Model Combination in Risk Sharing under Ambiguity

Abstract

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, characterized by a finite set of probability measures. 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, exploiting 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. We prove that the strategy we obtain is admissible and that the value function satisfies the appropriate verification conditions. Furthermore, we show that the model combination problem is equivalent to the monotone mean-variance problem of Maccheroni et al. (2009) under a composite probability measure that depends on the agent's reference probability measures. Finally, we apply the optimal strategy to an insurance setting in a simulation example and provide numerical illustrations of the results.