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Almost sure and mean square exponential stability of numerical solutions for neutral stochastic functional differential equations

In this paper, we investigate the almost sure and mean square exponential stability of the Euler method and the backward Euler method for neutral stochastic functional differential equations (NSFDEs). Moreover, the almost sure and pth moment exponential stability of exact solutions for NSFDEs are considered. It is shown that the Euler method and the backward Euler method can reproduce the property of almost sure and mean square exponential stability of exact solutions to NSFDEs under suitable conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.