Irreversible reinsurance: A singular control approach

Abstract

Reinsurance demand is often investigated using the classical stochastic control framework with the reinsurance level as a regular control variable. However, reinsurance contracts are not traded, and dynamic adjustment of the reinsurance strategy is impractically infeasible. Moreover, reinsurance contracts are long-term commitments and are considered irreversible as they are expensive to close before maturity. We propose a novel irreversible reinsurance framework, where the insurer enters into a sequence of reinsurance contracts at the most appropriate times and the contracts are never reversed afterwards. In each contract, the insurer specifies the amount of reinsurance. We model the insurer's risk exposure using a mean-reverting process as inspired by stochastic mortality modeling and the property loss data from US catastrophes, and then formulate the reinsurance decision as a two-dimensional degenerate singular control problem. Our mathematical problem is closely related to the well-known finite-fuel problem due to the boundedness of the insurer's risk retention. The optimal reinsurance purchase rule is triggered by a free boundary that characterizes the optimal relationship between risk exposure and reinsurance demand. We conduct a numerical analysis to investigate the features of the singular reinsurance strategy and its dependency on the model parameters.