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, 2020, 44(7).

• backward Euler–Maruyama method
• exponential mean‐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
  华南理工大学