Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations

Evelyn Buckwar , Rosza Horvath Bokor , Renate Winkler

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2004-25, 22 pages

MSC 2000

60H35 Computational methods for stochastic equations

65C30 Stochastic differential and integral equations

Abstract
We deal with linear multi-step methods for SDEs and study when the numerical appro\-xi\-mation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams-Bashforth- and Adams-Moulton-methods, the Milne-Simpson method and the BDF method.