Twostep Maruyama schemes for nonlinear stochastic differential delay equations
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, SouthCentral University for Nationalities, Wuhan, 430074, China.
Abstract
This work concerns the twostep Maruyama schemes for nonlinear stochastic differential delay equations (SDDEs). We first examine the strong convergence rates of the split twostep Maruyama scheme and linear twostep Maruyama scheme (including AdamsBashforth and AdamsMoulton 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 twostep Maruyama schemes. These results are important for three reasons: first, the convergence rates are established under the nonglobal Lipschitz condition; second, these stability results show that these twostep 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 twostep Maruyama methods of stochastic ordinary differential equations (SODEs).
Keywords
 Stochastic differential equations (SDEs)
 twostep Maruyama schemes
 strong convergence rate
 exponential mean square stability
MSC
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