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We propose a model in which, in exchange to the payment of a fixedtransaction cost, an insurance company can choose the retention level as wellas the time at which subscribing a perpetual reinsurance contract. The surplusprocess of the insurance company evolves according to the diffusiveapproximation of the Cram\'er-Lundberg model, claims arrive at a fixed constantrate, and the distribution of their sizes is general. Furthermore, we do notspecify any specific functional form of the retention level. The aim of thecompany is to take actions in order to minimize the sum of the expected valueof the total discounted flow of capital injections needed to avoid bankruptcyand of the fixed activation cost of the reinsurance contract. We provide anexplicit solution to this problem, which involves the resolution of a staticnonlinear optimization problem and of an optimal stopping problem for areflected diffusion. We then illustrate the theoretical results in the case ofproportional and excess-of-loss reinsurance, by providing a numerical study ofthe dependency of the optimal solution with respect to the model's parameters.

Irreversible reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost