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Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

Part of: Stochastic processes Markov processes Special processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

A. E. Kyprianou  and
Z. Palmowski
A. E. Kyprianou*
Affiliation:
The University of Bath
Z. Palmowski*
Affiliation:
Utrecht University and Wrocław University
*
∗ Postal address: Department of Mathematical Sciences, The University of Bath, Claverton Down, Bath BA2 7AY, UK. Email address: a.kyprianou@bath.ac.uk
∗∗ Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: zpalma@math.uni.wroc.pl
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Abstract

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We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

Keywords

Reflected Lévy processpassage problemintegrated exponential Lévy processinsurance risk processruin

MSC classification

Primary: 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes 91B30: Risk theory, insurance
Secondary: 60G70: Extreme value theory; extremal processes 60J55: Local time and additive functionals
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 2 , June 2007 , pp. 428 - 443
Copyright
Copyright © Applied Probability Trust 2007 

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