## 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)
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### 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
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