Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments

This work develops the Milstein scheme for commutative stochastic differential equations with piecewise continuous arguments (SDEPCAs), which can be viewed as stochastic differential equations with time-dependent and piecewise continuous delay. As far as we know, although there have been several papers investigating the convergence and stability for different numerical methods on SDEPCAs, all of these methods are Euler-type methods and the convergence orders do not exceed 1/2. Accordingly, we first construct the Milstein scheme for SDEPCAs in this work and then show its convergence order can reach 1. Moreover, we prove that the Milstein method can preserve the stability of SDEPCAs. In the last section, we provide several numerical examples to verify the theoretical results.