Strong convergence of implicit numerical methods for nonlinear stochastic functional differential equations

The main aim of this work is to prove that the backward Euler–Maruyama approximate solutions converge strongly to the true solutions for stochastic functional differential equations with superlinear growth coefficients. The paper also gives the boundedness and mean-square exponential stability of the exact solutions, and shows that the backward Euler–Maruyama method can preserve the boundedness of mean-square moments. Finally, a highly nonlinear example is provided to illustrate the main results.