Stability Analysis of Numerical Schemes for Stochastic Differential Equations
SIAM Journal on Numerical Analysis
Mean-Square and Asymptotic Stability of the Stochastic Theta Method
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Asymptotic moment boundedness of the numerical solutions of stochastic differential equations
Journal of Computational and Applied Mathematics
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Global almost sure asymptotic stability of stochastic @q-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 R^1. 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 @q-methods with variable step sizes or variable parameter @q governed by certain conditions can successfully be used to guarantee almost sure asymptotic stability while discretizing nonautonomous SDEs.