Foundations of Stochastic Analysis

Foundations of Stochastic Analysis

1st Edition - September 28, 1981

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  • Author: M. M. Rao
  • eBook ISBN: 9781483269313

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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

  • Preface

    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



Product details

  • No. of pages: 310
  • Language: English
  • Copyright: © Academic Press 1981
  • Published: September 28, 1981
  • Imprint: Academic Press
  • eBook ISBN: 9781483269313

About the Author

M. M. Rao

About the Editors

Z. W. Birnbaum

E. Lukacs

Affiliations and Expertise

Bowling Green State University

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