Maximality and numeraires in convex sets of nonnegative random variables

TL;DR

This paper introduces the concepts of max-closedness and numeraires in convex sets of nonnegative random variables, exploring their properties and relationships within the topology of convergence in probability.

Contribution

It defines max-closedness and numeraires in the context of convex sets of nonnegative random variables, providing new insights into their structure and density properties.

Findings

Max-closedness ensures maximal elements of the closure are in the set.

Numeraires are strictly positive optimizers in certain maximisation problems.

Numeraires are dense in the set of maximal elements under specified conditions.

Abstract

We introduce the concepts of max-closedness and numeraires of convex subsets in the nonnegative orthant of the topological vector space of all random variables built over a probability space, equipped with a topology consistent with convergence in probability. Max-closedness asks that maximal elements of the closure of a set already lie on the set. We discuss how numeraires arise naturally as strictly positive optimisers of certain concave monotone maximisation problems. It is further shown that the set of numeraires of a convex, max-closed and bounded set of of nonnegative random variables that contains at least one strictly positive element is dense in the set of its maximal elements.