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In this paper we investigate explicit numerical approximations for stochasticdifferential delay equations (SDDEs) under a local Lipschitz condition byemploying the adaptive Euler-Maruyama (EM) method. Working in both finite andinfinite horizons, we achieve strong convergence results by showing theboundedness of the pth moments of the adaptive EM solution. We also obtain theorder of convergence infinite horizon. In addition, we show almost sureexponential stability of the adaptive approximate solution for both SDEs andSDDEs.

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