Poisson Equation and Application to Multi-Scale SDEs with State-Dependent Switching

In this paper, we study the averaging principle and central limit theorem for multi-scale stochastic differential equations with state-dependent switching. To accomplish this, we first study the Poisson equation associated with a Markov chain and the regularity of its solutions. As applications of the results on the Poisson equations, we prove three averaging principle results and two central limit theorems results. The first averaging principle result is a strong convergence of order $1/2$ of the slow component $X^{\varepsilon}$ in the space $C([0,T],\mathbb{R}^n)$. The second averaging principle result is a weak convergence of $X^{\varepsilon}$ in $C([0,T],\mathbb{R}^n)$. The third averaging principle result is a weak convergence of order $1$ of $X^{\varepsilon}_t$ in $\mathbb{R}^n$ for any fixed $t\ge 0$. The first central limit theorem type result is a weak convergence of $(X^{\varepsilon}-\bar{X})/\sqrt{\varepsilon}$ in $C([0,T],\mathbb{R}^n)$, where $\bar{X}$ is the solution of the averaged equation. The second central limit theorem type result is a weak convergence of order $1/2$ of $(X^{\varepsilon}_t-\bar{X}_t)/\sqrt{\varepsilon}$ in $\mathbb{R}^n$ for fixed $t\ge 0$. Several examples are given to show that all the achieved orders are optimal.