Geometric versus non-geometric rough paths
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, p. 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 γ-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 𝐗 ¯ définie au-dessus d’une trajectoire étendue X ¯, de manière à ce que 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 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 γ-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 𝐗 ¯ lying above an extended path X ¯, such that 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 𝐗 ¯. One could think of this as a generalisation of the Itô–Stratonovich correction formula.
DOI : https://doi.org/10.1214/13-AIHP564
Classification:  60H10,  34K28,  16T05
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     author = {Hairer, Martin and Kelly, David},
     title = {Geometric versus non-geometric rough paths},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     pages = {207-251},
     doi = {10.1214/13-AIHP564},
     zbl = {06412903},
     mrnumber = {3300969},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_1_207_0}
}
Hairer, Martin; Kelly, David. Geometric versus non-geometric rough paths. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 207-251. doi : 10.1214/13-AIHP564. http://www.numdam.org/item/AIHPB_2015__51_1_207_0/

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