Numerical Solution of Stochastic Differential Equations

The aim of this book is to provide an accessible introduction to stochastic differ­ ential equations and their applications together with a systematic presentation of methods available for their numerical solution. During the past decade there has been an accelerating interest in the de­ velopment of numerical methods for stochastic differential equations (SDEs). This activity has been as strong in the engineering and physical sciences as it has in mathematics, resulting inevitably in some duplication of effort due to an unfamiliarity with the developments in other disciplines. Much of the reported work has been motivated by the need to solve particular types of problems, for which, even more so than in the deterministic context, specific methods are required. The treatment has often been heuristic and ad hoc in character. Nevertheless, there are underlying principles present in many of the papers, an understanding of which will enable one to develop or apply appropriate numerical schemes for particular problems or classes of problems.

1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. Introduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.