Numerical methods for strong solutions of stochastic differential equations: An overview

K. Burrage, P. M. Burrage, T. Tian

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Abstract

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Original languageEnglish
Pages (from-to)373-402
Number of pages30
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume460
Issue number2041
DOIs
Publication statusPublished - 8 Jan 2004

Keywords

  • Numerical methods
  • Stochastic differential equations
  • Strong solutions

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