Exponential stability of implicit numerical solution for nonlinear neutral stochastic differential equations with time-varying delay and poisson jumps

摘要

The aim of this work is to investigate the exponential mean-square stability for neutral stochastic differential equations with time-varying delay and Poisson jumps. When all the drift, diffusion, and jumps coefficients are allowed to be nonlinear, the exponential mean-square stability of the analytic solution to the equation is obtained. It is revealed that the implicit backward Euler-Maruyama numerical solution can reproduce the corresponding stability of the analytic solution under some given nonlinear conditions. It is different from the explicit Euler-Maruyama numerical solution whose stability depends on the linear growth condition. With some requirements related to the delayed function and the property of compensated Poisson process, we deal with time-varying delay and Poisson jumps. One highly nonlinear example is given to confirm the effectiveness of our theory.