Young and rough differential inclusions release_m74u6ji5kjd7fb7dbzkpbfvzpm

Abstract

We define in this work a notion of Young differential inclusion dz_t ∈ F(z_t)dx_t, for an α-Holder 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)d X_t, for an α-Holder 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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