Worst-case reinsurance strategy with likelihood ratio uncertainty

In this paper, we explore a non-cooperative optimal reinsurance problem incorporating likelihood ratio uncertainty, aiming to minimize the worst-case risk of the total retained loss for the insurer. We establish a general relation between the optimal reinsurance strategy under the reference probability measure and the strategy in the worst-case scenario. This relation can further be generalized to insurance design problems quantified by tail risk measures. We also characterize distortion risk measures for which the insurer’s optimal strategy remains the same in the worst-case scenario. As an application, we determine the optimal policies for the worst-case scenario using an expectile risk measure. Additionally, we propose and explore a cooperative problem, which can be viewed as a general risk sharing problem between two agents in a comonotonic market. We determine the risk measure value and the optimal reinsurance strategy in the worst-case scenario for the insurer and compare the results from the non-cooperative and cooperative models.