The Split-Step Method For Stochastic Delay Differential Equation With Jumps

In this paper we discuss the Split-Step methods for solving three different kinds of stochastic delay differential equations with poisson jumps in lto sence. First of all, considering the linear stochastic delay differential equations with jumps, the Split-Step method are constructed based on the Euler-Maruyama method. It is proved that the Split-Step method is convergent with order of1.5in the mean sense and order of1in the mean-square sense. We also obtain the mean-square stable of the Split-Step method under some conditions. Next, we expend the conclusion from linear equation to the nonlinear equation and prove that Split-Step method is convergent under the Lipschitz condition and the Linear growth condition. Two numerical examples are presented to demonstrate the effectiveness and the means-quare stability of the Split-Step method. Finally, considering the variable delay, former conclusions are expended. It is proved that the truncation error of the numerical approximation is convergent in the mean sense and the mean-square sense under some assumptions of the variable delay and the initial function. We also present one numerical experiment for demonstrating the calculation results of the Split-Step method.