Robust quasi-convex risk measures and applications

TL;DR

This paper introduces a unified framework for creating robust quasi-convex risk measures that account for ambiguity, extending classical models and analyzing their dual representations and implications for acceptance and capital allocation.

Contribution

It develops a comprehensive approach to robust quasi-convex risk measures on Lp spaces, including dual representations and applications to acceptance and capital allocation.

Findings

Dual representations for robust quasi-convex risk measures derived

Robust measures extend classical convex cash-additive models

Ambiguity impacts acceptability and capital distribution

Abstract

This paper develops a unified framework for the robustification of risk measures beyond the classical convex and cash-additive setting. We consider general risk measures on Lp spaces and construct their robust counterparts through families of uncertainty sets that capture ambiguity. Two complementary mechanisms generate robust quasi-convex measures: in the first, quasi-convexity is inherited from the initial risk measure under convex uncertainty sets; in the second it comes from the quasi-convex (or c-quasi-convex) structure of the uncertainty sets themselves. Building on Cerreia-Vioglio et al. (2011); Frittelli and Maggis (2011), we derive dual (penalty-type) representations for robust quasi-convex and cash-subadditive risk measures, showing that the classical convex cash-additive case arises as a special instance. We further analyze acceptance families and capital allocation rules under robustification, highlighting how ambiguity affects acceptability and the distribution of capital.