Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations

This paper is concerned with pth moment and almost sure exponential stability of the exact and numerical solutions of neutral stochastic delay differential equations (NSDDEs). Moment exponential stability criteria of the continuous and discrete solutions are established by virtue of the Lyapunov method. Then the almost sure exponential stability criterion is derived by the Chebyshev inequality and the Borel–Cantelli lemma. We also examine conditions under which the numerical solution can reproduce the exponential stability of exact solution. It is shown that the linear growth condition is necessary for Euler–Maruyama (EM) method to maintain the moment exponential stability of the exact solution. If the drift coefficient of NSDDE satisfies the one-sided Lipschitz condition, EM method may break down, but we show that the backward EM (BEM) method can share the mean square exponential stability of the exact solution.