Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Geometric versus non-geometric rough paths

Martin Hairer and David Kelly

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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 (J. Differential Equations 248 (2010) 693–721). We first show that branched rough paths can equivalently be defined as $\gamma$-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 $\mathbf{X}$ 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 Itô–Stratonovich correction formula.

Résumé

Dans cet article, nous considérons des équations différentielles conduites par des trajectoires rugueuses non-géométriques en utilisant le concept de trajectoire rugueuse ramifiée introduit dans (J. Differential Equations 248 (2010) 693–721). Nous montrons d’abord que celles-ci peuvent être définies de manière équivalente comme une fonction $\gamma$-Hölderienne à valeurs dans un certain groupe de Lie, comme c’est le cas pour les trajectoires rugueuses dites « géométriques » . Nous montrons ensuite que toute trajectoire rugueuse ramifiée peut être encodée par une trajectoire rugueuse géométrique. Plus précisément, pour toute trajectoire rugueuse ramifiée $\mathbf{X}$ définie au-dessus d’une trajectoire $X$, il existe une trajectoire rugueuse géométrique $\bar{\mathbf{X}}$ définie au-dessus d’une trajectoire étendue $\bar{X}$, de manière à ce que $\bar{\mathbf{X}}$ contienne toute l’information de $\mathbf{X}$. Il en suit que toute équation différentielle conduite par $\mathbf{X}$ peut être reformulée comme une équation différentielle modifiée conduite par $\bar{\mathbf{X}}$. On peut interpréter ceci comme une généralisation de la formule de correction Itô–Stratonovich.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 1 (2015), 207-251.

Dates
First available in Project Euclid: 14 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1421244404

Digital Object Identifier
doi:10.1214/13-AIHP564

Mathematical Reviews number (MathSciNet)
MR3300969

Zentralblatt MATH identifier
1314.60115

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 34K28: Numerical approximation of solutions 16T05: Hopf algebras and their applications [See also 16S40, 57T05]

Keywords
Rough paths Hopf algebra Integration

Citation

Hairer, Martin; Kelly, David. Geometric versus non-geometric rough paths. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 207--251. doi:10.1214/13-AIHP564. https://projecteuclid.org/euclid.aihp/1421244404


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