Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations

In this paper, we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients. This condition admits that the diffusion coefficient can be highly nonlinear, i.e., it does not necessarily satisfy a linear growth or global Lipschitz condition. It is proved that, for all positive stepsizes, the classical stochastic theta method with θ≥0.5θ≥0.5 is asymptotically mean square stable and the split-step theta method with θ>0.5θ>0.5 is exponentially mean square stable. Conditional stability results for the methods with θ<0.5θ<0.5 are also obtained under a stronger assumption. Finally, we further investigate the mean square dissipativity of the split-step theta method with θ>0.5θ>0.5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data.