Foundations of Stochastic Analysis - 1st Edition - ISBN: 9780125808507, 9781483269313

Foundations of Stochastic Analysis

1st Edition

Authors: M. M. Rao
Editors: Z. W. Birnbaum E. Lukacs
eBook ISBN: 9781483269313
Imprint: Academic Press
Published Date: 28th September 1981
Page Count: 310
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Foundations of Stochastic Analysis deals with the foundations of the theory of Kolmogorov and Bochner and its impact on the growth of stochastic analysis. Topics covered range from conditional expectations and probabilities to projective and direct limits, as well as martingales and likelihood ratios. Abstract martingales and their applications are also discussed.

Comprised of five chapters, this volume begins with an overview of the basic Kolmogorov-Bochner theorem, followed by a discussion on conditional expectations and probabilities containing several characterizations of operators and measures. The applications of these conditional expectations and probabilities to Reynolds operators are also considered. The reader is then introduced to projective limits, direct limits, and a generalized Kolmogorov existence theorem, along with infinite product conditional probability measures. The book also considers martingales and their applications to likelihood ratios before concluding with a description of abstract martingales and their applications to convergence and harmonic analysis, as well as their relation to ergodic theory.

This monograph should be of considerable interest to researchers and graduate students working in stochastic analysis.

Table of Contents


Chapter I Introduction and Generalities

1.1 Introducing a Stochastic Process

1.2 Résumé of Real Analysis

1.3 The Basic Existence Theorem

1.4 Some Results from Abstract Analysis and Vector Measures

1.5 Remarks on Measurability and Localizability

Complements and Problems

Chapter II Conditional Expectations and Probabilities

2.1 Introduction of the Concept

2.2 Some Characterizations of Conditional Expectations

2.3 Conditional Probabilities

2.4 Some Characterizations of Conditional Probabilities

2.5 Relations with Rényi's New Axiomatic Approach

2.6 Applications to Reynolds Operators

Complements and Problems

Chapter III Projective and Direct Limits

3.1 Definition and Immediate Consequences

3.2 Some Characterizations of Projective Limits

3.3 Direct Limits and a Generalized Kolmogorov Existence Theorem

3.4 Infinite Product Conditional Probability Measures

3.5 A Multidimensional Extension

Complements and Problems

Chapter IV Martingales and Likelihood Ratios

4.1 Definition and Fundamental Inequalities

4.2 Convergence Theory

4.3 Extensions to Infinite Measures

4.4 Applications to Likelihood Ratios

4.5 Asymptotic Martingales

Complements and Problems

Chapter V Abstract Martingales and Applications

5.1 Introduction

5.2 Abstract Martingales and Convergence

5.3 Martingales and Ergodic Theory

5.4 A Unified Formulation of Some Ergodic and Martingale Theorems

5.5 Martingales in Harmonic Analysis

Complements and Problems




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© Academic Press 1981
Academic Press
eBook ISBN:

About the Author

M. M. Rao

About the Editor

Z. W. Birnbaum

E. Lukacs

Affiliations and Expertise

Bowling Green State University

Ratings and Reviews