Stochastic Differential Equations: An Introduction with ApplicationsAn 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
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 No preview available - 2010 |
Stochastic Differential Equations: An Introduction with Applications Bernt Øksendal No preview available - 2003 |
Common terms and phrases
1-dimensional Brownian motion admissible portfolio apply arbitrage assume B₁ Berlin Heidelberg 2013 Bernt Øksendal Borel bounded Chapter choose condition constant continuous function Corollary define Definition denotes Dirichlet problem Dynkin's formula e¯ps Example Exercise exists filtering problem geometric Brownian motion Girsanov theorem given Hence Hint inf{t Itô diffusion Itô integral Itô process Itô's formula L²(P Lemma Let Bt linear lower semicontinuous Markov control Markov property martingale w.r.t. mathematical o-algebra obtain Øksendal optimal stopping optimal stopping problem probability measure probability space process Xt Proof prove random variable satisfies solution solve stochastic differential equation stochastic process Stratonovich superharmonic supermeanvalued Suppose T-claim unique X₁ Xt(w მი მყ