This work focuses on the numerical approximations of neutral stochastic delay differential equations with their drift and diffusion coefficients growing super-linearly with respect to both delay variables and state variables. Under generalized monotonicity conditions, we prove that the backward Euler method not only converges strongly in the mean square sense with order $1/2$, but also inherit the mean square exponential stability of the original equations. As a byproduct, we obtain the same results on convergence rate and exponential stability of the backward Euler method for stochastic delay differential equations with generalized monotonicity conditions. These theoretical results are finally supported by several numerical experiments.

翻译：本文研究了漂移系数和扩散系数关于延迟变量和状态变量均呈超线性增长的中立型随机延迟微分方程的数值逼近问题。在广义单调性条件下，我们证明了向后欧拉方法不仅以$1/2$阶在均方意义下强收敛，而且继承了原方程的均方指数稳定性。作为副产品，我们获得了广义单调性条件下随机延迟微分方程向后欧拉方法在收敛速度和指数稳定性方面的相同结论。最后，若干数值实验验证了这些理论结果。