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On the optimal dividend problem for a spectrally negative Lévy process. (English) Zbl 1136.60032
In classical collective risk theory the surplus process of an insurance company is described by the Cramer-Lundberg model, has positive first moment and has therefore the unrealistic property that it converges to infinity with probability 1. In answer to this objection, De Finetti (1957) introduced the divident barrier model, in which all surpluses a given level are transferred to a beneficiary. In mathematical finance and actuarial literature there is a good deal of work on divident barrier models. A drawback of the divident barrier model is that under this model the risk process will down-cross the level zero with probability 1.
In this paper, the authors approach the divident problem from the point of view of a general spectrally negative Lévy process. Drawing on the fluctuation theory of spectrally negative Lévy processes they give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function. They conclude the paper with some explicit examples in the classical and ‘bail-out’ setting.

MSC:
60G51 Processes with independent increments; Lévy processes
60J99 Markov processes
93E20 Optimal stochastic control
91B30 Risk theory, insurance (MSC2010)
91G50 Corporate finance (dividends, real options, etc.)
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