Convergence Rate Of Numerical Solutions To SDEs With Jumps

Three numerical methods for solving a stochastic differential equation with jump are given in this paper. Based on the Euler-Maruyama method, we introduce the Split-step method to a class of the stochastic differential equation with jump. We prove that it is convergence under the Lipschitz condition, the linear growth condition and some continuous conditions of the delay function. The convergence rate of Split-step numerical solutions to the stochastic differential equation with jump has order α^γ^1/2. Then we discuss the SSθ method of the stochastic differential equation without delay. We also show that its convergence rate has order1/2under the Lipschitz condition and the linear growth condition. Next, we expand the conclusion to the stochastic differential equation with delay. We obtain that it is convergence rate1/2of SSθ numerical solutions to the stochastic delay differential equation with jump. Finally, using lto-Taylor expansion, we obtain the first-order approximate numerical solutions of the stochastic differential equation with jump. And the rate of convergence is1. Numerical experiments are simulated to testify the performance and the effectiveness of the above method.