  In this paper, we consider the equivalence of the $p$th moment exponential
stability for stochastic differential equations (SDEs), stochastic differential
equations with piecewise continuous arguments (SDEPCAs) and the corresponding
Euler-Maruyama methods EMSDEs and EMSDEPCAs. We show that if one of the
SDEPCAs, SDEs, EMSDEs and EMSDEPCAs is $p$th moment exponentially stable, then
any of them is $p$th moment exponentially stable for a sufficiently small step
size $h$ and $\tau$ under the global Lipschitz assumption on the drift and
diffusion coefficients
