Mean-square and asymptotic stability of the stochastic theta method

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 stu dy a linear test equation with a multiplicative noise term, and consider me an-square and asymptotic stability of a stochastic version of the theta met hod. We extend some mean-square stability results in [Saito and Mitsui, SIA M. 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 fi nding the expected value of a parametrized random variable. We combine anal ytical and numerical techniques to get insights into the stability properti es. For a varian of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region. Th is allows us to prove a number of results. The technique introduced is wide ly applicable, and we use it to show that a fully implicit method suggested by [Kloeden and Platen, Numerical Solution of Stochastic Differential Equa tions, Springer-Verlag, 1992] has an asymptotic stability extension of the deterministic A-stability property. We also use the approach to explain som e numerical results reported in [Milstein, Platen, and Schurz, SIAM J. Nume r. Anal., 35 (1998), pp. 1010 1019.].