Extended Convolution Bounds on the Fréchet Problem: Robust Risk Aggregation and Risk Sharing

Paper: arXiv 2511.21929

Authors: Peng Liu, Yang Liu, Houhan Teng

Abstract

In this paper, we provide extended convolution bounds for the Fréchet problem and discuss related implications in quantitative risk management. First, we establish a new form of inequality for the Range-Value-at-Risk (RVaR). Based on this inequality, we obtain bounds for robust risk aggregation with dependence uncertainty for (i) RVaR, (ii) inter-RVaR difference and (iii) inter-quantile difference, and provide sharpness conditions. These bounds are called extended convolution bounds, which not only complement the results in the literature (convolution bounds in Blanchet et al. (2025)) but also offer results for some variability measures. Next, applying the above inequality, we study the risk sharing for the averaged quantiles (corresponding to risk sharing for distortion risk measures with special inverse S-shaped distortion functions), which is a non-convex optimization problem. We obtain the expression of the minimal value of the risk sharing and the explicit expression for the corresponding optimal allocation, which is comonotonic risk sharing for large losses and counter-comonotonic risk sharing for small losses or large gains. Finally, we explore the dependence structure for the optimal allocations, showing that the optimal allocation does not exist if the risk is not bounded from above.

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 rigorous mathematical proofs establishing new extended convolution bounds for the Fréchet problem, with applications to RVaR risk aggregation and non-convex risk sharing. It is a purely theoretical probability and quantitative risk management paper with high technical density but no empirical backtesting or data analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Fréchet Problem Bounds & Risk Sharing Optimization] --> B[Input: Risk Measures RVaR, Quantiles & Marginal Distributions]
    B --> C[Establish New RVaR Inequality Framework]
    C --> D[Compute Extended Convolution Bounds under Dependence Uncertainty]
    C --> E[Solve Non-Convex Risk Sharing for Averaged Quantiles]
    D --> F[Outcome 1: Sharp Bounds for RVaR & Variability Measures]
    E --> G[Outcome 2: Explicit Optimal Allocations & Existence Conditions]