Asymptotical stability of differential equations driven by H\"older--continuous paths

TL;DR

This paper proves the local exponential stability of solutions to differential equations driven by H"older--continuous paths with exponent greater than 1/2, including those driven by fractional Brownian motion, expanding understanding of stability in such systems.

Contribution

It establishes asymptotic local exponential stability for differential equations driven by H"older paths with exponent > 1/2, including fractional Brownian motion, which was not previously well-understood.

Findings

Proves local exponential stability for H"older paths with exponent > 1/2

Includes applications to fractional Brownian motion with Hurst > 1/2

Provides example of scalar equation with global stability

Abstract

In this manuscript, we establish asymptotic local exponential stability of the trivial solution of differential equations driven by H\"older--continuous paths with H\"older exponent greater than $1/2$. This applies in particular to stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than $1/2$. We motivate the study of local stability by giving a particular example of a scalar equation, where global stability of the trivial solution can be obtained.