Jointly Exchangeable Collective Risk Models: Interaction, Structure, and Limit Theorems

We introduce a framework for systemic risk modeling in insurance portfolios using jointly exchangeable arrays, extending classical collective risk models to account for interactions. Joint exchangeability is a more general probabilistic symmetric than de Finetti’s exchangeability, characterized by the Aldous-Hoover-Kallenberg representation. We establish central limit theorems that asymptotically capture total portfolio losses, providing a theoretical foundation for approximations in large portfolios and over long time horizons. These approximations are validated through simulation-based numerical experiments. Additionally, we analyze the impact of dependence on portfolio loss distributions, with a particular focus on tail behavior.

💡 Research Summary

The paper proposes a novel probabilistic framework for modeling systemic risk in insurance portfolios by employing jointly exchangeable arrays, thereby extending classical collective risk models to incorporate network‑type interactions among loss events. The authors begin by highlighting the limitations of traditional collective risk models, which typically separate idiosyncratic and common risk components but ignore contagion mechanisms that arise in interconnected systems such as cyber‑insurance, power‑grid coverage, or digital infrastructure policies. To capture these mechanisms, they introduce an infinite N×N array of random variables and impose joint exchangeability: the joint distribution remains invariant under any finite permutation of the index set N. This symmetry is weaker than de Finetti’s full exchangeability but still powerful enough to admit a concrete structural representation via the Aldous‑Hoover‑Kallenberg theorem.

According to this representation, each array entry X_{ij} can be written as X_{ij}=h(ξ,ξ_i,ξ_j,ξ_{ij}), where ξ, ξ_i, ξ_j, and ξ_{ij} are independent Uniform