Quasigeometric rough paths and rough change of variable formula
Abstract
Using some basic notions from the theory of Hopf algebras and quasishuffle algebras, we introduce rigorously a new family of rough paths: the quasigeometric 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 $\gamma$Hölder continuous path $x\colon [0, T]\to \mathbb{R}^d$, i.e. an explicit expression of $\varphi(x_t)$ in terms of rough integrals.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 DOI:
 10.48550/arXiv.2009.00903
 arXiv:
 arXiv:2009.00903
 Bibcode:
 2020arXiv200900903B
 Keywords:

 Mathematics  Probability;
 60L20;
 60L70