Conditional cores and conditional convex hulls of random sets

TL;DR

This paper introduces the concepts of conditional core and convex hull for random sets in Banach spaces, along with a generalized conditional expectation, providing new tools for analyzing multivariate risks in finance.

Contribution

It defines and studies the properties of conditional cores, convex hulls, and a generalized conditional expectation for random sets, extending existing set operations to non-linear, non-closed random sets in Banach spaces.

Findings

Conditional core is sublinear in set inclusion.

Conditional convex hull is superlinear in set inclusion.

Generalized conditional expectation is bounded between the core and convex hull.

Abstract

We define two non-linear operations with random (not necessarily closed) sets in Banach space: the conditional core and the conditional convex hull. While the first is sublinear, the second one is superlinear (in the reverse set inclusion ordering). Furthermore, we introduce the generalised conditional expectation of random closed sets and show that it is sandwiched between the conditional core and the conditional convex hull. The results rely on measurability properties of not necessarily closed random sets considered from the point of view of the families of their selections. Furthermore, we develop analytical tools suitable to handle random convex (not necessarily compact) sets in Banach spaces; these tools are based on considering support functions as functions of random arguments. The paper is motivated by applications to assessing multivariate risks in mathematical finance.