Approximation for stochastic functional differential equation with past-dependent switching

This work is devoted to the convergence of the Euler-Maruyama approximation for stochastic functional differential equation with past-dependent random switching. Based on the Skorokhod representation of switching, we examine the existence and uniqueness of the exact solution as well as its moment boundedness. The weak convergence of the approximate solution is established by using the martingale problem formulation. Furthermore, we reveal the strong convergence in the L ¹ -norm, and give the convergence rate close to 0.5 order under bounded conditions. Two numerical examples are also provided to illustrate our results.