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Optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes: an alternative approach. (English) Zbl 1176.60034

Summary: The optimal dividend problem proposed in [B. de Finetti, Su un’impostazion alternativa dell teoria collecttiva del rischio, Transactions of the XVth international Congress of Actuaries 2, 433–443 (1957)] is to find the dividend-payment strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the company is ruined. F. Avram, Z. Palmowski and M. R. Pistorius [Ann. Appl. Probab. 17, No. 1, 156–180 (2007; Zbl 1136.60032)] studied the case when the risk process is modelled by a general spectrally negative Lévy process and R. L. Loeffen [Ann. Appl. Probab. 18, No. 5, 1669–1680 (2008; [Zbl 1152.60344)] gave sufficient conditions under which the optimal strategy is of the barrier type. Recently A. E. Kyprianou, V. Rivero and R. Song [J. Theor. Probab. 23, No. 2, 547–564 (2010; Zbl 1188.93115; arxiv:0801.1951] strengthened the result of Loeffen [loc. cit.] which established a larger class of Lévy processes for which the barrier strategy is optimal among all admissible ones. In this paper we use an analytical argument to re-investigate the optimality of barrier dividend strategies considered in the three recent papers.

MSC:

60G51 Processes with independent increments; Lévy processes
93E20 Optimal stochastic control
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