Asymptotic stability of balanced methods for stochastic jump-diffusion differential equations

Positive results are proved here about the ability of balanced methods to reproduce the asymptotic stability of the stochastic differential equation with jumps. Balanced methods including strong balanced methods and weak balanced methods, which possess implicitness in the diffusion term, have the potential to overcome some of the numerical instabilities that are often experienced when using the explicit methods. The paper shows that the asymptotic stability for stochastic jump-diffusion differential equations is inherited by the two kinds of balanced methods with sufficiently small stepsizes. Some numerical experiments included in the paper illustrate the theoretical results.