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.

Contents

• Introduction
• Some Mathematical Preliminaries
• Itô Integrals
• Itô Formula and the Martingale Representation Theorem
• Stochastic Differential Equations
• The Filtering Problem
• Diffusions: Basic Properties
• Other Topics in Diffusion Theory
• Applications to Boundary Value Problems
• Applications to Optimal Stopping
• Application to Stochastic Control
• Application to Mathematical Finance

• Appendix A: Normal Random Variables
• Appendix B: Conditional Expectation
• Appendix C: Uniform Integrability and Martingale Convergence
• Appendix D: An Approximation Result

• Solutions and Additional Hints to Some of the Exercises
• References
• List of Frequently Used Notation and Symbols
• Index