Foundations of 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