Paper: arXiv 2412.19546

Authors: Peng Liu, Tiantian Mao, Ruodu Wang

Abstract

Choquet capacities and integrals are central concepts in decision making under ambiguity or model uncertainty, pioneered by Schmeidler. Motivated by risk optimization problems for quantiles under ambiguity, we study the subclass of Choquet integrals, called Choquet quantiles, which generalizes the usual (probabilistic) quantiles, also known as Value-at-Risk in finance, from probabilities to capacities. Choquet quantiles share many features with probabilistic quantiles, in terms of axiomatic representation, optimization formulas, and risk sharing. We characterize Choquet quantiles via only one axiom, called ordinality. We prove that the inf-convolution of Choquet quantiles is again a Choquet quantile, leading to explicit optimal allocations in risk sharing problems for quantile agents under ambiguity. A new class of risk measures, Choquet Expected Shortfall, is introduced, which enjoys most properties of the coherent risk measure Expected Shortfall. Our theory is complemented by optimization algorithms, numerical examples, and a stylized illustration with financial data.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework extending Value-at-Risk and Choquet integrals under model ambiguity, backed by rigorous axiomatic characterizations and optimal allocation proofs. While it includes numerical examples and stylized financial data, the contribution is primarily advanced mathematical finance rather than an empirical backtest.

Research Flowchart

  flowchart TD
    A[Goal: Optimize Quantile Risk Sharing under Ambiguity] --> B[Generalize VaR to Choquet Quantiles using Choquet Capacities]
    B --> C[Axiomatic Characterization via Ordinality Axiom]
    C --> D[Compute Inf-Convolution of Choquet Quantiles]
    D --> E[Develop Choquet Expected Shortfall & Optimization Algorithms]
    E --> F[Inputs: Multi-Asset Financial Data & Uncertainty Models]
    F --> G[Outcomes: Explicit Optimal Allocations & Coherent Risk Measures]