Michael selections and Castaing representations with cadlag functions

TL;DR

This paper characterizes when closed convex set-valued mappings on the Sorgenfrey line can be represented by cadlag selections, with applications to stochastic optimization and integral functionals of cadlag processes.

Contribution

It provides necessary and sufficient conditions for representing set-valued mappings via cadlag selections, extending Michael's selection theorem to this context.

Findings

Characterization of cadlag selection representations

Application to stochastic optimization problems

Results on integral functionals of cadlag functions

Abstract

Michael's selection theorem implies that a closed convex nonempty-valued mapping from the Sorgenfrey line to a euclidean space is inner semicontinuous if and only if the mapping can be represented as the image closure of right-continuous selections of the mapping. This article gives necessary and sufficient conditions for the representation to hold for cadlag selections, i.e., for selections that are right-continuous and have left limits. The characterization is motivated by continuous time stochastic optimization problems over cadlag processes. Here, an application to integral functionals of cadlag functions is given.