Wong–Zakai approximations with convergence rate for stochastic differential equations with regime switching

We construct Wong–Zakai approximations of time-inhomogeneous stochastic differential equations with regime switching (RSSDEs), and provide a convergence rate. In the proposed approximations, the standard Brownian motion driving the time-inhomogeneous RSSDEs is replaced by a family {Fλ}λ>0 of processes assumed to be continuous and of bounded variation on compact intervals. We show that, when these driving processes approximate Brownian motion strongly in the classical Wong–Zakai sense, the corresponding RSSDE approximations converge strongly to the solution, with a quantitative rate essentially inherited from the driving approximation. This is the first paper on Wong–Zakai approximations for time-inhomogeneous RSSDEs, and significantly extends the counterparts for time-homogeneous SDEs without regime switching in Römisch and Wakolbinger [W. Röomisch and A. Wakolbinger, On Lipschitz dependence in systems with differentiated inputs, Math. Ann. 272 (1985), pp. 237–248].