Paper: arXiv 2403.06188

Authors: Mücahit Aygün, Fabio Bellini, Roger J. A. Laeven

Abstract

Geometrically convex functions constitute an interesting class of functions obtained by replacing the arithmetic mean with the geometric mean in the definition of convexity. As recently suggested, geometric convexity may be a sensible property for financial risk measures ([7,13,4]). We introduce a notion of GG-convex conjugate, parallel to the classical notion of convex conjugate introduced by Fenchel, and we discuss its properties. We show how GG-convex conjugation can be axiomatized in the spirit of the notion of general duality transforms introduced in [2,3]. We then move to the study of GG-convex risk measures, which are defined as GG-convex functionals defined on suitable spaces of random variables. We derive a general dual representation that extends analogous expressions presented in [4] under the additional assumptions of monotonicity and positive homogeneity. As a prominent example, we study the family of Orlicz risk measures. Finally, we introduce multiplicative versions of the convex and of the increasing convex order and discuss related consistency properties of law-invariant GG-convex risk measures.

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 theoretical exploration of geometrically convex risk measures and general dual representations using advanced functional analysis. However, it is purely theoretical with no empirical backtesting, datasets, or real-world market applications.

Research Flowchart

  flowchart TD
    A[Research Goal: Formalize & Analyze<br>GG-Convex Risk Measures] --> B[Introduce GG-Convex Conjugation<br>& Axiomatize Duality Transforms]
    B --> C[Input: Spaces of Random Variables<br>& Geometric Convexity Framework]
    C --> D[Derive Dual Representation<br>without monotonicity/homogeneity]
    C --> E[Analyze Multiplicative Stochastic<br>Orders & Law-Invariance]
    D --> F[Outcome 1: General Dual Representation<br>for GG-Convex Risk Measures]
    E --> F
    D --> G[Outcome 2: Characterization of<br>Orlicz Risk Measures]