Averaging principle for slow-fast fractional stochastic differential equations

TL;DR

This paper extends the averaging principle to slow-fast systems where the slow component is governed by a fractional differential equation of order $\alpha\in(0,1)$ and the fast component is an SDE driven by Brownian motion. By constructing auxiliary time-discretized processes and leveraging ergodicity of the frozen fast dynamics, it proves convergence of the slow variable $X^{\varepsilon}$ to the solution $\overline{X}$ of the averaged fractional equation with coefficient $\overline{f}(x)=\int f(x,y)\,d\mu^{x}(y)$. In the special case where the fast dynamics do not depend on the slow variable, the paper provides a rate of convergence of order $\varepsilon^{\alpha/2}$ in mean-square norm. The results rely on moment bounds, regularity properties, and a careful decomposition of the error via an auxiliary system, offering a rigorous homogenization framework for slow-fast fractional stochastic models. This advances understanding of multiscale stochastic systems with fractional dynamics and broadens potential applications in physics, biology, and finance.

Abstract

We prove the averaging principle for a class of stochastic systems. The slow component is solution to a fractional differential equation, which is coupled with a fast component considered as solution to an ergodic stochastic differential equation driven by a standard Brownian motion. We establish the convergence of the slow component when the time-scale separation vanishes to the solution of the so-called averaged equation, which is an autonomous fractional differential equation, in the mean-square sense. Moreover, when the fast component does not depend on the slow component, we provide a rate of convergence depending on the order of the fractional derivative.

Theorems & Definitions (26)

• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Theorem 3.1
• Theorem 3.2
• Remark 3.3
• Proposition 4.1
• proof : Proof of Proposition \ref{['propo:boundXepsYeps']}
• Proposition 4.2
• proof : Proof of Proposition \ref{['propo:boundXbar']}