Explicit Numerical Approximations for SDDEs in Finite and Infinite Horizons using the Adaptive EM Method: Strong Convergence and Almost Sure Exponential Stability

TL;DR

This paper develops an adaptive Euler-Maruyama method for SDDEs, proving strong convergence, order of convergence over infinite horizons, and almost sure exponential stability in both finite and infinite time frames.

Contribution

It introduces an explicit adaptive EM scheme for SDDEs with proven strong convergence, stability, and convergence order results, extending previous methods to delay equations.

Findings

Proved boundedness of moments for the adaptive EM solution.

Established strong convergence and convergence order for SDDEs.

Demonstrated almost sure exponential stability of the numerical scheme.

Abstract

In this paper we investigate explicit numerical approximations for stochastic differential delay equations (SDDEs) under a local Lipschitz condition by employing the adaptive Euler-Maruyama (EM) method. Working in both finite and infinite horizons, we achieve strong convergence results by showing the boundedness of the pth moments of the adaptive EM solution. We also obtain the order of convergence infinite horizon. In addition, we show almost sure exponential stability of the adaptive approximate solution for both SDEs and SDDEs.