Convergence and stability in mean square of the stochastic θ-methods for systems of NSDDEs under a coupled monotonicity condition

Abstract

Our research is devoted to investigating the convergence and stability in mean square of the stochastic θ-methods applied to neutral stochastic differential delay equations (NSDDEs) with super-linearly growing coefficients. Under a coupled monotonicity condition, we show that the numerical approximations of the stochastic θ-methods with $\theta \in [\frac{1}{2},1]$ converge to the exact solution of NSDDEs strongly with order $\frac{1}{2}$. Moreover, it is shown that the stochastic θ-methods are capable of preserving the stability of the exact solution of original equations for any given stepsize $h>0$. Finally, several numerical examples are presented to illustrate the theoretical findings.