Towards a Systematic Linear Stability Analysis of Numerical Methods for Systems of Stochastic Differential Equations

We develop two classes of test equations for the linear stability analysis of numerical methods applied to systems of stochastic ordinary differential equations of Ito type (SODEs). Motivated by the theory of stochastic stabilization and destabilization, these test equations capture certain fundamental effects of stochastic perturbation in systems of SODEs, while remaining amenable to analysis before and after discretization. We then carry out a linear stability analysis of the $\theta$-Maruyama method applied to these test equations, investigating mean-square and almost sure asymptotic stability of the test equilibria. We discuss the implications of our work for the notion of A-stability of the $\theta$-Maruyama method and use numerical simulation to suggest extensions of our results to test systems with nonnormal drift coefficients.