Numerical Approximation To SDEs With Jumps

Two numerical methods for stochastic differential equations with jumps are discussed in this paper. Firstly, based on Euler-Maruyama method, we introduced the Split-step method of stochastic functional differential equations with Poisson jumps and variable delay. It is proved that the convergence of the Split-step numerical solutions for the stochastic functional differential equations has order 0.5 in the mean-square sense under the global Lipschitz condition, the linear growth condition and the continuity of initial data. Secondly, using Ito - Taylor expansion, we consider the Split-step θ method of stochastic differential equation with jumps. We prove that the rate of convergence of the numerical solution is 1. Mean-square stability of the method is also proved in the special case of the linear equations. Numerical experiments are simulated to testify the performance of the above methods.