Pathwise Uniqueness for Multiplicative Young and Rough Differential Equations Driven by Fractional Brownian Motion

TL;DR

This paper proves pathwise uniqueness for multiplicative stochastic differential equations driven by fractional Brownian motion with Hurst parameter between 1/3 and 1, under weaker regularity conditions on the volatility coefficient than previously known.

Contribution

It introduces a novel combination of stochastic averaging estimates and refined stochastic sewing techniques to establish uniqueness under less restrictive regularity assumptions.

Findings

Established pathwise uniqueness for Hurst parameter in (1/3,1).

Reduced regularity requirement on volatility coefficient from 1/H-Hölder to a weaker condition.

Extended results to arbitrary dimensions for fractional Brownian motion-driven SDEs.

Abstract

We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ with volatility coefficient $\sigma$ that is at least $\gamma$-H\"older continuous for $\gamma > \frac{1}{2H} \vee \frac{1-H}{H}$. This improves upon the long-standing results of [Lyo94 , Lyo98 , Dav08] which cover the same regime but require $\sigma$ to be at least $\frac{1}{H}$-H\"older continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to [L\^e20, Ger22, MP22].