Distributionally Robust Insurance under Bregman-Wasserstein Divergence

Paper: arXiv 2604.27837

Authors: Wenjun Jiang, Qingqing Zhang, Yiying Zhang

Abstract

This paper investigates two optimal insurance contracting problems under distributional uncertainty from the perspective of a potential policyholder, utilizing a Bregman-Wasserstein (BW) ball to characterize the ambiguity set of loss distributions. Unlike the $p$-Wasserstein distance, BW divergence enables asymmetric penalization of deviations from the benchmark distribution. The first problem examines an insurance demand model where the policyholder adopts an $α$-maxmin preference with Value-at-Risk (VaR). We derive the optimal indemnity function in closed form and study, both analytically and numerically, how the asymmetry inherent in BW divergence influences the optimal indemnity structure. The second problem employs a robust optimization framework, where the policyholder aims to secure robust insurance indemnity by minimizing the worst-case convex distortion risk measure while adhering to a guaranteed VaR constraint. In this context, we provide explicit characterizations of both the optimal indemnity and the worst-case distribution in closed form through a combined approach using the Lagrangian method and modification arguments. To illustrate the practical implications of our theoretical findings, we include a concrete example based on Tail Value-at-Risk (TVaR).

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated mathematical framework for distributionally robust insurance contracting using Bregman-Wasserstein divergence, deriving closed-form optimal indemnity functions. However, it is entirely theoretical and lacks empirical validation, real-world market datasets, or backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Optimal Insurance Contracting under Distributional Uncertainty] --> B[Input: Loss Distribution Benchmark & Bregman-Wasserstein Ball]
    B --> C[Framework 1: Alpha-Maxmin Preference with VaR Constraint]
    B --> D[Framework 2: Robust Optimization via Convex Distortion Risk Measure & VaR]
    C --> E[Derive Closed-Form Optimal Indemnity & Analyze Asymmetry Effects]
    D --> F[Apply Lagrangian Method & Modification Arguments]
    E --> G[Outcomes: Explicit Indemnity Functions & TVaR Practical Application]
    F --> G

Cited References (228)

1. Optimal dynamic reinsurance policies under a generalized Denneberg’s absolute deviation principle (2020)
2. Optimal dynamic reinsurance under heterogeneous beliefs and CARA utility (2022)
3. Optimal reinsurance arrangement under heterogeneous beliefs: A unified method with piecewise modification (2024)
4. Convergence of probability measures (2013)
5. Optimal insurance design under distortion risk measures with variance constraint (2026)
6. Quantifying distributional model risk via optimal transport (2019)
7. Distributionally robust stochastic optimization with Wasserstein distance (2023)
8. Inf-convolution, optimal allocations, and model uncertainty for tail risk measures (2022)

4 of 228 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.