Strong convergence in the infinite horizon of numerical methods for stochastic delay differential equations

In this work, we present a general technique for establishing the strong convergence of numerical methods for stochastic delay differential equations (SDDEs) in the infinite horizon. This technique can also be extended to analyze certain continuous function-valued segment processes associated with the numerical methods, facilitating the numerical approximation of invariant measures of SDDEs. To illustrate the application of these results, we specifically investigate the backward and truncated Euler-Maruyama methods. Several numerical experiments are provided to demonstrate the theoretical results.