A Theorem on Martingale Selection for Relatively Open Convex Set-Valued Random Sequences

For set-valued random sequences $(G_n)_{n=0}^N$ with relatively open convex values $G_n(\omega)$, we prove a new test for the existence of a sequence $(x_n)_{n=0}^N$ of selectors adapted to the filtration and admitting an equivalent martingale measure. The statement is formulated in terms of the supports of regular upper conditional distributions of $G_n$. This is a strengthening of the main result proved in our previous paper [1], where the openness of the set $G_n(\omega)$ was assumed and a possible weakening of this condition was discussed.