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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:259

Article; Proceedings Paper

Huang, Chengming

关键词: Stochastic delay differential equations;    Mean square stability;    Exponential stability;    Theta method;    Dissipativity;   

DOI  :  10.1016/j.cam.2013.03.038

来源: Elsevier

【 摘 要 】

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 theta >= 0.5 is asymptotically mean square stable and the split-step theta method with theta < 0.5 is exponentially mean square stable. Conditional stability results for the methods with theta < 0.5 are also obtained under a stronger assumption. Finally, we further investigate the mean square dissipativity of the split-step theta method with theta > 0.5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data. (C) 2013 Elsevier By. All rights reserved.

【 授权许可】

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