Quasi-geometric rough paths and rough change of variable formula

TL;DR

This paper introduces quasi-geometric rough paths using Hopf algebra concepts, relating them to Brownian integrals and establishing a rough change of variable formula for Hölder continuous paths.

Contribution

It rigorously defines quasi-geometric rough paths and connects them with iterated Brownian integrals and simple bracket extensions, providing a new framework in rough path theory.

Findings

Introduction of quasi-geometric rough paths

Relation to iterated Brownian integrals

Rough change of variable formula for Hölder paths

Abstract

Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them with iterated Brownian integrals and the concept of "simple bracket extension", developed in the PhD thesis of David Kelly. As a consequence of these results, we have a sufficient criterion to show for any $\gamma\in (0,1)$ and any sufficiently smooth function $\varphi \colon \mathbb{R}^d\to \mathbb{R}$ a rough change of variable formula on any $\gamma$-H\"older continuous path $x\colon [0, T]\to \mathbb{R}^d$, i.e. an explicit expression of $\varphi(x_t)$ in terms of rough integrals.