Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions

Abstract

We study the differential inclusion $\text{d}z_t\in F(z_t)\text{d}x_t$, with initial condition $z_0=ξ$, where $F$ is a nonconvex-valued multifunction, and $x$ a path of bounded $q$-variation, for some $1\leqslant q<2$, extending the work of Bailleul, Brault, and Coutin (2020). We obtain existence of local and global solutions to this inclusion without assuming $F$ bounded. If $z(ξ,x)$ denotes such a solution, we obtain measurability of $z$ with respect to $x$ and $ξ$. To establish this, we introduce a Skorokhod-type distance and prove that Young integration is continuous with respect to it. By the way, we prove that a compact-valued $γ$-H{ö}lder map $F$ has, for any $p>1/γ$ and $ξ\in F(0)$, a selection $f(ξ)$ of bounded $p$-variation, started at $ξ$, such that $f$ is measurable in $ξ$.