Journal of Applied Probability

Optimal control with absolutely continuous strategies for spectrally negative Lévy processes

Andreas E. Kyprianou, Ronnie Loeffen, and José-Luis Pérez

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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.

Article information

J. Appl. Probab., Volume 49, Number 1 (2012), 150-166.

First available in Project Euclid: 8 March 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J99: None of the above, but in this section
Secondary: 93E20: Optimal stochastic control 60G51: Processes with independent increments; Lévy processes

Scale function ruin problem de Finetti dividend problem complete monotonicity


Kyprianou, Andreas E.; Loeffen, Ronnie; Pérez, José-Luis. Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. J. Appl. Probab. 49 (2012), no. 1, 150--166. doi:10.1239/jap/1331216839.

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