Set risk measures

TL;DR

This paper introduces set risk measures (SRMs), extending traditional risk measures to entire sets of random variables, with a new axiomatic framework and dual representation.

Contribution

It develops an axiomatic framework for convex SRMs, including dual representation and applications to systemic risk and uncertainty.

Findings

Dual representation of convex SRMs via the extit{strict} topology.

Characterization of worst-case SRMs.

Examples related to systemic risk and Knightian uncertainty.

Abstract

We introduce set risk measures (SRMs), real-valued maps defined on the family of non-empty closed bounded sets of essentially bounded random variables. SRMs extend traditional scalar risk measures by assigning a single capital requirement to an entire set of positions. We develop an axiomatic framework for SRMs, adapting classical properties such as monotonicity, translation invariance, convexity, and positive homogeneity to set arithmetic. The main technical contribution is a dual representation of convex SRMs through the \strict{} topology and regular $\tau$-additive unit-mass measures. We also characterize worst-case SRMs and present examples related to systemic risk, Knightian uncertainty, and preference representations.