Rate of strong convergence to solutions of regime-switching stochastic differential equations

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 $\mathcal{B}$. Our results significantly extend the pathwise approximation for non-regime-switching, time-homogeneous SDE in R\"omisch et al. (1985). Here, two key techniques are the Lamperti transform for regime-switching SDEs and a pathwise analysis of their solutions.