Maxitive monetary risk measures: worst-case risk assessment and sharp large deviations

TL;DR

This work investigates maxitive monetary risk measures on $L^0$, establishing that maxitivity plus continuity from below on $L^\infty$ is equivalent to a penalized maximum loss representation ${\rm ML}_I(f)={\rm ess.sup}_X(f-I)$, and that the law-invariant, maxitive case reduces to the plain maximum loss ${\rm ML}$. It provides a comprehensive set of equivalences showing that such risk measures can be represented as penalized maximum losses, with $I_{min}$ playing a central role, and it extends these results to unbounded positions via $L^\phi$. The paper then applies these structural insights to sharp large deviation theory by deriving a non-topological criterion that yields sharp LDP and a Laplace principle in general measurable spaces, and to distorted expectations, obtaining a precise asymptotic form for the distortion-exponential premium under risk pooling. Collectively, the results deliver a robust, topology-free framework for worst-case risk assessment, linking maxitive risk measures to large deviation principles and to premium principles under distortion, with clear implications for both theory and risk-management practice.

Abstract

In decision making under uncertainty and risk, worst-case risk assessments are often conducted using maxitive monetary risk measures. In this article, we study maxitive monetary risk measures on the space $L^0$ of all random variables identified modulo almost sure equality. We prove that a monetary risk measure is maxitive and continuous from below if and only if it is a penalized maximum loss. Furthermore, we characterize the maximum loss as the unique maxitive and law-invariant monetary risk measure. We apply the results to large deviation theory by providing a general criterion to establish a sharp large deviation estimate for sequences of probability measures. We use these findings to provide a formula for the asymptotics of the distortion-exponential insurance premium principle under risk pooling.

Theorems & Definitions (46)

• Remark 2.1
• Definition 2.2
• Definition 3.1
• Remark 3.2
• Proposition 3.3
• Remark 3.4
• proof
• Theorem 4.1
• Remark 4.2
• Remark 4.3