Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences

We investigate an optimal reinsurance problem for an insurance company taking into
account subscription costs: that is, a constant fixed cost is paid when the reinsurance contract is
signed. Differently from the classical reinsurance problem, where the insurer has to choose an optimal
retention level according to some given criterion, in this paper, the insurer needs to optimally choose
both the starting time of the reinsurance contract and the retention level to apply. The criterion is
the maximization of the insurer’s 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 Cramér–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.