Stochastic Analysis and Diffusion Processes
1st Edition
Published on 26. November 2013
365 pages
978-0-19-163144-3 (ISBN)
Description
Beginning with the concept of random processes and Brownian motion and building on the theory and research directions in a self-contained manner, this book provides an introduction to stochastic analysis for graduate students, researchers and applied scientists interested in stochastic processes and their applications.
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Gopinath Kallianpur | P. Sundar
Stochastic Analysis and Diffusion Processes
Book
01/2014
Oxford University Press
€80.60
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Content
- Cover
- Preface
- Contents
- 1 Introduction to Stochastic Processes
- 1.1 The Kolmogorov Consistency Theorem
- 1.2 The Language of Stochastic Processes
- 1.3 Sigma Fields, Measurability, and Stopping Times
- Exercises
- 2 Brownian Motion
- 2.1 Definition and Construction of Brownian Motion
- 2.2 Essential Features of a Brownian Motion
- 2.3 The Reflection Principle
- Exercises
- 3 Elements of Martingale Theory
- 3.1 Definition and Examples of Martingales
- 3.2 Wiener Martingales and the Markov Property
- 3.3 Essential Results on Martingales
- 3.4 The Doob-Meyer Decomposition
- 3.5 The Meyer Process for L2-martingales
- 3.6 Local Martingales
- Exercises
- 4 Analytical Tools for Brownian Motion
- 4.1 Introduction
- 4.2 The Brownian Semigroup
- 4.3 Resolvents and Generators
- 4.4 Pregenerators and Martingales
- Exercises
- 5 Stochastic Integration
- 5.1 The Itô Integral
- 5.2 Properties of the Integral
- 5.3 Vector-valued Processes
- 5.4 The Itô Formula
- 5.5 An Extension of the Itô Formula
- 5.6 Applications of the Itô Formula
- 5.7 The Girsanov Theorem
- Exercises
- 6 Stochastic Differential Equations
- 6.1 Introduction
- 6.2 Existence and Uniqueness of Solutions
- 6.3 Linear Stochastic Differential Equations
- 6.4 Weak Solutions
- 6.5 Markov Property
- 6.6 Generators and Diffusion Processes
- Exercises
- 7 The Martingale Problem
- 7.1 Introduction
- 7.2 Existence of Solutions
- 7.3 Analytical Tools
- 7.4 Uniqueness of Solutions
- 7.5 Markov Property of Solutions
- 7.6 Further Results on Uniqueness
- 8 Probability Theory and Partial Differential Equations
- 8.1 The Dirichlet Problem
- 8.2 Boundary Regularity
- 8.3 Kolmogorov Equations: The Heuristics
- 8.4 Feynman-Kac Formula
- 8.5 An Application to Finance Theory
- 8.6 Kolmogorov Equations
- Exercises
- 9 Gaussian Solutions
- 9.1 Introduction
- 9.2 Hilbert-Schmidt Operators
- 9.3 The Gohberg-Krein Factorization
- 9.4 Nonanticipative Representations
- 9.5 Gaussian Solutions of Stochastic Equations
- Exercises
- 10 Jump Markov Processes
- 10.1 Definitions and Basic Results
- 10.2 Stochastic Calculus for Processes with Jumps
- 10.3 Jump Markov Processes
- 10.4 Diffusion Approximation
- Exercises
- 11 Invariant Measures and Ergodicity
- 11.1 Introduction
- 11.2 Ergodicity for One-dimensional Diffusions
- 11.3 Invariant Measures for d-dimensional Diffusions
- 11.4 Existence and Uniqueness of Invariant Measures
- 11.5 Ergodic Measures
- Exercises
- 12 Large Deviations Principle for Diffusions
- 12.1 Definitions and Basic Results
- 12.2 Large Deviations and Laplace-Varadhan Principle
- 12.3 A Variational Representation Theorem
- 12.4 Sufficient Conditions for LDP
- Exercises
- Notes on Chapters
- References
- Index
