The -finite case is the key part of this result; others can then be reduced to this case and can be further generalized.

The following result on the structure of certain measurable functions is particularly useful in probability and is due to Doob [1] and (in the form stated) to Dynkin [1]. We include its proof.

3. Theorem (Doob–Dynkin lemma) Let (Ω, Σ) and be measurable spaces and be measurable for . Let and be a mapping. Then g is -measurable (relative to the Borel algebra of ) iff there exists a measurable such that .

Proof Since , only the converse is nontrivial.

It suffices to prove the relation with the additional assumption that g is a step function.† Indeed, if this is known and g is any -measurable function, then there exists a sequence of -measurable step functions gn such that , and by the special case for some measurable . Let . Then and . Define , and = 0 for . It follows that ; so we need to establish the special case.

Thus let , disjoint. Then there exist such that . We disjunctify . Let T1=S1, and for j > 1, Tj = . Then

by disjointness of Ai. If , then h is measurable,

and the result follows.

The importance of this becomes clear from a specialization:

4. Corollary Let , the Borelian n-space. If is -measurable, and , then is -measurable iff there is a measurable h: such that g = h(f1, …, fn).

We shall see later that the case , gives an interesting application, related to the Kolmogorov existence theorem noted in Section 1. The Lebesgue limit theorems are deducible from the following monotone convergence criterion:

5. Theorem Let on (Ω,Σ) be a measure (or an -outer measure). If 0 ≤ fn ≤  fn + 1 ≤ … are -measurable (or arbitrary) functions on Ω, then

where the integral is defined in the usual manner for the measurable case, or more generally (for both cases) if one sets (region under f+) − (region under f−) when this makes sense for [This integral is evidently subadditive. The region under f+ is the set

The dominated convergence theorem and Fatou’s lemma are deduced from this result immediately.

We next present the (Lebesgue-)Radon-Nikodým and the (Μ. H. Stone extension of) Fubini and Tonelli theorems, which are used often in our work.

6. Definition (a) Let be a complete measure space such that has the finite subset property, i.e., for each , there is , . Then is said to be a localizable measure if each nonempty collection has a supremum , in the sense that for each and (ii) if , then .

(b) The measure has the direct sum property (or is strictly localizable) if there exists a collection

, such that each , satisfies where and is countable.

It can be shown that if has the direct sum property, then is localizable (cf. Zaanen [1, p. 263]). [Thus each -finite measure (special case of the direct sum property) is localizable.] However, it is not known whether or not these two concepts are equivalent.† In case the cardinality of the algebra Σ is at most of the continuum, McShane [1, p. 333] shows that the two notions are equivalent. The concept of localizability was introduced by Segal [1], to whom the precise form of the Radon–Nikodým theorem (given as part (i)) below is due.

7. Theorem (i) (Radon–Nikodým) Let (Ω, Σ) be a measurable space and v, two measures on Σ with finite subset properties. Let v vanish on each null set ( or v is absolutely continuous for ). Then there exists a -unique -measurable function such that iff is localizable. Moreover, a.e., if v is -finite (or only has the direct sum property) and is integrable if .

(ii) (Lebesgue–Radon–Nikodým) Let , v be arbitrary finite measures on Σ. Then v = v1 + v2 uniquely, where and v2 is -singular, in the sense that there exists , and a -unique integrable such that

For a detailed treatment of the Radon–Nikodým theorem, the reader may consult the textbook by Zaanen [1].

8. Theorem (i) (Fubini–Stone) Let , i = 1, 2, be two measure spaces and , their product. Let be a -measurable function. If , then the functions and are measurable relative to 2 and 1, respectively, and moreover,

(ii) (Tonelli) Let 1, 2 be -finite and be just -measurable. Then the conclusions of (i) hold. If 1, 2 are not restricted but if there exist -measurable a.e., such that for each n, then again (5) holds.

