Rao / Birnbaum / Lukacs Foundations of Stochastic Analysis

1. Auflage 2014
ISBN: 978-1-4832-6931-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 310 Seiten, Web PDF

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.

Autoren/Hrsg.

Rao, M. M.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover
;1
2;Foundations of Stochastic Analysis;4
3;Copyright Page
;5
4;Dedication;6
5;Table of Contents;8
6;Preface;10
7;Chapter I. Introduction and Generalities;14
7.1;1.1 INTRODUCING A STOCHASTIC PROCESS;14
7.2;1.2 RÉSUMÉ OF REAL ANALYSIS;16
7.3;1.3 THE BASIC EXISTENCE THEOREM;21
7.4;1.4 SOME RESULTS FROM ABSTRACT ANALYSIS AND VECTOR MEASURES;29
7.5;1.5 REMARKS ON MEASURABILITY AND LOCALIZABILITY;43
8;Chapter II. Conditional Expectations and Probabilities;50
8.1;2.1 INTRODUCTION OF THE CONCEPT;50
8.2;2.2 SOME CHARACTERIZATIONS OF CONDITIONAL EXPECTATIONS;57
8.3;2.3 CONDITIONAL PROBABILITIES;75
8.4;2.4 SOME CHARACTERIZATIONS OF CONDITIONAL PROBABILITIES;95
8.5;2.5 RELATIONS WITH RÉNYI'S NEW AXIOMATIC APPROACH;100
8.6;2.6 APPLICATIONS TO REYNOLDS OPERATORS;108
9;Chapter III. Projective and Direct Limits;129
9.1;3.1 DEFINITION AND IMMEDIATE CONSEQUENCES;129
9.2;3.2 SOME CHARACTERIZATIONS OF PROJECTIVE LIMITS;136
9.3;3.3 DIRECT LIMITS AND A GENERALIZED KOLMOGOROV EXISTENCE THEOREM;152
9.4;3.4 INFINITE PRODUCT CONDITIONAL PROBABILITY MEASURES;165
9.5;3.5 A MULTIDIMENSIONAL EXTENSION;169
10;Chapter IV. Martingales and Likelihood Ratios;183
10.1;4.1 DEFINITION AND FUNDAMENTAL INEQUALITIES;183
10.2;4.2 CONVERGENCE THEORY;193
10.3;4.3 EXTENSIONS TO INFINITE MEASURES;205
10.4;4.4 APPLICATIONS TO LIKELIHOOD RATIOS;211
10.5;4.5 ASYMPTOTIC MARTINGALES;236
11;Chapter V. Abstract Martingales and Applications;254
11.1;5.1 INTRODUCTION;254
11.2;5.2 ABSTRACT MARTINGALES AND CONVERGENCE;258
11.3;5.3 MARTINGALES AND ERGODIC THEORY;269
11.4;5.4 A UNIFIED FORMULATION OF SOME ERGODIC AND MARTINGALE THEOREMS;272
11.5;5.5 MARTINGALES IN HARMONIC ANALYSIS;278
12;Bibliography;296
13;Index;304
14;Probability and Mathematical Statistics;309