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De Finetti’s optimal dividends problem with an affine penalty function at ruin. (English) Zbl 1231.91212
Summary: In a Lévy insurance risk model, under the assumption that the tail of the Lévy measure is log-convex, we show that either a horizontal barrier strategy or the take-the-money-and-run strategy maximizes, among all admissible strategies, the dividend payments subject to an affine penalty function at ruin. As a key step for the proof, we prove that, under the aforementioned condition on the jump measure, the scale function of the spectrally negative Lévy process has a log-convex derivative.

MSC:
91B30 Risk theory, insurance (MSC2010)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60G51 Processes with independent increments; Lévy processes
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[1] Albrecher, H.; Thonhauser, S., Optimality results for dividend problems in insurance, RACSAM. revista de la real academia de ciencias exactas, fisicas y naturales. serie A. matematicas, 103, 2, 295-320, (2009) · Zbl 1187.93138
[2] Alvarez, L.H.R.; Rakkolainen, T.A., Optimal payout policy in presence of downside risk, Mathematical methods of operations research, 69, 1, 27-58, (2009) · Zbl 1189.90104
[3] An, M.Y., Logconcavity versus logconvexity: A complete characterization, Journal of economic theory, 80, 2, 350-369, (1998) · Zbl 0911.90071
[4] Asmussen, S., ()
[5] Avanzi, B., Strategies for dividend distribution: A review, North American actuarial journal, 13, 2, 217-251, (2009)
[6] Avram, F.; Kyprianou, A.E.; Pistorius, M.R., Exit problems for spectrally negative Lévy processes and applications to (canadized) Russian options, The annals of applied probability, 14, 1, 215-238, (2004) · Zbl 1042.60023
[7] Avram, F.; Palmowski, Z.; Pistorius, M.R., On the optimal dividend problem for a spectrally negative Lévy process, The annals of applied probability, 17, 1, 156-180, (2007) · Zbl 1136.60032
[8] Azcue, P.; Muler, N., Optimal reinsurance and dividend distribution policies in the cramér – lundberg model, Mathematical finance, 15, 2, 261-308, (2005) · Zbl 1136.91016
[9] Biffis, E., Kyprianou, A.E., A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math. Econom., in press (doi:10.1016/j.insmatheco.2009.04.005) · Zbl 1231.91145
[10] Bingham, N.H., Fluctuation theory in continuous time, Advances in applied probability, 7, 4, 705-766, (1975) · Zbl 0322.60068
[11] Boguslavskaya, E., 2008. Optimization problems in financial mathematics: Explicit solutions for diffusion models. Ph.D. Thesis, University of Amsterdam
[12] Chan, T., Kyprianou, A.E., Savov, M., 2009. Smoothness of scale functions for spectrally negative Lévy processes. arXiv:0903.1467v1 [math.PR] · Zbl 1259.60050
[13] De Bruijn, N.G.; Erdös, P., On a recursion formula and on some Tauberian theorems, Journal of research of the national bureau of standards, 50, (1953) · Zbl 0053.36903
[14] de Finetti, B., 1957. Su un’ impostazione alternativa dell teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries 2, pp. 433-443
[15] Dickson, D.C.M.; Waters, H.R., Some optimal dividends problems, Astin bulletin, 34, 1, 49-74, (2004) · Zbl 1097.91040
[16] Feller, W., An introduction to probability theory and its applications, Vol. II, (1971), John Wiley & Sons Inc. New York · Zbl 0219.60003
[17] Gerber, H.U.; Lin, X.S.; Yang, H., A note on the dividends-penalty identity and the optimal dividend barrier, Astin bulletin, 36, 2, 489-503, (2006) · Zbl 1162.91374
[18] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998) · Zbl 1081.60550
[19] Gerber, H.U.; Shiu, E.S.W.; Smith, N., Maximizing dividends without bankruptcy, Astin bulletin, 36, 1, 5-23, (2006) · Zbl 1162.91375
[20] Hansen, B.G.; Frenk, J.B.G., Some monotonicity properties of the delayed renewal function, Journal of applied probability, 28, 4, 811-821, (1991) · Zbl 0746.60083
[21] Kulenko, N.; Schmidli, H., Optimal dividend strategies in a cramér – lundberg model with capital injections, Insurance: mathematics and economics, 43, 2, 270-278, (2008) · Zbl 1189.91075
[22] Kyprianou, A.E., ()
[23] 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 (in press) arXiv:0801.1951v3 [math.PR] · Zbl 1188.93115
[24] Loeffen, R.L., Optimality of the barrier strategy in de finetti’s dividend problem for spectrally negative Lévy processes, The annals of applied probability, 18, 5, 1669-1680, (2008) · Zbl 1152.60344
[25] Loeffen, R.L., An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, Journal of applied probability, 46, 1, 85-98, (2009) · Zbl 1166.60051
[26] Loeffen, R.L., An optimal dividends problem with transaction costs for spectrally negative Lévy processes, Insurance: mathematics and economics, 45, 1, 41-48, (2009) · Zbl 1231.91211
[27] Müller, A.; Stoyan, D., ()
[28] Pistorius, M.R., On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum, Journal of theoretical probability, 17, 1, 183-220, (2004) · Zbl 1049.60042
[29] Prékopa, A., On logarithmic concave measures and functions, Acta universitatis szegediensis. acta scientiarum mathematicarum, 34, 335-343, (1973) · Zbl 0264.90038
[30] Protter, P.E., (), Version 2.1, Corrected third printing
[31] Roberts, A.W.; Varberg, D.E., ()
[32] Shreve, S.E.; Lehoczky, J.P.; Gaver, D.P., Optimal consumption for general diffusions with absorbing and reflecting barriers, SIAM journal on control and optimization, 22, 1, 55-75, (1984) · Zbl 0535.93071
[33] Thonhauser, S.; Albrecher, H., Dividend maximization under consideration of the time value of ruin, Insurance: mathematics and economics, 41, 1, 163-184, (2007) · Zbl 1119.91047
[34] Yin, C.; Wang, C., Optimality of the barrier strategy in de finetti’s dividend problem for spectrally negative Lévy processes: an alternative approach, Journal of computational and applied mathematics, 233, 2, 482-491, (2009) · Zbl 1176.60034
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