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In this paper, a general theorem on the equivalence of pth moment stabilitybetween stochastic differential delay equations (SDDEs) and their numericalmethods is proved under the assumptions that the numerical methods are stronglyconvergent and have the bouneded $p$th moment in the finite time. The truncatedEuler-Maruyama (EM) method is studied as an example to illustrate that thetheorem indeed covers a large ranges of SDDEs. Alongside the investigation ofthe truncated EM method, the requirements on the step size of the method aresignificantly released compared with the work, where the method was initiallyproposed.

Equivalence of pth moment stability between stochastic differential delay equations and their numerical methods