Foundations of Stochastic Analysis

By M. M. Rao

Stochastic analysis involves the study of a process involving a randomly determined sequence of observations, each of which represents a sample of one element of probability distribution. This volume considers fundamental theories and contrasts the natural interplay between real and abstract methods.
Starting with the introduction of the basic Kolmogorov-Bochner existence theorem, the text explores conditional expectations and probabilities as well as projective and direct limits. Subsequent chapters examine several aspects of discrete martingale theory, including applications to ergodic theory, likelihood ratios, and the Gaussian dichotomy theorem. Prerequisites include a standard measure theory course. No prior knowledge of probability is assumed; therefore, most of the results are proved in detail. Each chapter concludes with a problem section that features many hints and facts, including the most important results in information theory.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 17, 2013
• ISBN: 9780486296531

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Index

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

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.

(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 . If t1, …,tn are n points from T and x1, …,xn 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 , from (1) we get at once the following pair of relations:

where (i1, …, in) is any permutation of (1, …, n. The relations (2) and (3) are called the Kolmogorov compatibility conditions . 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.

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 there. A less simple collection is the Gaussian family of distribution functions given by

where K = (kij, det(K) = determinant(K), and a prime denotes the transpose. An easy computation, which we omit, shows that the family {Gn, n . 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, 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].)

on Ω by

is generated is an outer measure. Let

. 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 (biff 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.]

is Carathéodory regular-outer measure. In this case, if An↑Ais 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.

-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 -measurable step functions gn ; so we need to establish the special case.

. Let T1=S1, and for j > 1, Tj . Then

by disjointness of Ai, 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 hsuch that g = h(f1, …, fn).

, 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 has the finite subset propertyis said to be a localizable .

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

is countable.

-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 vtwo 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. Moreovera.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 2 and 1, respectively, and moreover,

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

-measurable f2 be localizable? What can we say about these measures? The answers to these questions have interest in real analysis.

C compact} for all open G. (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 is inner regular, or a Radon measure, if it is locally finite (i.e., every point of Ω is in some open set G we have

.]

is called regular (in the sense of Dunford–Schwartz, or D–S senseis the closure of E and int(F) is the interior of F.

is the Baire -algebra 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 such that

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

is the Lebesgue–Stieltjes probability determined by the distribution Ft1… tn, so that

of probabilities where D is the set of all finite subsets of T. are a pair of elements of Dso that D is directed, i.e., (D, <) is partially ordered and for any two elements there is a third (namely, their union) in D. Thus if

. Then the compatibility conditions (2) and (3) of . The conclusion of the above theorem thus becomes the existence of a probability P , being a Lebesgue–Stieltjes probability, is a Radon measure. This formulation admits extensions.

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 basesgives the process.)

Remark 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 )-measurable. If there is a probability P ,

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

, we have

-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 , (ii) the PT (i.e., (3) is true), and (iii) PT -additive. Then by Theorem 2.2, PT has a unique extension P satisfying all the conditions of the theorem. Let us prove these three assertions.

(i) To define PT . By directedness of D, we have

then gives

, the function PT is unambiguously defined and is nonnegative. Also if A, B are disjoint, there exists (by directedness of Dsuch that

. Then

.

. Then by definition of PT, one has

Since PT. Hence, with (8), one has

.

-additivity of PT : If

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

(by directedness of D). , and

Since Cn is also an approximating sequence. In fact, the additivity of PT implies

But

. Thus

.

Similarly, using B2 and C, we get

,

and by induction,

for some m, one has

Thus limn -additivity of PT and hence has a unique