Young and rough differential inclusions

Abstract

We define in this work a notion of Young differential inclusion $$ dz\_t \in F(z\_t),dx\_t, $$ for an $\alpha$-Hölder control $x$, with $\alpha > 1/2$, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, $\gamma$-Hölder continuous set-valued map on the interval $\[0,1]$ has a selection with finite $p$-variation, for $p > 1/\gamma$. We also give a notion of solution to the rough differential inclusion $$ dz\_t \in F(z\_t),dt + G(z\_t),d{\mathbf X}\_t, $$ for an $\alpha$-Hölder rough path $\mathbf X$ with $\alpha\in(1/3,1/2]$, 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.