Quasi-geometric rough paths and rough change of variable formula

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 $\phi :{\mathbb{R}}^d\to \mathbb{R}$ a rough change of variable formula on any $\gamma$-Hölder continuous path $x:[0,T]\to {\mathbb{R}}^d$, i.e. an explicit expression of $\phi (x_t)$ in terms of rough integrals.

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• Planarly branched rough paths and rough differential equations on homogeneous spaces
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Cited by in corpus (3)

• Smooth rough paths, their geometry and algebraic renormalization
• Branched Itô formula and natural Itô-Stratonovich isomorphism
• Flows driven by multi-indices Rough Paths