Almost sure asymptotic stability of drift-implicit θ-methods for bilinear ordinary stochastic differential equations in R1

Global almost sure asymptotic stability of stochastic θ-methods with nonrandom variable step sizes when applied to bilinear, nonautonomous, homogeneous test systems of ordinary stochastic differential equations (SDEs) is investigated. Sufficient conditions for almost sure asymptotic stability are proved for both analytical and numerical solutions in R1. The results of Saito and Mitsui (World Sci. Ser. Appl. Math. 2 (1993) 333, SIAM J. Numer. Anal. 33 (1996) 2254), Higham (SIAM J. Numer. Anal. 38 (2001) 753) and Schurz (Stochastic Anal. Appl. 14 (1996) 313, Handbook of Stochastic Analysis and Applications, 2002) for the constant step sizes are carried over to the case with variable step sizes and nonautonomous linear test equations. The investigations indicate that θ-methods with variable step sizes or variable parameter θ governed by certain conditions can successfully be used to guarantee almost sure asymptotic stability while discretizing nonautonomous SDEs.