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In this paper, we consider the equivalence of the $p$th moment exponentialstability for stochastic differential equations (SDEs), stochastic differentialequations with piecewise continuous arguments (SDEPCAs) and the correspondingEuler-Maruyama methods EMSDEs and EMSDEPCAs. We show that if one of theSDEPCAs, SDEs, EMSDEs and EMSDEPCAs is $p$th moment exponentially stable, thenany of them is $p$th moment exponentially stable for a sufficiently small stepsize $h$ and $\tau$ under the global Lipschitz assumption on the drift anddiffusion coefficients

Stability equivalence among stochastic differential equations and  stochastic differential equations with piecewise continuous arguments and  corresponding Euler-Maruyama methods