Paper: arXiv 2603.17954

Authors: Francesca Centrone, Asmerilda Hitaj, Elisa Mastrogiacomo, Emanuela Rosazza Gianin

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous and advanced mathematical framework for robust quasi-convex risk measures on Lp spaces, extending foundational dual representation theorems. However, it is entirely theoretical with no empirical backtesting, simulation analysis, or historical data implementation provided.

Research Flowchart

  flowchart TD
    A["Research Goal: Robustify general risk measures<br>on Lᵖ spaces beyond cash-additivity"] --> B["Inputs: Initial risk measures on Lᵖ spaces<br>& families of uncertainty sets"]
    B --> C["Construct Robust Counterparts:<br>Inherited via convex uncertainty OR<br>quasi-convex uncertainty structures"]
    C --> D["Derive Dual Representations:<br>Penalty-type duals for robust quasi-convex<br>& cash-subadditive measures"]
    D --> E["Analyze Decision & Allocation Impacts:<br>Evaluate acceptance families &<br>capital allocation under ambiguity"]
    E --> F["Key Outcomes: Unified quasi-convex framework,<br>generalizing classical convex cash-additive case<br>& clarifying ambiguity impacts"]