An introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text, in order to motivate and illustrate the theory and show its importance for many applications in e.g. economics, biology and physics. The basic idea of the presentation is to start from some basic results (without proofs) of the easier cases and develop the theory from there, and to concentrate on the proofs of the easier case in order to quickly progress to the parts of the theory that are most important for the applications. For the 6th edition the author has added further exercises and, for the first time, solutions to many of the exercises are provided.
Author(s): Bernt Øksendal
Series: Universitext
Edition: 6
Publisher: Springer-Verlag
Year: 2013
Language: English
Commentary: Sixth Corrected Printing of the Sixth Edition
Pages: 403
Tags: Stochastic Differential Equations; Stochastic Processes
Preface to the Sixth Corrected Printing of the Sixth Edition
Preface to the Fifth Corrected Printing of the Sixth Edition
Preface to the Fourth Corrected Printing of the Sixth Edition
Preface to the Third Corrected Printing of the Sixth Edition
Preface to the Sixth Edition
Preface to Corrected Printing, Fifth Edition
Preface to the Fifth Edition
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction
1.1 Stochastic Analogs of Classical Differential Equations
1.2 Filtering Problems
1.3 Stochastic Approach to Deterministic Boundary Value Problems
1.4 Optimal Stopping
1.5 Stochastic Control
1.6 Mathematical Finance
2 Some Mathematical Preliminaries
2.1 Probability Spaces, Random Variables and StochasticProcesses
2.2 An Important Example: Brownian Motion
Exercises
3 Itô Integrals
3.1 Construction of the Itô Integral
3.2 Some properties of the Itô integral
3.3 Extensions of the Itô integral
Exercises
4 The Itô Formula and the Martingale Representation Theorem
4.1 The 1-dimensional Itô formula
4.2 The Multi-dimensional Itô Formula
4.3 The Martingale Representation Theorem
Exercises
5 Stochastic Differential Equations
5.1 Examples and Some Solution Methods
5.2 An Existence and Uniqueness Result
5.3 Weak and Strong Solutions
Exercises
6 The Filtering Problem
6.1 Introduction
6.2 The 1-Dimensional Linear Filtering Problem
6.3 The Multidimensional Linear Filtering Problem
Exercises
7 Diffusions: Basic Properties
7.1 The Markov Property
7.2 The Strong Markov Property
7.3 The Generator of an Itô Diffusion
7.4 The Dynkin Formula
7.5 The Characteristic Operator
Exercises
8 Other Topics in Diffusion Theory
8.1 Kolmogorov's Backward Equation. The Resolvent
8.2 The Feynman-Kac Formula. Killing
8.3 The Martingale Problem
8.4 When is an Itô Process a Diffusion?
8.5 Random Time Change
8.6 The Girsanov Theorem
Exercises
9 Applications to Boundary Value Problems
9.1 The Combined Dirichlet-Poisson Problem. Uniqueness
9.2 The Dirichlet Problem. Regular Points
9.3 The Poisson Problem
Exercises
10 Application to Optimal Stopping
10.1 The Time-Homogeneous Case
10.2 The Time-Inhomogeneous Case
10.3 Optimal Stopping Problems Involving an Integral
10.4 Connection with Variational Inequalities
Exercises
11 Application to Stochastic Control
11.1 Statement of the Problem
11.2 The Hamilton-Jacobi-Bellman Equation
11.3 Stochastic control problems with terminal conditions
Exercises
12 Application to Mathematical Finance
12.1 Market, portfolio and arbitrage
12.2 Attainability and Completeness
12.3 Option Pricing
Exercises
Appendix A: Normal Random Variables
Appendix B: Conditional Expectation
Appendix C: Uniform Integrability and MartingaleConvergence
Appendix D: An Approximation Result
Solutions and Additional Hints to Some of the Exercises
References
List of Frequently Used Notation and Symbols
Index
