Diffusion Approximation for Slow-Fast SDEs with State-Dependent Switching

In this paper, we study the diffusion approximation for slow-fast stochastic differential equations with state-dependent switching, where the slow component $X^{\epsilon }$ is the solution of a stochastic differential equation with additional homogenization term, while the fast component ${\alpha }^{\epsilon }$ is a switching process. We first prove the weak convergence of $\{X^{\epsilon }{\}}_{0<\epsilon \le 1}$ to $\bar{X}$ in the space of continuous functions, as $\epsilon \to 0$. Using the martingale problem approach and Poisson equation associated with a Markov chain, we identify this weak limiting process as the unique solution $\bar{X}$ of a new stochastic differential equation, which has new drift and diffusion terms that differ from those in the original equation. Next, we prove the order $1\mathrm{/}2$ of weak convergence of $X_t^{\epsilon }$ to ${\bar{X}}_t$ by applying suitable test functions $\phi$, for any $t\in [0,T]$. Additionally, we provide an example to illustrate that the order we achieve is optimal.