Bipolar Theorems for Sets of Non-negative Random Variables

TL;DR

This paper establishes comprehensive bipolar theorems for non-negative random variables within a general, non-dominated probabilistic framework, extending previous results and applicable to robust financial modeling.

Contribution

It provides necessary and sufficient conditions for bipolar representations without restrictions on the measure space, unifying and generalizing prior theorems.

Findings

Generalizes bipolar theorems to non-dominated frameworks

Unifies existing results under weaker assumptions

Applicable to robust financial models

Abstract

This paper assumes a robust, in general not dominated, probabilistic framework and provides necessary and sufficient conditions for a bipolar representation of subsets of the set of all quasi-sure equivalence classes of non-negative random variables, without any further conditions on the underlying measure space. This generalizes and unifies existing bipolar theorems proved under stronger assumptions on the robust framework. Applications are in areas of robust financial modeling.