Implicit numerical approximation for stochastic delay differential equations with the nonlinear diffusion term in the infinite horizon

This paper investigates the approximation of stochastic delay differential equations (SDDEs) via the backward Euler-Maruyama (BEM) method under generalized monotonicity and Khasminskii-type conditions in the infinite horizon. First, by establishing the uniform moment boundedness and finite-time strong convergence of the BEM method, we prove that for sufficiently small step sizes, the numerical approximations strongly converge to the underlying solution in the infinite horizon with a rate of $1/2$, which coincides with the optimal finite-time strong convergence rate. Next, we establish the uniform boundedness and convergence in probability for the segment processes associated with the BEM method. This analysis further demonstrates that the probability measures of the numerical segment processes converge to the underlying invariant measure of the SDDEs. Finally, a numerical example and simulations are provided to illustrate the theoretical results.