Large deviation principle for two-time scale FSDEs with fractional Brownian motion

Li Feng; Haibo Gu; Juan Chen; Li Feng; Haibo Gu; Juan Chen

doi:10.3934/era.2026356

Electronic Research Archive

2026, Volume 34, Issue 11: 8397-8427. doi: 10.3934/era.2026356

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Large deviation principle for two-time scale FSDEs with fractional Brownian motion

School of Mathematical Sciences, Xinjiang Normal University, Urumqi 830017, China

Received: 05 August 2026 Revised: 13 September 2026 Accepted: 17 September 2026 Published: 24 September 2026

This paper investigates the large deviation of two-time-scale fractional stochastic systems perturbed by fractional Brownian motion. The stochastic integrals with respect to fractional Brownian motion are understood in the sense of the generalized Riemann–Stieltjes integral, and Itô integration is employed for the standard Brownian motion component. The slow variable is governed by a Caputo fractional derivative, whereas the fast variable is described by an integer-order derivative. By combining the classical weak convergence approach with the averaging principle, a large deviation principle is established for this class of systems. Finally, the theoretical findings are illustrated through a numerical example.
- large deviation principle,
- two-time scale,
- fractional stochastic systems,
- fractional Brownian motion,
- weak convergence
Citation: Li Feng, Haibo Gu, Juan Chen. Large deviation principle for two-time scale FSDEs with fractional Brownian motion[J]. Electronic Research Archive, 2026, 34(11): 8397-8427. doi: 10.3934/era.2026356

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Abstract

This paper investigates the large deviation of two-time-scale fractional stochastic systems perturbed by fractional Brownian motion. The stochastic integrals with respect to fractional Brownian motion are understood in the sense of the generalized Riemann–Stieltjes integral, and Itô integration is employed for the standard Brownian motion component. The slow variable is governed by a Caputo fractional derivative, whereas the fast variable is described by an integer-order derivative. By combining the classical weak convergence approach with the averaging principle, a large deviation principle is established for this class of systems. Finally, the theoretical findings are illustrated through a numerical example.

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