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

Optimal reinsurance problem under fixed cost and exponential preferences