Young and rough differential inclusions
Résumé
We define in this work a notion of Young differential inclusion dz_t ∈ F(z_t) dx_t , for an α-Hölder control x, with α > 1/2, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, γ-Hölder continuous set-valued map on the interval [0,1] has a selection with finite p-variation, for p > 1/γ. We also give a notion of solution to the rough differential inclusion dz_t ∈ F(z_t) dt+G(z_t) dX_t , for an α-Hölder rough path X with α ∈ (1/3 , 1/2] , a set-valued map F and a single-valued one form G. Then, we prove the existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.
Domaines
Probabilités [math.PR]Format du dépôt | Fichier |
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Type de dépôt | Article dans une revue |
Titre |
en
Young and rough differential inclusions
|
Résumé |
en
We define in this work a notion of Young differential inclusion dz_t ∈ F(z_t) dx_t , for an α-Hölder control x, with α > 1/2, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, γ-Hölder continuous set-valued map on the interval [0,1] has a selection with finite p-variation, for p > 1/γ. We also give a notion of solution to the rough differential inclusion dz_t ∈ F(z_t) dt+G(z_t) dX_t , for an α-Hölder rough path X with α ∈ (1/3 , 1/2] , a set-valued map F and a single-valued one form G. Then, we prove the existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.
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Auteur(s) |
Ismaël Bailleul
1
, Antoine Brault
2, 3
, Laure Coutin
4
1
IRMAR -
Institut de Recherche Mathématique de Rennes
( 75 )
- Campus de Beaulieu, bâtiments 22 et 23,
263 avenue du Général Leclerc, CS 74205
35042 RENNES Cédex
- France
2
MAP5 - UMR 8145 -
Mathématiques Appliquées Paris 5
( 1004645 )
- UFR Mathématiques et Informatique,
45 rue des Saints-Pères
75270 PARIS CEDEX 06
- France
3
CMM -
Center for Mathematical Modeling
( 227891 )
- Av. Blanco Encalada 2120 Piso 7 Santiago de Chile
- Chili
4
UT3 -
Université Toulouse III - Paul Sabatier
( 217752 )
- 118 route de Narbonne - 31062 Toulouse
- France
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Langue du document |
Anglais
|
Vulgarisation |
Non
|
Comité de lecture |
Oui
|
Audience |
Internationale
|
Date de publication |
2021
|
Volume |
37
|
Numéro |
4
|
Nom de la revue |
|
Page/Identifiant |
1489 - 1512
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Date de publication électronique |
2020-11-23
|
Domaine(s) |
|
Projet(s) ANR |
|
DOI | 10.4171/rmi/1236 |
Origine :
Fichiers produits par l'(les) auteur(s)
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