Stochastic Differential Equations: An Introduction with Applications

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Springer Science & Business Media, Nov 9, 2010 - Mathematics - 379 pages
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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. Apart from several minor corrections and improvements, based on useful comments from readers and experts, the most important change in the corrected 5th printing of the 6th edition is in Theorem 10.1.9, where the proof of part b has been corrected and rewritten. The corrected 5th printing of the 6th edition is forthcoming and expected in September 2010.
 

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Contents

1 Introduction
1
2 Some Mathematical Preliminaries
6
3 Itô Integrals
21
4 The Itô Formula and the Martingale Representation Theorem
43
5 Stochastic Differential Equations
65
6 The Filtering Problem
85
Basic Properties
115
8 Other Topics in Diffusion Theory
141
12 Application to Mathematical Finance
269
Normal Random Variables
315
Conditional Expectation
318
Uniform Integrability and MartingaleConvergence
323
An Approximation Result
327
Solutions and Additional Hints to Some of the Exercises
331
References
361
List of Frequently Used Notation and Symbols
370

9 Applications to Boundary Value Problems
181
10 Application to Optimal Stopping
212
11 Application to Stochastic Control
243

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