# Young and rough differential inclusions

## TL;DR

This paper introduces a new framework for Young and rough differential inclusions, proving existence of solutions under certain regularity and continuity conditions, and establishing properties of set-valued maps with applications to differential systems.

## Contribution

It defines Young and rough differential inclusions with existence results, and shows that set-valued maps with certain regularity have selections with finite p-variation.

## Key findings

- Existence of solutions for Young differential inclusions with -Hb6lder controls.
- Existence of solutions for rough differential inclusions with -Hb6lder rough paths.
- Set-valued maps with -5 ext{H"older} regularity have selections with finite p-variation.

## Abstract

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