Optimal reinsurance problem under fixed cost and exponential preferences

TL;DR

This paper addresses an optimal reinsurance strategy considering fixed costs and exponential utility, solving a complex control and stopping problem to determine when and how to initiate reinsurance for maximum expected utility.

Contribution

It introduces a novel approach combining control and stopping problems under fixed costs, providing explicit strategies and economic insights within the classical risk model.

Findings

Optimal strategy is deterministic and parameter-dependent.

Existence of a maximum fixed cost for contract activation.

Numerical simulations illustrate the theoretical results.

Abstract

We investigate an optimal reinsurance problem for an insurance company facing a constant fixed cost when the reinsurance contract is signed. The insurer needs to optimally choose both the starting time of the reinsurance contract and the retention level in order to maximize the expected utility of terminal wealth. This leads to a mixed optimal control/optimal stopping time problem, which is solved by a two-step procedure: first considering the pure reinsurance stochastic control problem and next discussing a time-inhomogeneous optimal stopping problem with discontinuous reward. Using the classical Cram\'er-Lundberg approximation risk model, we prove that the optimal strategy is deterministic and depends on the model parameters. In particular, we show that there exists a maximum fixed cost that the insurer is willing to pay for the contract activation. Finally, we provide some economical interpretations and numerical simulations.