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Alternative approach to the optimality of the threshold strategy for spectrally negative Lévy processes. (English) Zbl 1286.60043

Summary: Consider the optimal dividend problem for an insurance company whose uncontrolled surplus precess evolves as a spectrally negative Lévy process. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. In this paper, we show that a threshold strategy (also called refraction strategy) forms an optimal strategy under the condition that the Lévy measure has a completely monotone density.

MSC:

60G51 Processes with independent increments; Lévy processes
93E20 Optimal stochastic control
91B30 Risk theory, insurance (MSC2010)
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[1] Albrecher, H., Thonhauser, S. Optimal dividend strategies for a risk process under force of interest. Insurance: Math. Econ., 43: 134–149 (2008) · Zbl 1140.91371
[2] Albrecher, H., Thonhauser, S. Optimality results for dividend problems in insurance. Rev. R. Acad. Cien. Serie A. Mat., 103: 295–320 (2009) · Zbl 1187.93138
[3] An, Y. Logconcavity versus logconvexity: a complete characterization. J. Econom. Theory, 80: 350–369 (1998) · Zbl 0911.90071
[4] Asmussen, S., Taksar, M. Controlled diffusion models for optimal dividend pay-out. Insurance: Math. Econom., 20: 1–15 (1997) · Zbl 1065.91529
[5] Avram, F., Palmowski, Z., Pistorius, M.R. On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab., 17: 156–180 (2007) · Zbl 1136.60032
[6] Avanzi, B. Strategies for dividend distribution: a review. North Amer. Actuar. J., 13: 217–249 (2009)
[7] Azcue, P., Muler, N. Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Math. Finance, 15: 261–308 (2005) · Zbl 1136.91016
[8] Azcue, P., Muler, N. Optimal investment policy and dividend payment strategy in an insurance company. Ann. Appl. Probab., 20(4): 1253–1302 (2010) · Zbl 1196.91033
[9] Bagnoli, M., Bergstrom, T. Log-concave probability and its applications. Economic Theory, 26: 445–469 (2005) · Zbl 1077.60012
[10] Chan, T., Kyprianou, A.E., Savov, M. Smoothness of scale functions for spectrally negative Lévy processes. Probability Theory and Related Fields, 150: 691–708 (2011) · Zbl 1259.60050
[11] De Finetti, B. Su un’impostazion alternativa dell teoria collecttiva del rischio. Transactions of the XVth International Congress of Actuaries, 2: 433–443 (1957)
[12] Doney, A. D., Kyprianou, A.E. Overshoots and undershoots of Lévy processes. Ann. Appl. Probab., 16: 91–106 (2006) · Zbl 1101.60029
[13] Fang, Y., Wu, R. Optimal dividend strategy in the compound Poisson model with constant interest. Stochastic Models, 23: 149–166 (2007) · Zbl 1291.91105
[14] Fang, Y., Wu, R. Optimal dividends in the Brownian motion risk model with interest. J. Comput. Appl. Math., 229: 145–151 (2009) · Zbl 1162.91012
[15] Gerber, H. U. Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 69: 185–227 (1969)
[16] Gerber, H. U., Shiu, E.S.W. Optimal dividends: Analysis with Brownian motion. North Amer. Actuar. J., 8: 1–20 (2004) · Zbl 1085.62122
[17] Gerber, H. U., Shiu, E.S.W. On optimal dividend strategies in the compound Poisson model. North Amer. Actuar. J., 10: 76–93 (2006)
[18] Garrido, J., Morales, M. On the expected discounted penalty function for Lévy risk processes. North Amer. Actuar. J., 10: 196–218 (2006)
[19] Hubalek, F., Kyprianou, A.E. Old and new examples of scale functions for spectrally negative Lévy processes. In: Sixth Seminar on Stochastic Analysis, Random Fields and Applications, ed. by R. Dalang, M. Dozzi, F. Russo, Progress in Probability, Birkhäuser, 2010, 119–146 · Zbl 1274.60148
[20] Huzak, M. M., Perman, M., Šikić, H., Vondraček, Z. Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Probab., 14: 1378–1397 (2004) · Zbl 1061.60075
[21] Jeanblanc-Picqué, M., Shiryaev, A.N. Optimization of the flow of dividends. Russian Math. Surveys, 50: 257–277 (1995) · Zbl 0878.90014
[22] Jeannin, M., Pistorius, M. A transform approach to compute prices and Greeks of barrier options driven by a class of Lévy processes. Quantitative Finance, 10(6): 629–644 (2010) · Zbl 1192.91177
[23] Klüppelberg, C., Kyprianou, A.E., Maller, R.A. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab., 14: 1766–1801 (2004) · Zbl 1066.60049
[24] Kyprianou, A. E. Introductory Lectures on Fluctuations of Lévy processes with Applications. Springer-Verlag, Berlin, 2006 · Zbl 1104.60001
[25] Kyprianou, A. E., Palmowski, Z. Distributional study of de Finetti’s dividend problem for a general Lévy insurance risk process. J. Appl. Probab., 44: 428–443 (2007) · Zbl 1137.60047
[26] Kyprianou, A. E., Rivero, V., Song, R. Convexity and smoothness of scale functions with applications to de Finetti’s control problem. Journal of Theoretical Probability, 23: 547–564 (2010) · Zbl 1188.93115
[27] Kyprianou, A. E., Loeffen, R. Refracted Lévy processes. Annales de l’Institut Henri Poincaré-Probabilités et Statistiques, 46: 24–44 (2010) · Zbl 1201.60042
[28] Kyprianou, A. E., Loeffen, R., Pérez, J.L. Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. Arxiv preprint arXiv: 1008.2363, 2010-arxiv.org, to appear in the Journal of Applied Probability
[29] Loeffen, R. On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Probab., 18: 1669–1680 (2008) · Zbl 1152.60344
[30] Loeffen, R., Renaud, J. De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insurance: Mathematics and Economics, 46: 98–108 (2010) · Zbl 1231.91212
[31] Schmidli, H. Stochastic Control in Insurance, Springer-Verlag, 2008 · Zbl 1133.93002
[32] Shreve, S. E., Lehoczky, J.P., Gaver, D.P. Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optim., 22: 55–75 (1984) · Zbl 0535.93071
[33] Wan, N. Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion. Insurance: Mathematics and Economics, 40: 509–523 (2007) · Zbl 1183.91077
[34] Yang, H. L., Zhang, L.Z. Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Probab., 33: 281–291 (2001) · Zbl 0978.60104
[35] Yin, C. C., Wang, C.W. Optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes: An alternative approach. J. Comput. Appl. Math., 233: 482–491 (2009) · Zbl 1176.60034
[36] Yin, C. C., Yuen, K.C. Optimality of the threshold dividend strategy for the compound Poisson model. Statistics and Probability Letters, 81: 1841–1846 (2011) · Zbl 1225.91030
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