Stochastic Differential Equations

by Ksendal, B. K.; Oksendal, Bernt

• ISBN13: 9783540047582
• ISBN10: 3540047581
• Edition: 6th
• Format: Paperback
• Copyright: 2003-12-01
• Publisher: PALGRAVE MACMILLAN

Summary

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. For the 6th edition the author has added further exercises and, for the first time, solutions to many of the exercises are provided.

Table of Contents

1. Introduction	1	(6)
1.1 Stochastic Analogs of Classical Differential Equations	1	(1)
1.2 Filtering Problems	2	(1)
1.3 Stochastic Approach to Deterministic Boundary Value Problems	2	(1)
1.4 Optimal Stopping	3	(1)
1.5 Stochastic Control	4	(1)
1.6 Mathematical Finance	4	(3)
2. Some Mathematical Preliminaries	7	(14)
2.1 Probability Spaces, Random Variables and Stochastic Processes	7	(4)
2.2 An Important Example: Brownian Motion	11	(4)
Exercises	15	(6)
3. Itô Integrals	21	(22)
3.1 Construction of the Itô Integral	21	(9)
3.2 Some Properties of the Itô Integral	30	(4)
3.3 Extensions of the Itô Integral	34	(3)
Exercises	37	(6)
4. The Itô Formula and the Martingale Representation Theorem	43	(20)
4.1 The 1-dimensional Itô Formula	43	(5)
4.2 The Multi-dimensional Itô Formula	48	(1)
4.3 The Martingale Representation Theorem	49	(5)
Exercises	54	(9)
5. Stochastic Differential Equations	63	(20)
5.1 Examples and Some Solution Methods	63	(5)
5.2 An Existence and Uniqueness Result	68	(4)
5.3 Weak and Strong Solutions	72	(2)
Exercises	74	(9)
6. The Filtering Problem	83	(30)
6.1 Introduction	83	(2)
6.2 The 1-Dimensional Linear Filtering Problem	85	(19)
6.3 The Multidimensional Linear Filtering Problem	104	(1)
Exercises	105	(8)
7. Diffusions: Basic Properties	113	(26)
7.1 The Markov Property	113	(3)
7.2 The Strong Markov Property	116	(5)
7.3 The Generator of an Itô Diffusion	121	(3)
7.4 The Dynkin Formula	124	(2)
7.5 The Characteristic Operator	126	(2)
Exercises	128	(11)
8. Other Topics in Diffusion Theory	139	(36)
8.1 Kolmogorov's Backward Equation. The Resolvent	139	(4)
8.2 The Feynman-Kac Formula. Killing	143	(3)
8.3 The Martingale Problem	146	(2)
8.4 When is an Itô Process a Diffusion?	148	(5)
8.5 Random Time Change	153	(6)
8.6 The Girsanov Theorem	159	(9)
Exercises	168	(7)
9. Applications to Boundary Value Problems	175	(30)
9.1 The Combined Dirichlet-Poisson Problem. Uniqueness	175	(3)
9.2 The Dirichlet Problem. Regular Points	178	(12)
9.3 The Poisson Problem	190	(7)
Exercises	197	(8)
10. Application to Optimal Stopping	205	(30)
10.1 The Time-Homogeneous Case	205	(12)
10.2 The Time-Inhomogeneous Case	217	(5)
10.3 Optimal Stopping Problems Involving an Integral	222	(2)
10.4 Connection with Variational Inequalities	224	(4)
Exercises	228	(7)
11. Application to Stochastic Control	235	(26)
11.1 Statement of the Problem	235	(2)
11.2 The Hamilton-Jacobi-Bellman Equation	237	(14)
11.3 Stochastic Control Problems with Terminal Conditions	251	(1)
Exercises	252	(9)
12. Application to Mathematical Finance	261	(44)
12.1 Market, Portfolio and Arbitrage	261	(10)
12.2 Attainability and Completeness	271	(8)
12.3 Option Pricing	279	(19)
Exercises	298	(7)
Appendix A: Normal Random Variables	305	(4)
Appendix B: Conditional Expectation	309	(2)
Appendix C: Uniform Integrability and Martingale Convergence	311	(4)
Appendix D: An Approximation Result	315	(4)
Solutions and Additional Hints to Some of the Exercises	319	(26)
References	345	(8)
List of Frequently Used Notation and Symbols	353	(4)
Index	357