March 2012 Optimal control with absolutely continuous strategies for spectrally negative Lévy processes
Andreas E. Kyprianou, Ronnie Loeffen, José-Luis Pérez
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J. Appl. Probab. 49(1): 150-166 (March 2012). DOI: 10.1239/jap/1331216839


In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Lévy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picqué and Shiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.


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Andreas E. Kyprianou. Ronnie Loeffen. José-Luis Pérez. "Optimal control with absolutely continuous strategies for spectrally negative Lévy processes." J. Appl. Probab. 49 (1) 150 - 166, March 2012.


Published: March 2012
First available in Project Euclid: 8 March 2012

zbMATH: 1253.93001
MathSciNet: MR2952887
Digital Object Identifier: 10.1239/jap/1331216839

Primary: 60J99
Secondary: 60G51 , 93E20

Keywords: complete monotonicity , de Finetti dividend problem , ruin problem , scale function

Rights: Copyright © 2012 Applied Probability Trust


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Vol.49 • No. 1 • March 2012
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