Optimal time for the excess of loss reinsurance with fixed costs

We consider the optimal excess of loss reinsurance for an insurance company facing a constant fixed cost when reinsurance contract is signed. To maximize the survival probability, the company needs to determine the starting time of reinsurance and the claims covered by insurer. It concludes a combination of optimal stopping and regular stochastic control problems, and we solve it by employing the dynamic programming principle. The results indicate that the maximum amount of loss plays an important role for the optimal timing of excess of loss reinsurance. When the claims have a long tail, the insurer will launch an excess of loss reinsurance no matter how expensive of reinsurance is. When the claims have a finite maximum loss, the insurer will not take the excess of loss reinsurance if the reinsurance is too expensive. In addition, the fixed cost only affects the timing of excess of loss reinsurance, but the relative price between reinsurer and insurer affects both the timing and the retained amount of risks. To explain the results, we also give some numerical examples.