A remainder estimate for branched rough differential equations

TL;DR

This paper establishes an optimal order bound on the remainder of the Taylor expansion for controlled differential equations driven by branched rough paths, using Hopf algebra isomorphisms.

Contribution

It introduces a novel bound for the Taylor expansion remainder in branched rough differential equations leveraging Hopf algebra isomorphisms.

Findings

Provides an optimal order bound on the Taylor remainder

Utilizes two isomorphisms of Hopf algebras

Enhances understanding of controlled differential equations

Abstract

Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.