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Optimal Control with Absolutely Continuous Strategies for Spectrally Negative Lévy Processes

Part of: Stochastic systems and control Markov processes Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Andreas E. Kyprianou ,
Ronnie Loeffen  and
José-Luis Pérez
Andreas E. Kyprianou*
Affiliation:
University of Bath
Ronnie Loeffen*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
José-Luis Pérez*
Affiliation:
University of Bath
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
∗∗∗ Current address: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. R. L. gratefully acknowledges support from the AXA Research Fund.
∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
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Abstract

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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.

Keywords

Scale functionruin problemde Finetti dividend problemcomplete monotonicity

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 93E20: Optimal stochastic control 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 150 - 166
Copyright
© Applied Probability Trust 

Footnotes

J.-L. P. acknowledges financial support from CONACyT, grant number 000000000129326.

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