×

zbMATH — the first resource for mathematics

Distributional study of De Finetti’s dividend problem for a general Lévy insurance risk process. (English) Zbl 1137.60047
The authors provide a study of the solution to the classical control problem, which concerns the optimal payment of dividends from an insurance risk process prior to ruin. They build on recent work in the actuarial literature concerning calculations of the \(n\)-th moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations presented in the article are valid for a general spectrally negative Lévy process as opposed to the classical Cramer-Lundberg process with exponentially distributed jumps.

MSC:
60K10 Applications of renewal theory (reliability, demand theory, etc.)
91B30 Risk theory, insurance (MSC2010)
60K05 Renewal theory
60K15 Markov renewal processes, semi-Markov processes
60G51 Processes with independent increments; Lévy processes
60G70 Extreme value theory; extremal stochastic processes
60J55 Local time and additive functionals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albrecher, H., Claramunt, M. M. and Mármol, M. (2005). On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang\((n)\) interclaim times. Insurance Math. Econom. 37, 324–334. · Zbl 1117.91377 · doi:10.1016/j.insmatheco.2005.05.004
[2] Asmussen, S., Avram, F. and Pistorius, M. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111. · Zbl 1075.60037 · doi:10.1016/j.spa.2003.07.005
[3] 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. Prob. 14, 215–238. · Zbl 1042.60023 · doi:10.1214/aoap/1075828052
[4] Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156–180. · Zbl 1136.60032 · doi:10.1214/105051606000000709
[5] 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
[6] Bertoin, J. (1996). Lévy Processes . Cambridge University Press. · Zbl 0861.60003
[7] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Prob. Surveys 2, 191–212. · Zbl 1189.60096 · doi:10.1214/154957805100000122 · eudml:228237
[8] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation . Cambridge University Press. · Zbl 0617.26001
[9] De Finetti, B. (1957). Su un’impostazion alternativa dell teoria collecttiva del rischio. Trans. XVth Internat. Congr. Actuaries 2, 433–443.
[10] Dickson, D. C. M. and Dos Reis, A. E. (1996). On the distribution of the duration of negative surplus. Scand. Actuarial J. 2, 148–164. · Zbl 0864.62069 · doi:10.1080/03461238.1996.10413969
[11] Dickson, D. C. M. and Waters, H. R. (2004). Some optimal dividends problems. ASTIN Bull. 34, 49–74. · Zbl 1097.91040 · doi:10.2143/AST.34.1.504954
[12] Doney, R. A. (1995). Spitzer’s condition and ladder variables in random walks. Prob. Theory Relat. Fields 101, 577–580. · Zbl 0818.60060 · doi:10.1007/BF01202785
[13] Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, 91–106. · Zbl 1101.60029 · doi:10.1214/105051605000000647
[14] Dos Reis, A. E. (1993). How long is the surplus below zero? Insurance Math. Econom. 12, 23–38. · Zbl 0777.62096 · doi:10.1016/0167-6687(93)90996-3
[15] Furrer, H. (1998). Risk processes perturbed by \(\alpha\)-stable Lévy motion. Scand. Actuarial J. 1998, 59–74. · Zbl 1026.60516 · doi:10.1080/03461238.1998.10413992
[16] Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Schweiz. Verein. Versicherungsmath. Mitt. 69, 185–228. · Zbl 0193.20501
[17] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004a). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397. · Zbl 1061.60075 · doi:10.1214/105051604000000332
[18] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004b). Ruin probabilities for competing claim processes. J. Appl. Prob. 41, 679–690. · Zbl 1065.60100 · doi:10.1239/jap/1091543418
[19] Klüppelberg, C. and Kyprianou, A. E. (2006). On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Prob. 43, 594–598. · Zbl 1118.60071 · doi:10.1239/jap/1152413744
[20] Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801. · Zbl 1066.60049 · doi:10.1214/105051604000000927 · euclid:aoap/1099674077
[21] K\Huchler, U. and Sørensen, M. (1997) Exponential Families of Stochastic Processes. Springer, New York.
[22] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[23] Kyprianou, A. E. and Palmowski, Z. (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857 ), Springer, Berlin, pp. 16–29. · Zbl 1063.60071
[24] Lambert, A. (2000). Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré. Prob. Statist. 36 , 251–274. · Zbl 0970.60055 · doi:10.1016/S0246-0203(00)00126-6 · numdam:AIHPB_2000__36_2_251_0 · eudml:77658
[25] Lin, X. S. and Willmot, G. E. (2000). The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance Math. Econom. 27, 19–44. · Zbl 0971.91031 · doi:10.1016/S0167-6687(00)00038-X
[26] Lin, X. S., Willmot, G. E. and Drekic, S. (2003). The classical risk model with a constant dividend barrier: analysis of the Gerber–Shiu discounted penalty function. Insurance Math. Econom. 33, 551–566. · Zbl 1103.91369 · doi:10.1016/j.insmatheco.2003.08.004
[27] Renaud, J. and Zhou, X. (2007). Distribution of the present value of dividend payments in a Lévy risk model. J. Appl. Prob. 44, 420–427. · Zbl 1132.60041 · doi:10.1239/jap/1183667411
[28] Rivero, V. (2003). A law of iterated logarithm for increasing self-similar Markov processes. Stoch. Stoch. Reports 75, 443–472. · Zbl 1053.60027 · doi:10.1080/10451120310001646014
[29] Rogers, L. C. G. (2000). Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Prob. 37, 1173–1180. · Zbl 0981.60048 · doi:10.1239/jap/1014843099
[30] Surya, B. A. (2006). Evaluating scale functions of spectrally negative Lévy processes. · Zbl 1140.60027 · doi:10.1239/jap/1208358957
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.