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

Burrage, Kevin, Burrage, Pamela, & Tian, Tianhai (2004) Numerical methods for strong solutions of stochastic differential equations : an overview. Royal Society of London. Proceedings A. Mathematical, Physical and Engineering Sciences, 460(2041), pp. 373-402.

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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.

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ID Code: 57770
Item Type: Journal Article
Refereed: Yes
Keywords: Stochastic differential equations, Strong solutions, Numerical methods
DOI: 10.1098/rspa.2003.1247
ISSN: 1364-5021
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > STATISTICS (010400) > Stochastic Analysis and Modelling (010406)
Divisions: Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Deposited On: 04 Mar 2013 23:08
Last Modified: 22 Mar 2013 04:04

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