## Journal of Applied Probability

### An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density

R. L. Loeffen

#### Abstract

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.

#### Article information

Source
J. Appl. Probab. Volume 46, Number 1 (2009), 85-98.

Dates
First available in Project Euclid: 1 April 2009

https://projecteuclid.org/euclid.jap/1238592118

Digital Object Identifier
doi:10.1239/jap/1238592118

Mathematical Reviews number (MathSciNet)
MR2508507

#### Citation

Loeffen, R. L. An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density. J. Appl. Probab. 46 (2009), no. 1, 85--98. doi:10.1239/jap/1238592118. https://projecteuclid.org/euclid.jap/1238592118.

#### References

• Albrecher, H., Renaud, J.-F. and Zhou, X. (2008). A Lévy insurance risk process with tax. J. Appl. Prob. 45, 363--375.
• 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.
• 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.
• Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér--Lundberg model. Math. Finance 15, 261--308.
• Boguslavskaya, E. V. (2008). Optimization problems in financial mathematics: explicit solutions for diffusion models. Doctoral Thesis, University of Amsterdam.
• Chan, T. and Kyprianou, A. E. (2007). Smoothness of scale functions for spectrally negative Lévy processes. Preprint.
• De Finetti, B. (1957). Su un'impostazion alternativa della teoria collecttiva del rischio. Trans. XVth Internat. Congress Actuaries 2, 433--443.
• Dufresne, F. and Gerber, H. U. (1993). The probability of ruin for the inverse Gaussian and related processes. Insurance Math. Econom. 12, 9--22.
• Dufresne, F., Gerber, H. U. and Shiu, E. S. W. (1991). Risk theory with the gamma process. Astin Bull. 21, 177--192.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.
• Furrer, H. (1998). Risk processes perturbed by $\alpha$-stable Lévy motion. Scand. Acturial J. 1998, 59--74.
• Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Mit. Verein. Schweiz. Versicherungsmath. 69, 185--227.
• 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.
• 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.
• Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
• 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.
• Kyprianou, A. E. and Rivero, V. (2008). Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Prob. 13, 1672--1701.
• 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.
• 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.
• 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.
• Protter, P. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.
• Radner, R. and Shepp, L. (1996). Risk vs. profit potential: a model for corporate strategy. J. Econom. Dynamics Control 20, 1373--1393.
• 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.
• 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.
• Surya, B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135\nobreakdash--149.
• Thonhauser, S. and Albrecher, H. (2007). Dividend maximization under consideration of the time value of ruin. Insurance Math. Econom. 41, 163--184.
• Van Tiel, J. (1984). Convex Analysis. John Wiley, New York.