Multi-step methods for SDEs and their application to problems with small noise

Evelyn Buckwar , Renate Winkler

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

60H35 Computational methods for stochastic equations

65C30 Stochastic differential and integral equations

Abstract
In this paper the numerical approximation of solutions of Ito stochastic differential equations is considered, in particular for equations with a small parameter $\epsilon$ in the noise coefficient. We construct stochastic linear multi-step methods and develop the fundamental numerical analysis concerning their mean-square consistency, numerical stability in the mean-square sense and mean-square convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency. Further, for the small noise case we obtain expansions of the local error in terms of the stepsize and the small parameter $\epsilon$. Simulation results using several explicit and implicit stochastic linear k-step schemes, k=1,2, illustrate the theoretical findings.