This is the most general result known. For instance, if (5) holds for all nonnegative -measurable f, should 1, 2 be localizable? What can we say about these measures? The answers to these questions have interest in real analysis.

In case the basic space Ω is topological, some refinements are possible. We state the regularity of in that situation, as this is needed. To motivate the general definition, it is convenient to recall the Lebesgue–Stieltjes measure on . From the standard theory such a has the following properties: (a) open sets are -measurable, (b) compact sets (are measurable and) have finite measure, and (c) the outer measure of every set can be approximated from above by the measures of open sets, i.e., . Then also has an inner approximation property: (d) C compact} for all open G. Conversely, these properties characterize Lebesgue–Stieltjes measures on. (See, e.g., Sion [1].) This last property is called inner regularity (and (c) is outer regularity) and is the crucial requirement in the general study. It is taken as a definition. Let us state this precisely.

9. Definition (i) Let Ω be a Hausdorff topological space and its Borel algebra (i.e., the -algebra generated by the open or closed sets of Ω). Then a measure on is inner regular, or a Radon measure, if it is locally finite (i.e., every point of Ω is in some open set G of finite -measure, so that each compact set has finite -measure) and for each we have

[It can then be shown that must also be outer regular on .]

(ii) Let Ω be a topological space and Σ0 be an algebra of subsets of Ω. Then an additive set function on is called regular (in the sense of Dunford–Schwartz, or D–S sense) if for each and there exists a pair such that , where is the closure of E and int(F) is the interior of F, and for each , we have .

If in (ii) is a -additive measure and Σ0 is the Borel algebra, then it may be shown that this definition and that of (i) agree. (One needs the work of Schwartz [1, pp. 17–18].) Moreover, the variation measure of in (ii) is also regular. If in (i) Ω is a completely regular space and is the Baire -algebra (i.e., the smallest -algebra with respect to which each real continuous function on Ω is measurable), then of (i) is a regular Baire measure when all sets in the definition are restricted to the Baire compact and open sets.

The following result provides basic information on these (regular) measures and some interrelations.

10. Theorem (i) (Alexandroff) Let Ω be a compact space and be an additive bounded (real or complex) regular (D–S sense) set function on an algebra Σ0 of Ω. Then has a unique -additive extension to and the extended function, also denoted , is regular (D–S sense). [The -additivity part holds even if the boundedness hypothesis is suppressed—a result due to R. P. Langlands.]

(ii) Let Ω be a sigma compact Hausdorff space and a regular Baire measure on Ω. Then there exists a unique Radon measure on Ω such that its restriction to the Baire -algebra coincides with . Moreover, for each Borel set B of finite measure, there is a Baire set D and a Borel null set N such that (symmetric difference).

The result (i) is proved in Dunford–Schwartz [1, p. 138] and (ii) is in Royden [1, p. 314]. An extended discussion of regularity and of Radon measures is given in Schwartz [1]. We find an application of regularity in Chapter II when conditional probabilities are treated.

1.3 THE BASIC EXISTENCE THEOREM

It was noted in Section 1 that the first problem for stochastic processes, and then for an anlysis on them, is to establish their existence when the basic information can be put in terms of a (compatible) family of distribution functions. The solution, due to Kolmogorov [1], will be precisely stated here. To make the structure more explicit, we then establish a slightly more general version of this fundamental result. It will be sufficient for many applications.

1. Theorem (Kolmogorov) Let T be a subset of the real line and t1< … < tn be n points from it. With each such n-tuple let there be given a distribution function Ft1, …, tn. Suppose that the family thus given is compatible, i.e., equations (2) and (3) of Section 1 are satisfied. Let . Let be the -algebra generated by all the sets . Then there exists a unique probability P on and a stochastic process such that

The process is formed by the coordinate functions defined by the equation, .

