Geometric versus non-geometric rough paths

TL;DR

This paper demonstrates that non-geometric rough paths can be represented as geometric rough paths, allowing rough differential equations driven by non-geometric paths to be reformulated as geometric ones, extending classical correction formulas.

Contribution

It introduces a method to encode non-geometric branched rough paths as geometric rough paths, enabling the rewriting of associated RDEs in a geometric framework.

Findings

Non-geometric rough paths can be equivalently defined as paths in a Lie group.

Every branched rough path can be encoded into a geometric rough path.

RDEs driven by non-geometric rough paths can be reformulated as geometric RDEs.

Abstract

In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in Gubinelli (2004). We first show that branched rough paths can equivalently be defined as $\gamma$-H\"older continuous paths in some Lie group, akin to geometric rough paths. We then show that every branched rough path can be encoded in a geometric rough path. More precisely, for every branched rough path $\mathbf{X}$ lying above a path $X$, there exists a geometric rough path $\bar{\mathbf{X}}$ lying above an extended path $\bar X$, such that $\bar{\mathbf{X}}$ contains all the information of $\mathbf{X}$. As a corollary of this result, we show that every RDE driven by a non-geometric rough path $\mathbf{X}$ can be rewritten as an extended RDE driven by a geometric rough path $\bar{\mathbf{X}}$. One could think of this as a generalisation of the It\^o-Stratonovich correction formula.