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Robust quasi-convex risk measures and applications

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.

Robust quasi-convex risk measures and applications

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.
Paper Structure (14 sections, 11 theorems, 80 equations)

This paper contains 14 sections, 11 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and setting
  3. Setting and notations
  4. Risk measures
  5. Uncertainty sets and robustification
  6. Dual representation of robust quasi-convex risk measures
  7. First approach: quasi-convex $\rho$, convex $\mathcal{U}$
  8. Second approach: convex $\rho$ , quasi-convex $\mathcal{U}$
  9. Acceptance families under robustification/uncertainty
  10. Applications and examples
  11. Capital allocation under robustification /ambiguity
  12. Law invariant case under Wasserstein distance
  13. Examples of families of uncertainty sets
  14. Conclusions

Key Result

Lemma 1

i) Solidity of $\mathcal{U}$ is equivalent to $\mathcal{U}_X=\mathcal{U}_X + L^{p}_+$ for any $X \in L^{p}$. ii) If the family $\mathcal{U}$ is quasi-convex, then it is also c-quasi-convex. The converse implication holds under solidity of $\mathcal{U}$.

Theorems & Definitions (31)

  • Definition 1
  • Remark 1
  • Example 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 21 more