Geometric versus non-geometric rough paths

In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in [Gub10]. We first show that branched rough paths can equivalently be defined as γ-Hölder 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 X lying above a path X , there exists a geometric rough path X̄ lying above an extended path X̄ , such that X̄ contains all the information of X. As a corollary of this result, we show that every RDE driven by a non-geometric rough path X can be rewritten as an extended RDE driven by a geometric rough path X̄. One could think of this as a generalisation of the Itô-Stratonovich correction formula.