Homogenization principle and numerical analysis for fractional stochastic differential equations with different scales

TL;DR

This paper investigates fractional stochastic differential equations with multiple scales, establishing solution existence, proving a homogenization principle, and analyzing a numerical scheme with practical examples demonstrating computational benefits.

Contribution

It introduces a homogenization principle for fractional stochastic systems with different scales and provides a numerical scheme analysis with advantages over existing methods.

Findings

Homogenization principle holds in mean square sense.

Euler-Maruyama scheme's error is analyzed.

Numerical examples confirm theoretical results and computational advantages.

Abstract

This work is concerned with fractional stochastic differential equations with different scales. We establish the existence and uniqueness of solutions for Caputo fractional stochastic differential systems under the non-Lipschitz condition. Based on the idea of temporal homogenization, we prove that the homogenization principle (averaging principle) holds in the sense of mean square ($L^2$ norm) convergence under a novel homogenization assumption. Furthermore, an Euler-Maruyama scheme for the non-autonomous system is constructed and its numerical error is analyzed. Finally, two numerical examples are presented to verify the theoretical results. Different from the existing literature, we demonstrate the computational advantages of the homogenized autonomous system from a numerical perspective.