Mean square stability of second-order weak numerical methods for stochastic differential equations

Recent years have marked many significant advances in numerical treatments for stochastic differential equations. Emphasis was not only on the nature and order of convergence of numerical schemes, but also on their stability features [J. Comput. Appl. Math. 125 (2000) 171; BIT 33 (1993) 654; Comput. Math. Appl. 28 (1994) 45; SIAM J. Appl. Anal. 51 (1991) 542; SIAM J. Numer. Anal. 33 (1996) 2254]. In this article we discuss mean square stability of second-order weak numerical methods. The closed form of the second moment established by [D. Talay, Simulation and numerical analysis of stochastic differential systems, INRIA Report 1313, 1990] will be used to create a mean square stability criterion for these schemes. Numerical examples will be presented to support the theoretical analysis. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.