The truncated EM method for stochastic differential equations with Poisson jumps

Abstract

In this paper, we use the truncated Euler–Maruyama (EM) method to study the finite time strong convergence for SDEs with Poisson jumps under the Khasminskii-type condition. We establish the finite time L r ( r ≥ 2 ) -convergence order when the drift and diffusion coefficients satisfy the super-linear growth condition and the jump coefficient satisfies the linear growth condition. The result shows that the optimal L r - convergence order is close to 1. This is significantly different from the result on SDEs without jumps. When all the three coefficients of SDEs are allowing to grow super-linearly, the L r ( 0 < r < 2 ) -convergence results are also investigated and the optimal L r - convergence order is shown to be not greater than 1 ∕ 4 . Moreover, we prove that the truncated EM method preserves nicely the mean square exponential stability and asymptotic boundedness of the underlying SDEs with Poisson jumps. Several examples are given to illustrate our results.