Moderate Deviation Principle for Two-Time-Scale Caputo FSDEs Driven by Fractional Brownian Motion

This work investigates the moderate deviation principle for a class of two-time-scale Caputo fractional stochastic differential equations. The driving noise of the slow variable is fractional Brownian motion with Hurst index H∈(12,1). The driving noise of the fast variable is standard Brownian motion. The fractional derivative operator of the slow variable is defined by Caputo, and the derivative of the fast variable is of the integer order. The proof process is mainly based on the weak convergence method of fractional Brownian motion variational representation. We first establish the moderate deviation principle by proving the weak convergence of the single-time-scale controlled version. Subsequently, we combine Khasminskii time discretization technology to extend the theoretical framework to two-time-scale systems. Finally, a concrete computational case is offered to demonstrate the applicability of the theoretical framework.