Young and rough differential inclusions

We define in this work a notion of Young differential inclusion $dz_t \in F(z_t)dx_t,$ 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 $dz_t \in F(z_t)dt + G(z_t)d{\bf X}_t,$ for an $α$-Holder rough path $\bf X$ with $α\in \left(\frac{1}{3},\frac{1}{2}\right]$, 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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