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 Xε is the solution of a stochastic differential equation with additional homogenization term, while the fast component αε is a switching process. We first prove the weak convergence of ^ε\0<ε≤ 1 to X̄ in the space of continuous functions, as ε→ 0. Using the martingale problem approach and Poisson equation associated with a Markov chain, we identify this weak limiting process as the unique solution X̄ 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 $1/2$ of weak convergence of Xεₜ to X̄ₜ by applying suitable test functions φ, for any t∈ [0, T]. Additionally, we provide an example to illustrate that the order we achieve is optimal.