Convergence and stability of the backward Euler method for jump–diffusion SDEs with super-linearly growing diffusion and jump coefficients

Abstract

This paper firstly investigates convergence of the backward Euler method for stochastic differential equations (SDEs) driven by Brownian motion and compound Poisson process. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super-linear growth. It is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on SDEs, allowing for possibly super-linearly growing drift, diffusion and jump coefficients. An exact order 1 2 is recovered when further differentiability assumption is put on the coefficients. Furthermore, the considered method is able to inherit the mean square stability of a wider class of Lévy noise driven SDEs for all stepsizes. These results are finally supported by some numerical experiments.