The pth moment exponential stability and almost surely exponential stability of stochastic differential delay equations with Poisson jump

Abstract

This paper is devoted to study a class of nonlinear stochastic differential delay equations with Poisson jump. In comparison to the Brownian motion, the jump leads to the discontinuity of sample paths, which makes the analysis more difficult. We first introduce the local Lipschitz condition and a new nonlinear growth condition, which is weaker than those in the previous literature. Then by virtue of Lyapunov function and semi-martingale convergence theorem, we prove that the considered stochastic system has a unique global solution. Moreover, we also investigate the pth moment exponential stability and the almost surely exponential stability of solutions. Finally, an example is given to illustrate the effectiveness of our theoretical results.