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). As far as we know, although there have been several papers considering the convergence and stability for different types of numerical solutions on SDEPCAs, none of these numerical methods has a convergence order of more than one-half. However, it is well known that the Milstein-type method for stochastic differential equations (SDEs) or stochastic delay differential equations (SDDEs) has a strong convergence order of one, which is better than the convergence order of the Euler-type method. Accordingly, we first construct the Milstein method for SDEPCAs in this paper and then show its convergence order can also reach one. Finally, we prove that the Milstein method can preserve the stability of SDEPCAs. In the last section, we provide several illustrative examples to show that the convergence rate and the stability of the Milstein simulations are consistent with the theoretical results.