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. %Given a family of finite-variation processes $\{\mathcal{F}^{\lambda}\}_{\lambda\ge 0}$ that converge strongly to a standard Brownian motion $\mathcal{B}$, we construct pathwise approximations for regime-switching, time-inhomogeneous stochastic differential equations in the Wong-Zakai sense. Moreover, we determine the rate of strong convergence to the solutions of such regime-switching SDEs, showing that this rate is almost as good as that of $\{\mathcal{F}^{\lambda}\}_{\lambda\ge 0}$ to $\mathcal{B}$. In the proposed approximations, the standard Brownian motion driving the time-inhomogeneous RSSDEs is replaced by a family of finite--variation processes $\{\mathcal{F}^{\lambda}\}_{\lambda > 0}$. We show that if $\mathcal{F}^{\lambda}$ strongly converges to $\mathcal{B}$ at rate $\delta(\lambda)$, then the Wong--Zakai approximation strongly converges to the original solution of the time--inhomogeneous RSSDE at rate $\delta(\lambda) \lambda^{\varepsilon}$, for any $\varepsilon > 0$. 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 R\"{o}misch and Wakolbinger (1985).