Robust quasi-convex risk measures and applications

2026-03-18

AI Summary

This study addresses model uncertainty under non-convex and non-cash-additive risk measures by developing a robust quasiconvex risk measurement framework in general Lp spaces. By introducing an uncertainty set and integrating acceptance sets with capital allocation rules, the work employs functional analysis, duality theory, and c-quasiconvex analysis to propose two complementary mechanisms for generating robust risk measures. The main contribution lies in overcoming the classical limitations of convexity and cash additivity, establishing penalty-type dual representations for robust quasiconvex and cash-subadditive risk measures. Furthermore, the paper demonstrates that the structure of uncertainty itself can induce quasiconvexity, thereby revealing the fundamental impact of ambiguity on capital allocation and asset acceptability.

Technology Category

Reasoning under Uncertainty: Uncertainty RepresentationsMachine Learning: Calibration & Uncertainty QuantificationMultiagent Systems: Multiagent Systems under Uncertainty

Application Category

Security and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsEconomics, Online Markets and Human Computation: LLM based quality controls for crowd work

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.

Problem

• robust risk measures
• quasi-convexity
• model ambiguity
• cash-subadditivity
• Lp spaces

Innovation

• robust risk measures
• quasi-convexity
• uncertainty sets
• dual representation
• cash-subadditivity