Øksendal: Stochastic Differential Equations

Title: Stochastic Differential Euqations, An introduction with applications
Author: Bernd K. Øksendal
latest edition: 6th edition Springer Berlin (2003)
ISBN: 978-3540637202

Description

This book gives 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 (which nevertheless are often sufficiently general for many purposes) in order to be able to reach quickly the parts of the theory which is most important for the applications. The new feature of this 5th edition is an extra chapter on applications to mathematical finance.

Info

Presents the subject of stochastic differential equations in a manner approachable by the nonexpert in order to awaken further interest in the subject.

Table of contents

1. Introduction
   1. Stochastic Analogs of Classical Differential Equations
   2. Filtering Problems
   3. Stochastic Approach to deterministic Boundary Value Problems
   4. Optimal Stopping
   5. Stochastic Control
   6. Mathematical Finance
2. Some Mathematical Preliminaries
   1. Probability Spaces, Random Variables, Stochastic Processes
   2. An important example: Brownian Motion
   3. Exercises
3. Itô Integrals
4. The Itô formula and the Martingale Representation Theorem
   1. The 1-dim Fromula
   2. Multi-dim formula
   3. Martingale representation thm
   4. Exercises
5. Stochastic Differential Equations
   1. Examples and some Solution methods
   2. Existence and Uniqueness result
   3. Weak and Strong solutions
   4. Exercises
6. The filtering Problem
7. Diffusions: Basic Properties
8. Other topics in Diffusion thy
9. Applications to Boundary Value Problems
10. Application to Optimal stopping
11. Application to Stochastic Control
    1. Statement of the Problem
    2. The Hamilton-Jacobi-Bellman Equation
    3. Stochastic Control Problems with Terminal Conditions
    4. Exercises
12. Application to Mathematical Finance
    1. Market, Portfolio and Arbitrage
    2. Attainability and Completeness
    3. Option pricing
    4. Exercises

Appendices:

1. Normal Random Variables
2. Conditional Expectation
3. Uniform integrability and Martingale Convergence
4. An Approximation Result
5. Solutions and additional hints to some of the Exercises
6. References
7. List of frequently used Notation and Symbols
8. Index