A structural characterization of numeraires of convex sets of nonnegative random variables

TL;DR

This paper introduces the concept of numeraires in convex sets of nonnegative random variables, providing a necessary and sufficient condition for an element to serve as a numeraire, inspired by financial mathematics.

Contribution

It formalizes the notion of numeraires within convex sets of random variables and characterizes them mathematically, extending financial mathematics concepts.

Findings

Provides a necessary and sufficient condition for numeraires.

Formalizes the concept of numeraires in convex sets.

Connects the theory to financial mathematics applications.

Abstract

We introduce the concept of numeraires of convex sets in the nonnegative orthant of the topological vector space of all random variables built over a probability space. A necessary and sufficient condition for an element of a convex set to be its numeraire is given, inspired from ideas in financial mathematics.