Rough differential equations in the flow approach

TL;DR

This paper demonstrates how the flow approach using elementary differentials indexed by trees can establish well-posedness for rough stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/4.

Contribution

It introduces a novel coordinate system for the flow approach, indexed by trees instead of multi-indices, to prove well-posedness of rough SDEs driven by fractional Brownian motion.

Findings

Proves well-posedness for rough SDEs with H > 1/4.

Uses tree-indexed coordinates for the flow approach.

Extends the flow approach to fractional Brownian motion.

Abstract

We show how the flow approach of Duch, with elementary differentials as coordinates, can be used to prove well-posedness for rough stochastic differential equations driven by fractional Brownian motion with Hurst index $H > \frac{1}{4}$. A novelty appearing here is that we use coordinates for the flow that are indexed by trees rather than multi-indices.