Mean-square and asymptotic stability of numerical methods for stochastic ordinary differential equations

Stability analysis of numerical methods for ordinary differential equations (ODEs) is motivated by the question 'for what choices of stepsize does the numerical method reproduce the characteristics of the test equation?' We study a linear test equation with a multiplicative noise term, and consider mean-square and asymptotic stability of a stochastic version of the theta method. We extend some mean-square stability results in [Saito and Mitsui, SIAM. J. Numer. Anal., 33 (1996), pp. 2254--2267]. In particular, we show that an extension of the deterministic A-stability property holds. We also plot mean-square stability regions for the case where the test equation has real parameters. For asymptotic stability, we show that the issue reduces to finding the expected value of a parametrized random variable. We combine analytical and numerical techniques to get insights into the stability properties. For a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region. This allows us to prove a number of results. The technique introduced is widely applicable, and we use it to show that a fully implicit method suggested by [Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992] has an asymptotic stability extension of the deterministic A-stability property. We also use the approach to explain some numerical results reported in [Milstein, Platen, and Schurz, SIAM J. Numer. Anal., 35 (1998), pp. 1010--1019.