Continuous Dependence for Stochastic Functional Differential Equations with State-Dependent Regime-Switching on Initial Values

This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations (SFDEs) with state-dependent regime-switching. Due to the state-dependence, this problem is very different to the corresponding problem for SFDEs without switching or SFDEs with Markovian switching. We provide a method to overcome the intensive interaction between the continuous component and the discrete component based on a subtle application of Skorokhod's representation for jumping processes. Furthermore, we establish the strong convergence of Euler-Maruyama's approximations, and estimate the order of error. The continuous dependence on initial values of Euler-Maruyama's approximations is also investigated in the end.