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Foundations of Stochastic Analysis
1st Edition - September 28, 1981
Author: M. M. Rao
Editors: Z. W. Birnbaum, E. Lukacs
9 7 8 - 1 - 4 8 3 2 - 6 9 3 1 - 3
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… Read more
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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.
PrefaceChapter 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 ProblemsChapter 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 ProblemsChapter 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 ProblemsChapter 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 ProblemsChapter 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 ProblemsBibliographyIndex