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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: Stochastic processes Stochastic systems and control Markov 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 

References

[1] Albrecher, H., Renaud, J.-F. and Zhou, X. (2008). A Lévy insurance risk process with tax. J. Appl. Prob. 45, 363–375.Google Scholar
[2] Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–235.Google Scholar
[3] Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156–180.Google Scholar
[4] Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15, 261–308.Google Scholar
[5] Boguslavskaya, E. V. (2008). Optimization problems in financial mathematics: explicit solutions for diffusion models. , University of Amsterdam.Google Scholar
[6] Chan, T. and Kyprianou, A. E. (2007). Smoothness of scale functions for spectrally negative Lévy processes. Preprint.Google Scholar
[7] De Finetti, B. (1957). Su un'impostazion alternativa della teoria collecttiva del rischio. Trans. XVth Internat. Congress Actuaries 2, 433–443.Google Scholar
[8] Dufresne, F. and Gerber, H. U. (1993). The probability of ruin for the inverse Gaussian and related processes. Insurance Math. Econom. 12, 9–22.Google Scholar
[9] Dufresne, F., Gerber, H. U. and Shiu, E. S. W. (1991). Risk theory with the gamma process. Astin Bull. 21, 177–192.Google Scholar
[10] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
[11] Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Acturial J. 1998, 59–74.Google Scholar
[12] Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Mit. Verein. Schweiz. Versicherungsmath. 69, 185–227.Google Scholar
[13] Hubalek, F. and Kyprianou, A. E. (2007). Old and new examples of scale functions for spectrally negative Lévy processes. Preprint. Available at http://arxiv.org/abs/0801.0393v1.Google Scholar
[14] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397.Google Scholar
[15] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[16] Kyprianou, A. E. and Palmowski, Z. (2007). Distributional study of de Finetti's dividend problem for a general Lévy insurance risk process. J. Appl. Prob. 44, 428–443.Google Scholar
[17] Kyprianou, A. E. and Rivero, V. (2008). Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Prob. 13, 1672–1701.Google Scholar
[18] Kyprianou, A. E. and Surya, B. A. (2007). Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. Finance Stoch. 11, 131–152.Google Scholar
[19] Kyprianou, A. E., Rivero, V. and Song, R. (2008). Convexitity and smoothness of scale functions and de Finetti's control problem. Preprint. Available at http://arXiv.org/abs/0801.1951v2.Google Scholar
[20] Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 1669–1680.Google Scholar
[21] Protter, P. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
[22] Radner, R. and Shepp, L. (1996). Risk vs. profit potential: a model for corporate strategy. J. Econom. Dynamics Control 20, 1373–1393.Google Scholar
[23] Renaud, J. F. and Zhou, X. (2007). Distribution of the present value of dividend payments in a Lévy risk model. J. Appl. Prob. 44, 420–427.Google Scholar
[24] Shreve, S. E., Lehoczky, J. P. and Gaver, D. P. (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optimization 22, 55–75.Google Scholar
[25] Surya, B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135–149.Google Scholar
[26] Thonhauser, S. and Albrecher, H. (2007). Dividend maximization under consideration of the time value of ruin. Insurance Math. Econom. 41, 163–184.Google Scholar
[27] Van Tiel, J. (1984). Convex Analysis. John Wiley, New York.Google Scholar