  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.
