Strong convergence of explicit numerical schemes for stochastic differential equations with piecewise continuous arguments

In 2015, Mao (J. Comput. Appl. Math., 290, 370-384, 2015) proposed the truncated Euler-Maruyama (EM) method for stochastic differential equations (SDEs) under the local Lipschitz condition plus the Khasminskii-type condition. Adapting the truncation idea from Mao (J. Comput. Appl. Math., 290, 370-384, 2015) and Mao (Appl. Numer. Math., 296, 362-375, 2016), lots of modified truncated EM methods are proposed (see, e.g., Guo et al. (Appl. Numer. Math., 115, 235-251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1-17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847-892, 2019) and the references therein). These truncated-type EM methods Mao (J. Comput. Appl. Math., 290, 370-384, 2015) and Mao (Appl. Numer. Math., 296, 362-375, 2016) and Guo et al. (Appl. Numer. Math., 115, 235-251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1-17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847-892, 2019) construct the numerical solutions by defining an appropriate truncation projection, then applying the truncation projection to the numerical solutions before substituting them into the coefficients in each iteration. In this paper, we develop a new class of explicit schemes for superlinear stochastic differential equations with piecewise continuous arguments (SDEPCAs), which are defined by directly truncating the coefficients. Our method has a more simple structure and is easier to implement. We not only show the explicit schemes converge strongly to SDEPCAs but also demonstrate the convergence rate is optimal 1/2. A numerical example is provided to demonstrate the theoretical results.