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

Xiaobin Sun, Jue Wang, Yingchao Xie

TL;DR

We analyze diffusion approximation for slow-fast SDEs with state-dependent switching, showing the slow component $X^{\varepsilon}$ converges weakly to an averaged diffusion $\bar{X}$ as $\varepsilon\to0$, with coefficients derived via a Poisson equation tied to the fast Markov chain. The approach combines Poisson-equation techniques with martingale problem arguments to prove the limit is the unique solution of $d\bar{X}_t=\bar{B}(\bar{X}_t)dt+\bar{\Sigma}^{1/2}(\bar{X}_t)d\bar{W}_t$, and establishes a weak convergence rate of $O(\varepsilon^{1/2})$ for smooth test functions, with a 1D example showing optimality. The work also provides a priori moment bounds and demonstrates well-posedness for the averaged equation, along with tightness and martingale-problem-based proofs for convergence in the path space. Altogether, it clarifies how state-dependent switching interacts with homogenization to modify drift and diffusion in the diffusion-approximation limit, advancing multi-scale analysis for Markov-switching systems.

Abstract

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.

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

TL;DR

We analyze diffusion approximation for slow-fast SDEs with state-dependent switching, showing the slow component converges weakly to an averaged diffusion as , with coefficients derived via a Poisson equation tied to the fast Markov chain. The approach combines Poisson-equation techniques with martingale problem arguments to prove the limit is the unique solution of , and establishes a weak convergence rate of for smooth test functions, with a 1D example showing optimality. The work also provides a priori moment bounds and demonstrates well-posedness for the averaged equation, along with tightness and martingale-problem-based proofs for convergence in the path space. Altogether, it clarifies how state-dependent switching interacts with homogenization to modify drift and diffusion in the diffusion-approximation limit, advancing multi-scale analysis for Markov-switching systems.

Abstract

In this paper, we study the diffusion approximation for slow-fast stochastic differential equations with state-dependent switching, where the slow component is the solution of a stochastic differential equation with additional homogenization term, while the fast component is a switching process. We first prove the weak convergence of to in the space of continuous functions, as . Using the martingale problem approach and Poisson equation associated with a Markov chain, we identify this weak limiting process as the unique solution 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 of weak convergence of to by applying suitable test functions , for any . Additionally, we provide an example to illustrate that the order we achieve is optimal.

Paper Structure

This paper contains 8 sections, 8 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations and assumptions
  4. Main results
  5. A Priori estimates and well-posedness of Eq.(\ref{['bar']})
  6. Weak Convergence of $X^{\varepsilon}$ in $C([0,T],\mathbb{R}^n)$
  7. Tightness of $\{X^{\varepsilon}\}$ in $C([0,T];\mathbb{R}^n)$
  8. Proof of Theorem 2.3

Key Result

Proposition 2.2

Suppose that assumptions A2 and A3 hold. Let where $\alpha_{t}^{x,i}$ is the unique $\mathbb{S}$-valued Markov chain generated by generator $Q(x)$ with initial value $\alpha_{0}^{x,i}=i\in\mathbb{S}$. Then $\Phi(x,i)$ is a solution of Eq.(PE1), and there exists a constant $C > 0$ such that for any $x \in \mathbb{R}^n$, we have

Theorems & Definitions (16)

  • Remark 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Example 2.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 6 more