Exponential stability of numerical solution to neutral stochastic functional differential equation

Stability of the solution to neutral stochastic functional differential equation (NSFDE) has received a great deal of attention, but there is so far little work on stability of numerical solution. To close the gap, the paper develops new criteria on stability of numerical solutions to linear and nonlinear NSFDEs. We show that the backward Euler-Maruyama(EM) method can reproduce the almost surely exponential stability of the exact solution to highly nonlinear NSFDE, and EM method can preserve the almost surely exponential stability of NSFDE with linear growing coefficients. Two examples are provided to illustrate the main results.