The compatibility conditions may be expressed more symmetrically using the following abstration due to Bochner [1]. If is the Borel algebra of , and is the Lebesgue–Stieltjes probability determined by the distribution Ft1… tn, so that

then the family of distributions is equivalent to the set of probabilities where D is the set of all finite subsets of T. If , are a pair of elements of D, let < stand for so that D is directed, i.e., (D, <) is partially ordered and for any two elements there is a third (namely, their union) in D, which dominates both. If < , let be the coordinate projection of onto . Thus if

, and in , we have . Then the compatibility conditions (2) and (3) of Section 1 take the form: for < , . The conclusion of the above theorem thus becomes the existence of a probability P on such that where is the coordinate projection. Note that each , being a Lebesgue–Stieltjes probability, is a Radon measure. This formulation admits extensions.

The following result is a generalization of Theorem 1 in that is replaced by a more general space, though the proof still uses the basic ideas of Kolmogorov. On the other hand, it is a specialization of a more inclusive theorem due to Bochner [1], which will be given in Chapter III along with other generalizations. The present intermediate version fits in here.

2. Theorem (Kolmogorov–Bochner) Let T be an index set and D be the directed (by inclusion) family of all finite subsets of T. Let be a family of measurable spaces where Ωt is a Hausdorff space and is the Borel algebra of

, let be a Radon probability on , , where all product spaces are endowed with product topologies. Let ; then there exists a probability P (=PT) on such that , where is the coordinate projection (and are also such projections), iff for each , in D with or. . When this holds, P is restrictedly regular in that for each , we have

where

, the cylinders with compact bases. (Again, gives the process.)

Remark If , , and is given by (2), this result becomes Theorem 1. We also must note that P is not necessarily a Radon measure. Its restricted regularity can be extended slightly, as shown in the corollary that follows, but nothing more can be asserted without further hypothesis and (deeper) analysis. The latter is therefore postponed to Chapter III.

Proof The necessity is simple. In fact, the coordinate projections clearly satisfy the composition rules, for =identity. Moreover, for each open set is an open set in since are coordinate projections. Hence each is continuous and ( , )-measurable. If there is a probability P on such that , then for any , we have, since ,

So , and is a compatible family of probabilities, proving this part. The converse is nontrivial.

Let be the class of cylinders with compact bases, as in the statement. Denote by the class of all cylinders of Ω whose bases are in, , . Then is an algebra since it is closed under unions and complements. Also for , if , so that for an , we have

since . Thus and is an algebra. Moreover, , by the definition of an infinite product -algebra, which in fact is generated by all the cylinders. To prove the sufficiency, and the theorem, it is only required to show that (i) an additive function PT can be unambiguously defined on in terms of the , (ii) the PT is restrictedly regular on (i.e., (3) is true), and (iii) PT is -additive. Then by Theorem 2.2, PT has a unique extension P to satisfying all the conditions of the theorem. Let us prove these three assertions.

(i) To define PT on , let . Then there exist , such that . By directedness of D, there is a such that . Since , we have

But , so is one-to-one. Hence (6) yields the relation . The compatibility of then gives

If we now set , where , the function PT on is unambiguously defined and is nonnegative. Also if A, B are disjoint, there exists (by directedness of D) a such that

. Then

. This implies the first statement. Note that .

(ii) For the restricted regularity, let , so , . Then by definition of PT, and the regularity of , one has

Since PT(·) is increasing by its (sub-)additivity, we also have for all . Hence, with (8), one has

.

(iii) To prove the -additivity of PT on , we use the following property of : If

for each , then . This is a consequence of the topological lemma below, and it will be used now.

For (iii) it suffices to establish that for any . Thus let be given and be such a sequence. We may and do assume that and (by directedness of D). Then by (ii) there exists a , and

Since Cn need not be monotone, let . Clearly, , and we assert that is also an approximating sequence. In fact, the additivity of PT on implies

But

. Thus

.

Similarly, using B2 and C3 and noting that , we get

,

and by induction,

However, , and by (*), for some m0. Hence , and by (10) for , one has

Thus limn . Since is arbitrary, this proves the -additivity of PT on and hence has a unique extension