Mean-square convergence and stability of the backward Euler method for SDDEs with highly nonlinear growing coefficients

By virtue of the novel technique, this paper focuses on the mean-square convergence of the backward Euler method (BEM) for stochastic differential delay equations (SDDEs) without using the moment boundedness of numerical solutions. The convergence rate for SDDE whose drift and diffusion coefficients can both grow polynomially is shown. Furthermore, under fairly general conditions, the novel technique is applied to prove that the BEM can inherit the exponential mean-square stability with a simple proof. At last, some numerical experiments are implemented to illustrate the reliability of the theories.