This new, thoroughly revised and expanded 3rd edition of a classic gives a comprehensive coverage of modern probability in a single book. It is a truly modern text, providing not only classical results but also material that will be important for future research. Much has been added to the previous edition, including eight entirely new chapters on subjects like random measures, Malliavin calculus, multivariate arrays, and stochastic differential geometry. Apart from important improvements and revisions, some of the earlier chapters have been entirely rewritten. To help the reader, the material has been grouped together into ten major areas, each arguably indispensable to any serious graduate student and researcher, regardless of their specialization.
Each chapter is largely self-contained and includes plenty of exercises, making the book ideal for self-study and for designing graduate-level courses and seminars in different areas and at different levels. Extensive notes and a detailed bibliography make it easy to go beyond the presented material if desired.
Author(s): Olav Kallenberg
Series: Probability Theory and Stochastic Modelling 99
Edition: 3
Language: English
Pages: 946
Tags: Probability, Measure, Martingale, Markov Process, Stochastic Calculus
Preface to the First Edition
Preface to the Second Edition
Preface to the Third Edition
Acknowledgments
Words of Wisdom and Folly
Contents
Introduction and Reading Guide
I. Measure Theoretic Prerequisites
Chapter 1 Sets and Functions, Measures and Integration
Chapter 2 Measure Extension and Decomposition
Chapter 3 Kernels, Disintegration, and Invariance
II. Some Classical Probability Theory
Chapter 4 Processes, Distributions, and Independence
Chapter 5 Random Sequences, Series, and Averages
Chapter 6 Gaussian and Poisson Convergence
Chapter 7 Infinite Divisibility and General Null Arrays
III. Conditioning and Martingales
Chapter 8 Conditioning and Disintegration
Chapter 9 Optional Times and Martingales
Chapter 10 Predictability and Compensation
IV. Markovian and Related Structures
Chapter 11 Markov Properties and Discrete-Time Chains
Chapter 12 Random Walks and Renewal Processes
Chapter 13 Jump-Type Chains and Branching Processes
V. Some Fundamental Processes
Chapter 14 Gaussian Processes and Brownian Motion
Chapter 15 Poisson and Related Processes
Chapter 16 Independent-Increment and Lévy Processes
Chapter 17 Feller Processes and Semi-groups
VI. Stochastic Calculus and Applications
Chapter 18 Itô Integration and Quadratic Variation
Chapter 19 Continuous Martingales and Brownian Motion
Chapter 20 Semi-Martingales and Stochastic Integration
Chapter 21 Malliavin Calculus
VII. Convergence and Approximation
Chapter 22 Skorohod Embedding and Functional Convergence
Chapter 23 Convergence in Distribution
Chapter 24 Large Deviations
VIII. Stationarity, Symmetry and Invariance
Chapter 25 Stationary Processes and Ergodic Theory
Chapter 26 Ergodic Properties of Markov Processes
Chapter 27 Symmetric Distributions and Predictable Maps
Chapter 28 Multi-variate Arrays and Symmetries
IX. Random Sets and Measures
Chapter 29 Local Time, Excursions, and Additive Functionals
Chapter 30 Random Measures, Smoothing and Scattering
Chapter 31 Palm and Gibbs Kernels, Local Approximation
X. SDEs, Diffusions, and Potential Theory
Chapter 32 Stochastic Equations and Martingale Problems
Chapter 33 One-Dimensional SDEs and Diffusions
Chapter 34 PDE Connections and Potential Theory
Chapter 35 Stochastic Differential Geometry
Appendices
1. Measurable maps
2. General topology
3. Linear spaces
4. Linear operators
5. Function and measure spaces
6. Classes and spaces of sets
7. Differential geometry
Notes and References
Bibliography
Indices
Authors
Topics
Symbols
