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

作者：Mo, Haoyi; Liu, Linna; Xing, Mali^*; Deng, Feiqi; Zhang, Bo

来源：MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 2021, 44(7): 5574-5592.

DOI：10.1002/mma.7132

• backward Euler&#8211
• Maruyama method
• exponential mean&#8208
• square stability
• nonlinear condition
• stochastic differential equations with jumps
• time delay

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.

• 单位
  广东工业大学