On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps

Abstract

This paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps. By the Itô formula, stochastic inequality and semi-martingale convergence theorem, we study the asymptotic stability in the pth moment and almost sure exponential stability of solutions under the local Lipschitz condition and nonlinear growth condition. On the other hand, we also show the convergence in probability of numerical schemes under nonlinear growth condition. Finally, an example is provided to illustrate the theory.