Convergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equations

ABSTRACT

In this paper, we are interested in numerical methods with variable step-size for stochastic pantograph differential equations (SPDEs). SPDEs are very special stochastic delay differential equations (SDDEs) with unbounded memory. The problem of computer memory hold, when the numerical methods with constant step-size are applied to the SPDEs. In this work, we construct split-step theta (SSθ) methods with variable step-size for SPDEs. The boundedness and strong convergence of the numerical methods are investigated under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. It is proved that, the SSθ methods with variable step-size for $\theta \in [\frac{1}{2},1]$ converge strongly to the exact solution. In addition, the strong order 0.5 is given under mild assumptions. The mean-square stability (MS-Stability) of the numerical methods with $\theta \in (\frac{1}{2},1]$ is given. Finally, some illustrative numerical examples are presented to show the efficiency of the methods, and how MS-Stability of SSθ methods depends on the parameter theta for both linear and nonlinear models.

KEYWORDS:

• Stochastic pantograph differential equations
• variable step-size
• mean-square stability
• split-step theta methods
• convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 60H10
• 65C20
• 65C30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors are grateful to Fengyu Guo for helpful discussions. We also thanks anonymous referees for careful reading and many helpful suggestions to improve the presentation of this paper. This research was partly financed by National Natural Science Foundation of China grant no. 91646106. Additionally, this work was supported by Higher Education Commission of Egypt.