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

Authors:
Chengming Huang
Affiliations:
-
Venue:
Journal of Computational and Applied Mathematics
Year:
2014

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Abstract

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 @q=0.5 is asymptotically mean square stable and the split-step theta method with @q0.5 is exponentially mean square stable. Conditional stability results for the methods with @q0.5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data.