In this paper, we develop the truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), and consider the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The order of convergence is obtained. Moreover, we show that the truncated EM method can preserve the exponential mean square stability of SDEPCAs. Numerical examples are provided to support our conclusions.

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