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An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density

Part of: Markov processes Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  14 July 2016

R. L. Loeffen
R. L. Loeffen*
Affiliation:
Austrian Academy of Sciences
*
∗Postal address: Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria. Email address: ronnie.loeffen@oeaw.ac.at
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Abstract

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We consider a modified version of the classical optimal dividends problem of de Finetti in which the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Lévy process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Lévy measure has a completely monotone density. As a prerequisite for the proof, we show that, under the aforementioned condition on the Lévy measure, the q-scale function of the spectrally negative Lévy process has a derivative which is strictly log-convex.

Keywords

Lévy processstochastic controldividend problemscale functioncomplete monotonicity

MSC classification

Secondary: 60J99: None of the above, but in this section 93E20: Optimal stochastic control 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 85 - 98
Copyright
Copyright © Applied Probability Trust 2009 

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