Geometric versus non-geometric rough paths
By Martin Hairer, David Kelly. 2012
117 citations recorded by the source corpus.
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.
Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/db54e1a785ca91bbd222a7581245f51c2c94a9c2
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In Mathematical Physics
- Huilin Zhang
- Christian Bayer
- Benjamin Stemper
- Jorg Martin
- Paul Gassiat
- Peter K. Friz
- Nicolas Victoir
- Massimiliano Gubinelli
In Nuclear and High Energy Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Differential equations driven by rough paths with jumps (2017, methodology): “For example, as pointed out in the works of Gubinelli and Hairer–Kelly, [18, 20], the 2In absence of jumps forward and geometric solutions coincide.”
- A regularity structure for rough volatility (2017, background): “e a tree notation (omnipresent in the works of Hairer) would come in handy and we refer to [9] (and the references therein) for a recent attempt to reconcile the tree formalism of branched rough path [29, 34] and the most recent algebraic formalism of regularity structures. (In a nutshell, the simpl”