文摘
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 is asymptotically mean square stable and the split-step theta method with is exponentially mean square stable. Conditional stability results for the methods with are also obtained under a stronger assumption. Finally, we further investigate the mean square dissipativity of the split-step theta method with and prove that the method possesses a bounded absorbing set in mean square independent of initial data.
