The randomly distorted Choquet integrals with respect to a G-randomly distorted capacity and risk measures

Paper: arXiv 2509.17555

Authors: Ohood Aldalbahi, Miryana Grigorova

Abstract

We study randomly distorted Choquet integrals with respect to a capacity c on a measurable space (Ω,F), where the capacity c is distorted by a G-measurable random distortion function (with G a sub-σ-algebra of F). We establish some fundamental properties, including the comonotonic additivity of these integrals under suitable assumptions on the underlying capacity space. We provide a representation result for comonotonic additive conditional risk measures which are monotone with respect to the first-order stochastic dominance relation (with respect to the capacity c) in terms of these randomly distorted Choquet integrals. We also present the case where the random distortion functions are concave. In this case, the G-randomly distorted Choquet integrals are characterised in terms of comonotonic additive conditional risk measures which are monotone with respect to the stop-loss stochastic dominance relation (with respect to the capacity c). We provide examples, extending some well-known risk measures in finance and insurance, such as the Value at Risk and the Average Value at Risk.

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 is a rigorous mathematical development in measure theory and non-additive probability, extending Choquet integrals and risk measures to conditional settings with random distortions. However, it is purely theoretical with no empirical data, backtesting, or code, placing it firmly in the ‘Lab Rats’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Define & analyze G-randomly distorted Choquet integrals and risk measures] --> B[Inputs: Capacity space Ω, F, c and G-measurable random distortion function]
    B --> C[Core Process: Establish fundamental properties & comonotonic additivity]
    C --> D[Case Analysis: General vs. Concave Distortion Functions]
    D --> E[Outcome 1: Representation of 1st-order stochastic dominance monotone risk measures]
    D --> F[Outcome 2: Representation of stop-loss stochastic dominance monotone risk measures]
    E & F --> G[Financial Applications: Generalized VaR and AVaR models]