## Stochastic Differential Equations: An Introduction with ApplicationsThis 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

### Other editions - View all

Stochastic Differential Equations: An Introduction with Applications Bernt Øksendal Limited preview - 2003 |

Stochastic Differential Equations: An Introduction with Applications Bernt Oksendal Limited preview - 2013 |

Stochastic Differential Equations: An Introduction with Applications Bernt Oksendal Limited preview - 2013 |

### Common terms and phrases

1-dimensional Brownian motion admissible portfolio apply arbitrage assume Berlin Heidelberg 2013 Bernt Øksendal Borel sets bounded Chapter choose condition constant continuous function Corollary define Definition denotes Dirichlet problem Dynkin Dynkin's formula Example Exercise exists filtering problem geometric Brownian motion Girsanov theorem given Hence Hint inf{t Itô diffusion Itô integral Itó process Itô's formula Karatzas Lemma Let f Let Xt linear lower semicontinuous Markov control Markov property martingale w.r.t. mathematical measure Q normal O-algebra obtain Øksendal oo a.s. optimal control optimal stopping optimal stopping problem option probability measure probability space Proof prove random variable result satisfies Show solve stochastic control problem stochastic differential equation stochastic integral stochastic process Stratonovich subsets superharmonic supermeanvalued Suppose T-claim unique Universitext w)dB Xidt