Perturbed linear rough differential equations

Laure Coutin; Antoine Lejay

doi:10.5802/ambp.338

Perturbed linear rough differential equations
[Équations différentielles linéaires rugueuses perturbées]

Laure Coutin ¹ ; Antoine Lejay ²

¹ Institut de Mathématiques de Toulouse, F-31062 Toulouse Cedex 9, France.
² Université de Lorraine, Institut Élie Cartan, UMR 7502, Vandœuvre-lès-Nancy, F-54600, France CNRS, Institut Élie Cartan, UMR 7502, Vandœuvre-lès-Nancy, F-54600, France Inria, Villers-lès-Nancy, F-54600, France

Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 103-150.

Résumé
Abstract

Nous étudions les équations différentielles linéaires rugueuses et résolvons des équations linéaires rugueuses perturbées à l’aide du principe de Duhamel. Ces résultats donnent un argument technique pour étudier la différentiabilité de l’application d’Itô. La notion d’équation différentielle rugueuses nous condition à considérer des fonctionnelles multiplicatives à valeurs dans des algèbres de Banach plus générales que celle des algèbres tensorielles, ainsi que des extensions de résultats classiques tels que les formules de Magnus et Chen-Strichartz.

We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus and the Chen-Strichartz formula.

MR Zbl

DOI : 10.5802/ambp.338

Classification : 34A25, 60H10
Keywords: Rough paths, Rough differential equations, Banach algebra, Magnus formula Chen-Strichartz formula, perturbation formula, Duhamel’s principle
Mot clés : Trajectoires rugueuses, Équations différentielles rugueuses, algèbre de Banach, formule de Magnus, formule de Chen-Strichartz, formule de perturbation, principe de Duhamel

Affiliations des auteurs :

Laure Coutin ¹ ; Antoine Lejay ²

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Laure Coutin; Antoine Lejay. Perturbed linear rough differential equations. Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 103-150. doi : 10.5802/ambp.338. https://ambp.centre-mersenne.org/articles/10.5802/ambp.338/

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