# Two-step Maruyama schemes for nonlinear stochastic differential delay equations

Volume 10, Issue 10, pp 5245--5260
Publication Date: October 19, 2017 Submission Date: August 23, 2017
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### Authors

Dongxia Lei - School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China. Xiaofeng Zong - School of Automation, China University of Geosciences, Wuhan, 430074, China. Junhao Hu - School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China.

### Abstract

This work concerns the two-step Maruyama schemes for nonlinear stochastic differential delay equations (SDDEs). We first examine the strong convergence rates of the split two-step Maruyama scheme and linear two-step Maruyama scheme (including Adams-Bashforth and Adams-Moulton schemes) for nonlinear SDDEs with highly nonlinear delay variables, then we investigate the exponential mean square stability and exponential decay rates of the two classes of two-step Maruyama schemes. These results are important for three reasons: first, the convergence rates are established under the non-global Lipschitz condition; second, these stability results show that these two-step Maruyama schemes can not only reproduce the exponential mean square stability, but also preserve the bound of Lyapunov exponent for sufficient small stepsize; third, they are also suitable for the corresponding two-step Maruyama methods of stochastic ordinary differential equations (SODEs).

### Keywords

• Stochastic differential equations (SDEs)
• two-step Maruyama schemes
• strong convergence rate
• exponential mean square stability

•  65C30
•  93E15
•  60H35

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