×

zbMATH — the first resource for mathematics

An optimal dividends problem with transaction costs for spectrally negative Lévy processes. (English) Zbl 1231.91211
Summary: We consider an optimal dividends problem with transaction costs where the reserves are modeled by a spectrally negative Lévy process. We make the connection with the classical de Finetti problem and show in particular that when the Lévy measure has a log-convex density, then an optimal strategy is given by paying out a dividend in such a way that the reserves are reduced to a certain level \(c_{1}\) whenever they are above another level \(c_{2}\). Further we describe a method to numerically find the optimal values of \(c_{1}\) and \(c_{2}\).

MSC:
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albrecher, H.; Renaud, J.-F.; Zhou, X., A Lévy insurance risk process with tax, Journal of applied probability, 45, 2, 363-375, (2008) · Zbl 1144.60032
[2] Alvarez, L.H.R.; Rakkolainen, T.A., Optimal payout policy in presence of downside risk, Mathematical methods of operations research, 69, 27-58, (2009) · Zbl 1189.90104
[3] Avram, F.; Palmowski, Z.; Pistorius, M.R., On the optimal dividend problem for a spectrally negative Lévy process, Annals of applied probability, 17, 156-180, (2007) · Zbl 1136.60032
[4] Azcue, P.; Muler, N., Optimal reinsurance and dividend distribution policies in the cramér-lundberg model, Mathematical finance, 15, 261-308, (2005) · Zbl 1136.91016
[5] Bather, J.A., A continuous time inventory model, Journal of applied probability, 3, 538-549, (1966) · Zbl 0199.23202
[6] Bertoin, J., On the first exit time of a completely asymmetric Lévy process from a finite interval, Bulletin of the London mathematical society, 28, 514-520, (1995) · Zbl 0863.60068
[7] de Finetti, B., Su un’impostazione alternativa dell teoria collecttiva del rischio, Transactions of the xvth international congress of actuaries, 2, 433-443, (1957)
[8] Dufresne, F.; Gerber, H.U., The probability of ruin for the inverse Gaussian and related processes, Insurance: mathematics and economics, 12, 9-22, (1993) · Zbl 0768.62097
[9] Dufresne, F.; Gerber, H.U.; Shiu, E.S.W., Risk theory with the gamma process, Astin bulletin, 21, 177-192, (1991)
[10] Furrer, H., Risk processes perturbed by \(\alpha\)-stable Lévy motion, Scandinavian actuarial journal, 59-74, (1998) · Zbl 1026.60516
[11] Gerber, H.U., Entscheidungskriterien für den zusammengesetzten Poisson-prozess, Mitteilungen der vereinigung schweizerischer versicherungsmathematiker, 69, 185-227, (1969) · Zbl 0193.20501
[12] Hubalek, F., Kyprianou, A.E., Old and new examples of scale functions for spectrally negative Lévy processes, Preprint, 2007 · Zbl 1274.60148
[13] Huzak, M.; Perman, M.; Šikić, H.; Vondraček, Z., Ruin probabilities and decompositions for general perturbed risk processes, The annals of applied probability, 14, 1378-1397, (2004) · Zbl 1061.60075
[14] Jeanblanc-Picqué, M.; Shiryaev, A.N., Optimization of the flow of dividends, Russian math. surveys, 50, 257-277, (1995) · Zbl 0878.90014
[15] Kyprianou, A.E., Introductory lectures on fluctuations of Lévy processes with applications, (2006), Springer · Zbl 1104.60001
[16] Kyprianou, A.E.; Palmowski, Z., Distributional study of de finetti’s dividend problem for a general Lévy insurance risk process, Journal of applied probability, 44, 349-365, (2007) · Zbl 1137.60047
[17] Kyprianou, A.E.; Rivero, V., Special, conjugate and complete scale functions for spectrally negative Lévy processes, Electronic journal of probability, 13, 57, 1672-1701, (2008) · Zbl 1193.60064
[18] Kyprianou, A.E., Rivero, V., Song, R., 2008. Convexity and smoothness of scale functions and de Finetti’s control problem. Journal of Theoretical Probability (in press) arXiv:0801.1951v2 [math.PR] · Zbl 1188.93115
[19] Kyprianou, A.E.; Surya, B.A., Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels, Finance and stochastics, 11, 1, 131-152, (2007) · Zbl 1143.91020
[20] Loeffen, R.L., On optimality of the barrier strategy in de finetti’s dividend problem for spectrally negative Lévy processes, Annals of applied probability, 18, 5, 1669-1680, (2008) · Zbl 1152.60344
[21] 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
[22] Paulsen, J., Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs, Advances in applied probability, 39, 669-689, (2007) · Zbl 1126.93058
[23] Protter, P., Stochastic integration and differential equations, (2005), Springer, version 2.1
[24] Renaud, J.F.; Zhou, X., Distribution of the present value of dividend payments in a Lévy risk model, Journal of applied probability, 44, 420-427, (2007) · Zbl 1132.60041
[25] Sulem, A., A solvable one-dimensional model of a diffusion inventory system, Mathematics of operations research, 11, 1, 125-133, (1986) · Zbl 0601.93069
[26] Surya, B.A., Evaluating scale functions of spectrally negative Lévy processes, Journal of applied probability, 45, 1, 135-149, (2008) · Zbl 1140.60027
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.