Some differential systems driven by a fBm with Hurst parameter greater than 1/4

TL;DR

This paper extends algebraic integration methods to analyze differential systems driven by fractional Brownian motion with Hurst parameter greater than 1/4, including delay equations, providing new theoretical insights.

Contribution

It advances the algebraic integration framework to handle noisy inputs with low regularity, specifically H>1/4, and applies it to delay differential equations driven by fractional Brownian motion.

Findings

Fractional Brownian motion with H>1/4 satisfies the assumptions of the new theorems.

The extended algebraic integration approach effectively treats delay differential systems.

The paper provides a rigorous analysis for systems driven by rough signals with H>1/4.

Abstract

This note is devoted to show how to push forward the algebraic integration setting in order to treat differential systems driven by a noisy input with H\"older regularity greater than 1/4. After recalling how to treat the case of ordinary stochastic differential equations, we mainly focus on the case of delay equations. A careful analysis is then performed in order to show that a fractional Brownian motion with Hurst parameter H>1/4 fulfills the assumptions of our abstract theorems.