×

zbMATH — the first resource for mathematics

An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density. (English) Zbl 1166.60051
The author considers a modified version of the classical optimal dividends problem of de Finetti by adding to the objective function an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Lévy process. It is shown that, in general, a barrier strategy is an optimal strategy under the condition that the Lévy measure has a completely monotone density. As a prerequisite, it is shown that under the latter condition, the \(q\)-scale function of a spectrally negative Lévy process has a derivative which is strictly non-convex.

MSC:
60J99 Markov processes
60G51 Processes with independent increments; Lévy processes
93E20 Optimal stochastic control
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albrecher, H., Renaud, J.-F. and Zhou, X. (2008). A Lévy insurance risk process with tax. J. Appl. Prob. 45, 363–375. · Zbl 1144.60032 · doi:10.1239/jap/1214950353
[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. · Zbl 1042.60023 · doi:10.1214/aoap/1075828052
[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. · Zbl 1136.60032 · doi:10.1214/105051606000000709
[4] Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15, 261–308. · Zbl 1136.91016 · doi:10.1111/j.0960-1627.2005.00220.x
[5] Boguslavskaya, E. V. (2008). Optimization problems in financial mathematics: explicit solutions for diffusion models. Doctoral Thesis, University of Amsterdam.
[6] Chan, T. and Kyprianou, A. E. (2007). Smoothness of scale functions for spectrally negative Lévy processes. · Zbl 1259.60050
[7] De Finetti, B. (1957). Su un’impostazion alternativa della teoria collecttiva del rischio. Trans. XVth Internat. Congress Actuaries 2, 433–443.
[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. · Zbl 0768.62097 · doi:10.1016/0167-6687(93)90995-2
[9] Dufresne, F., Gerber, H. U. and Shiu, E. S. W. (1991). Risk theory with the gamma process. Astin Bull. 21, 177–192.
[10] Feller, W. (1971). An Introduction to Probability Theory and Its Applications , Vol. 2, 2nd edn. John Wiley, New York. · Zbl 0219.60003
[11] Furrer, H. (1998). Risk processes perturbed by \(\alpha\)-stable Lévy motion. Scand. Acturial J. 1998, 59–74. · Zbl 1026.60516 · doi:10.1080/03461238.1998.10413992
[12] Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Mit. Verein. Schweiz. Versicherungsmath. 69, 185–227. · Zbl 0193.20501
[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. · Zbl 1274.60148
[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. · Zbl 1061.60075 · doi:10.1214/105051604000000332
[15] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[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. · Zbl 1137.60047 · doi:10.1239/jap/1183667412
[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. · Zbl 1193.60064 · emis:journals/EJP-ECP/_ejpecp/viewarticle12e5.html · eudml:230675
[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. · Zbl 1143.91020 · doi:10.1007/s00780-006-0028-y
[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.
[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. · Zbl 1152.60344 · doi:10.1214/07-AAP504
[21] Protter, P. (2005). Stochastic Integration and Differential Equations , 2nd edn. Springer, Berlin.
[22] Radner, R. and Shepp, L. (1996). Risk vs. profit potential: a model for corporate strategy. J. Econom. Dynamics Control 20, 1373–1393. · Zbl 0875.90045 · doi:10.1016/0165-1889(95)00904-3
[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. · Zbl 1132.60041 · doi:10.1239/jap/1183667411
[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. · Zbl 0535.93071 · doi:10.1137/0322005
[25] Surya, B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135\nobreakdash–149. · Zbl 1140.60027 · doi:10.1239/jap/1208358957
[26] Thonhauser, S. and Albrecher, H. (2007). Dividend maximization under consideration of the time value of ruin. Insurance Math. Econom. 41, 163–184. · Zbl 1119.91047 · doi:10.1016/j.insmatheco.2006.10.013
[27] Van Tiel, J. (1984). Convex Analysis . John Wiley, New York. · Zbl 0565.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.