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]| Origine | Fichiers produits par l'(les) auteur(s) |
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