zbMATH — the first resource for mathematics

On the optimal dividend problem for a spectrally negative Lévy process. (English) Zbl 1136.60032
In classical collective risk theory the surplus process of an insurance company is described by the Cramer-Lundberg model, has positive first moment and has therefore the unrealistic property that it converges to infinity with probability 1. In answer to this objection, De Finetti (1957) introduced the divident barrier model, in which all surpluses a given level are transferred to a beneficiary. In mathematical finance and actuarial literature there is a good deal of work on divident barrier models. A drawback of the divident barrier model is that under this model the risk process will down-cross the level zero with probability 1.
In this paper, the authors approach the divident problem from the point of view of a general spectrally negative Lévy process. Drawing on the fluctuation theory of spectrally negative Lévy processes they give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function. They conclude the paper with some explicit examples in the classical and ‘bail-out’ setting.

60G51 Processes with independent increments; Lévy processes
60J99 Markov processes
93E20 Optimal stochastic control
91B30 Risk theory, insurance (MSC2010)
91G50 Corporate finance (dividends, real options, etc.)
Full Text: DOI arXiv
[1] Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15 261–308. · Zbl 1136.91016 · doi:10.1111/j.0960-1627.2005.00220.x
[2] Asmussen, S. (2003). Applied Probability and Queues , 2nd ed. Springer, New York. · Zbl 1029.60001 · doi:10.1007/b97236
[3] Asmussen, S., Højgaard, B. and Taksar, M. (2000). Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation. Finance Stoch. 4 299–324. · Zbl 0958.91026 · doi:10.1007/s007800050075
[4] Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14 215–238. · Zbl 1042.60023 · doi:10.1214/aoap/1075828052
[5] Bertoin, J. (1995). On the first exit time of a completely asymmetric Lévy process from a finite interval. Bull. London Math. Soc. 28 514–520. · Zbl 0863.60068 · doi:10.1112/blms/28.5.514
[6] Bertoin, J. (1996). Lévy Processes . Cambridge Univ. Press. · Zbl 0861.60003
[7] Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7 156–169. · Zbl 0880.60077 · doi:10.1214/aoap/1034625257
[8] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. in Appl. Probab. 7 705–766. JSTOR: · Zbl 0322.60068 · doi:10.2307/1426397 · links.jstor.org
[9] Çinlar, E., Jacod, J., Protter, P. and Sharpe, M. J. (1980). Semimartingales and Markov process. Z. Wahrsch. Verw. Gebiete 54 161–219. · Zbl 0443.60074 · doi:10.1007/BF00531446
[10] De Finetti, B. (1957). Su un’impostazione alternativa dell teoria colletiva del rischio. Trans. XV Intern. Congress Act. 2 433–443.
[11] Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory . Hübner Foundation for Insurance Education, Philadelphia. · Zbl 0431.62066
[12] Gerber, H. U. and Shiu, E. S. W. (2004). Optimal dividends: Analysis with Brownian motion. North American Actuarial J. 8 1–20. · Zbl 1085.62122
[13] Harrison, J. M. and Taylor, A. J. (1978). Optimal control of a Brownian storage system. Stochastic Process. Appl. 6 179–194. · Zbl 0372.60116 · doi:10.1016/0304-4149(78)90059-5
[14] Irbäck, J. (2003). Asymptotic theory for a risk process with a high dividend barrier. Scand. Actuarial J. 2 97–118. · Zbl 1092.91043 · doi:10.1080/03461230110106345
[15] Jeanblanc, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50 257–277. · Zbl 0878.90014 · doi:10.1070/RM1995v050n02ABEH002054
[16] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[17] Kyprianou, A. E. and Palmowski, Z. (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. Séminaire de Probabilités XXXVIII . Lecture Notes in Math. 1857 16–29. Springer, Berlin. · Zbl 1063.60071
[18] Lambert, A. (2000). Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré Probab. Statist. 36 251–274. · Zbl 0970.60055 · doi:10.1016/S0246-0203(00)00126-6 · numdam:AIHPB_2000__36_2_251_0 · eudml:77658
[19] Løkka, A. and Zervos, M. (2005). Optimal dividend and issuance of equity policies in the presence of proportional costs. · Zbl 1141.91528
[20] Pistorius, M. R. (2003). On doubly reflected completely asymmetric Lévy processes. Stochastic Process. Appl. 107 131–143. · Zbl 1075.60573 · doi:10.1016/S0304-4149(03)00049-8
[21] Pistorius, M. R. (2004). On exit and ergodicity of the completely asymmetric Lévy process reflected at its infimum. J. Theoret. Probab. 17 183–220. · Zbl 1049.60042 · doi:10.1023/B:JOTP.0000020481.14371.37
[22] Pistorius, M. R. (2006). An excursion theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. Sém. Probab. · Zbl 1126.60039 · doi:10.1007/978-3-540-71189-6_15
[23] Protter, P. (1995). Stochastic Integration and Differential Equations . Springer, Berlin. · Zbl 1041.60005
[24] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions . Cambridge Univ. Press. · Zbl 0973.60001
[25] Zhou, X. (2005). On a classical risk model with a constant dividend barrier. North American Actuarial J. 9 1–14. · Zbl 1215.60051 · www.soa.org
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.