Asymptotic preserving schemes for SDEs driven by fractional Brownian motion in the averaging regime

TL;DR

This paper develops numerical schemes for slow-fast stochastic differential equations driven by fractional Brownian motion, ensuring they accurately approximate the averaged system as the time-scale parameter diminishes.

Contribution

It introduces asymptotic preserving schemes for SDEs with fractional Brownian motion, extending averaging techniques beyond the standard Wiener process case.

Findings

The schemes are asymptotic preserving as the time-scale parameter approaches zero.

The proposed methods accurately capture the averaged behavior of the system.

The approach highlights differences between fractional and standard Brownian motion cases.

Abstract

We design numerical schemes for a class of slow-fast systems of stochastic differential equations, where the fast component is an Ornstein-Uhlenbeck process and the slow component is driven by a fractional Brownian motion with Hurst index $H>1/2$. We establish the asymptotic preserving property of the proposed scheme: when the time-scale parameter goes to $0$, a limiting scheme which is consistent with the averaged equation is obtained. With this numerical analysis point of view, we thus illustrate the recently proved averaging result for the considered SDE systems and the main differences with the standard Wiener case.