Numerical experiments with S(P)DE's

Summary

Introduction

For the last thirty years, there has been interest in numerical simulation of solutions of stochastic differential equations (SDE's). More recently, along with the growing general interest in stochastic partial differential equations (SPDE's), there has been a desire to produce numerical solutions to SPDE's. There are very few SDE's for which analytical solutions can be obtained and this is naturally also true for SPDE's. One hope is that using numerical methods to generate solutions to such equations will lead to better understanding of the equations. Theory and numerical work often go hand in hand: while theory can provide useful numerical methods, pictures obtained numerically can lead to conjectures that the analyst may then succeed in proving. Another hope is that numerical methods will be of use to people outside mathematics. There are already SPDE's used in practical applications, such as in finance.

Our aim is to show how simple techniques, already in use for numerical solution of SDE's, can be extended to provide numerical solutions to SPDE's. We will concentrate on strong solutions, that is solutions obtained for one particular realisation of Brownian motion or of a Brownian sheet. We describe how Brownian paths and sheets can be generated and stored in tree-like structures, allowing for subsequent refinement and therefore comparison of solutions using different meshes.