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

TL;DR

This paper develops a general technique to prove strong convergence of numerical methods for stochastic delay differential equations over an infinite time horizon, and applies it to Euler-Maruyama methods with numerical validation.

Contribution

It introduces a novel approach for analyzing the strong convergence of numerical schemes for SDDEs over infinite horizons, including invariant measure approximation.

Findings

The technique successfully establishes strong convergence in the infinite horizon.

Application to Euler-Maruyama methods demonstrates practical effectiveness.

Numerical experiments confirm theoretical predictions.

Abstract

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.