In this work, an adaptive time-stepping Milstein method is constructed for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift is one-sided Lipschitz continuous and the diffusion does not impose the commutativity condition. It is widely recognized that explicit Euler or Milstein methods may blow up when the system exhibits superlinear growth, and modifications are needed. Hence we propose an adaptive variant to deal with the case of superlinear growth drift coefficient. To the best of our knowledge, this is the first work to develop a numerical method with variable step sizes for nonlinear SDEPCAs. It is proven that the adaptive Milstein method is strongly convergent in the sense of $L_p, p\ge 2$, and the convergence rate is optimal, which is consistent with the order of the explicit Milstein scheme with globally Lipschitz coefficients. Finally, several numerical experiments are presented to support the theoretical analysis.

翻译：本文针对具有分段连续参数的随机微分方程(SDEPCAs)构建了一种自适应时间步长的Milstein方法，其中漂移项满足单边Lipschitz连续条件，而扩散项不要求满足交换性条件。众所周知，当系统呈现超线性增长时，显式Euler或Milstein方法可能发散，因此需要进行修正。为此，我们提出了一种自适应变体来处理漂移系数超线性增长的情况。据我们所知，这是首次为非线性SDEPCAs开发具有可变步长的数值方法。研究证明，自适应Milstein方法在$L_p, p\ge 2$意义下具有强收敛性，且收敛速率达到最优，这与全局Lipschitz系数下显式Milstein格式的阶数一致。最后，通过若干数值实验验证了理论分析结果。