Geometric versus non-geometric rough paths

Hairer, Martin; Kelly, David

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 207-251.

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 $𝐗$ définie au-dessus d’une trajectoire $X$, il existe une trajectoire rugueuse géométrique $\overline{𝐗}$ définie au-dessus d’une trajectoire étendue $\overline{X}$, de manière à ce que $\overline{𝐗}$ contienne toute l’information de $𝐗$. Il en suit que toute équation différentielle conduite par $𝐗$ peut être reformulée comme une équation différentielle modifiée conduite par $\overline{𝐗}$. On peut interpréter ceci comme une généralisation de la formule de correction Itô–Stratonovich.

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 $𝐗$ lying above a path $X$, there exists a geometric rough path $\overline{𝐗}$ lying above an extended path $\overline{X}$, such that $\overline{𝐗}$ 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 $\overline{𝐗}$. One could think of this as a generalisation of the Itô–Stratonovich correction formula.

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