Model Combination in Risk Sharing under Ambiguity

We consider the problem of an agent who faces losses in continuous time over a finite time horizon and may choose to share some of these losses with a counterparty. The agent is uncertain about the true loss distribution and has multiple models for the losses. Their goal is to optimize a mean-variance type criterion with model combination under ambiguity through risk sharing. We construct such a criterion using the chi-squared divergence, adapting the monotone mean-variance preferences of Maccheroni et al. (2009) to the model combination setting and exploit a dual representation to expand the state space, yielding a time consistent problem. Assuming a Cramér-Lundberg loss model, we fully characterize the optimal risk sharing contract and the agent’s wealth process under the optimal strategy. Furthermore, we prove that the strategy we obtain is admissible and that the value function satisfies the appropriate verification conditions. Finally, we apply the optimal strategy to an insurance setting using data from a Spanish automobile insurance portfolio, where we obtain differing models using cross-validation and provide numerical illustrations of the results.

💡 Research Summary

This paper studies an optimal risk‑sharing problem faced by an insurer who experiences stochastic losses in continuous time over a finite horizon and may transfer part of those losses to a counterparty (e.g., a reinsurer). The insurer does not know which of several candidate loss‑distribution models is the true one; instead, she has access to a finite set of reference probability measures (P_1,\dots ,P_n) together with the counterparty’s own model (P_C). To capture both model ambiguity and the desire to combine information from all models, the authors introduce a novel preference functional that extends the monotone mean‑variance (MMV) preferences of Maccheroni et al. (2009) to a multi‑model setting.

The key ingredient is a chi‑squared divergence penalty. For a candidate decision measure (Q) the criterion is

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