Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.08047 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- In this paper, we study the diffusion approximation for slow-fast stochastic differential equations with state-dependent switching, where the slow component $X^{\varepsilon}$ is the solution of a stochastic differential equation with additional homogenization term, while the fast component $α^{\varepsilon}$ is a switching process. We first prove the weak convergence of $\{X^\varepsilon\}_{0<\varepsilon\leq 1}$ to $\bar{X}$ in the space of continuous functions, as $\varepsilon\rightarrow 0$. 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 $1/2$ of weak convergence of $X^{\varepsilon}_t$ to $\bar{X}_t$ by applying suitable test functions $ϕ$, for any $t\in [0, T]$. Additionally, we provide an example to illustrate that the order we achieve is optimal.