Random multifunctions as the set minimizers of infinitely many differentiable random functions

TL;DR

This paper demonstrates that any random multifunction can be represented as the set of minimizers of infinitely differentiable random functions, preserving convexity and enabling applications in approximation and characterization of integrable selections.

Contribution

It introduces a novel representation of random multifunctions via infinitely differentiable integrands, extending the theoretical understanding and practical applications in stochastic analysis.

Findings

Representation of random multifunctions as minimizers of smooth functions

Preservation of convexity in the representation

Characterization of integrable selections as minimizers

Abstract

Under mild assumptions, we prove that any random multifunction can be represented as the set of minimizers of an infinitely many differentiable normal integrand, which preserves the convexity of the random multifunction. We provide several applications of this result to the approximation of random multifunctions and integrands. The paper ends with a characterization of the set of integrable selections of a measurable multifunction as the set of minimizers of an infinitely many differentiable integral function.