Distributionally Robust Insurance under Bregman-Wasserstein Divergence

2026-04-30

AI Summary

This study addresses the decision-making risk arising from uncertainty in loss distributions within insurance contract design. From the policyholder’s perspective, it constructs an asymmetric distributional ambiguity set using a Bregman–Wasserstein (BW) ball to capture asymmetric penalties for deviations from a reference distribution. Under α-maxmin preferences with a Value-at-Risk (VaR) constraint and worst-case convex distortion risk measures, the authors combine Lagrangian methods with a refined duality argument to derive, for the first time, closed-form solutions for both the optimal indemnity function and the worst-case distribution within a robust insurance demand framework. Numerical experiments demonstrate that the asymmetry inherent in the BW divergence critically shapes the structure of optimal insurance contracts.

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).

Problem

• distributional uncertainty
• optimal insurance contracting
• Bregman-Wasserstein divergence
• Value-at-Risk
• robust optimization

Innovation

• Bregman-Wasserstein divergence
• distributionally robust optimization
• asymmetric ambiguity
• optimal insurance indemnity
• closed-form solution