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 |
Keywords:
insurance risk theory; optimal dividends; deficit at ruin; Gerber-Shiu functions; Lévy processes; stochastic control; log-convexityReferences:
| [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., (Applied Probability and Queues. Applied Probability and Queues, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (1987), John Wiley & Sons Ltd.: John Wiley & Sons Ltd. Chichester) · Zbl 0624.60098 |
| [5] | Avanzi, B., Strategies for dividend distribution: A review, North American Actuarial Journal, 13, 2, 217-251 (2009) · Zbl 1483.91177 |
| [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 |
| [10] | Bingham, N. H., Fluctuation theory in continuous time, Advances in Applied Probability, 7, 4, 705-766 (1975) · Zbl 0322.60068 |
| [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 |
| [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.: 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., (Introductory Lectures on Fluctuations of Lévy Processes with Applications. Introductory Lectures on Fluctuations of Lévy Processes with Applications, Universitext (2006), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1104.60001 |
| [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., (Comparison Methods for Stochastic Models and Risks. Comparison Methods for Stochastic Models and Risks, Wiley Series in Probability and Statistics (2002), John Wiley & Sons Ltd.: John Wiley & Sons Ltd. Chichester) · Zbl 0999.60002 |
| [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., (Stochastic Integration and Differential Equations. Stochastic Integration and Differential Equations, Stochastic Modelling and Applied Probability, vol. 21 (2005), Springer-Verlag: Springer-Verlag Berlin), Version 2.1, Corrected third printing · Zbl 1041.60005 |
| [31] | Roberts, A. W.; Varberg, D. E., (Convex Functions. Convex Functions, Pure and Applied Mathematics, vol. 57 (1973), Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers): Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers) New York, London) · Zbl 0271.26009 |
| [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 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
