Preserving exponential mean square stability and decay rates in two classes of theta approximations of stochastic differential equations

Abstract

This paper examines exponential mean square stability of the split-step theta approximation and the stochastic theta method for the stochastic differential delay equations and stochastic ordinary differential equations (SODEs) under a coupled monotone condition on drift and diffusion coefficients. It is shown that for the two classes of the theta approximations can preserve the exponential mean square stability when some conditions on the stepsize and drift coefficient are imposed, but for , without the globally Lipschitz continuity, these two classes of theta methods show exponentially mean square stability unconditionally. Moreover, for sufficiently small stepsize, the decay rate as measured by the bound of the Lyapunov exponent can be reproduced arbitrarily accurately. Some results in this paper extend the existing results for linear SODEs to nonlinear stochastic differential equations (SDEs), and also improve our previous results of numerical stability of nonlinear SDEs.

Keywords::

• stochastic differential delay equations
• stochastic ordinary differential equations
• exponential mean square stability
• exponential decay rate
• the split-step theta method
• stochastic theta method

Keywords::

• 60H35
• 65C20
• 65L20

Acknowledgements

The research of the second author was supported in part by the National Science Foundation of China (Grant No. 11001091) and the Program for New Century Excellent Talents in University. The research of the third author was supported in part by the National Science Foundation of China (Grant No. 91130003) and the Fundamental Research Funds for the Central Universities (Grant No. 2013TS137).