×

The moments of the discounted loss and the discounted dividends for a spectrally negative Lévy risk process. (English) Zbl 1326.60063

Summary: Consider a spectrally negative risk process where, on ruin, the deficit is immediately paid, and the process restarts from 0. When the process reaches a threshold \(b\), all the surplus above \(b\) is paid as dividend. Applying the theory of exit times for a spectrally negative Lévy process and its reflection at the maximum and at the minimum, we obtain recursive formulae for the following moments. (i) The moments of the discounted loss until the process reaches \(b\). This is equivalent to the moments of the discounted dividends in the dual model under the barrier strategy. (ii) The moments of the discounted loss for models with and without a dividend barrier for the infinite horizon. (iii) The moments of the discounted dividends for the infinite horizon.

MSC:

60G51 Processes with independent increments; Lévy processes
91B30 Risk theory, insurance (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1-15. · Zbl 1065.91529
[2] 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
[3] Avanzi, B. and Gerber, H. U. (2008) Optimal dividends in the dual model with diffusion. ASTIN Bull. 38, 653-667. · Zbl 1274.91463
[4] Avanzi, B., Gerber, H. U. and Shiu, E. W. S. (2007). Optimal dividends in the dual model. Insurance Math. Econom. 41, 111-123. · Zbl 1131.91026
[5] Avanzi, B., Shen, J. and Wong, B. (2011). Optimal dividends and capital injections in the dual model with diffusion. ASTIN Bull. 41, 611-644. · Zbl 1242.91089
[6] 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
[7] Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2013). On optimal dividends in the dual model. ASTIN Bull. 43, 359-372. · Zbl 1283.91192
[8] Bertoin, J. (1996). Lévy Processes . Cambridge University Press. · Zbl 0861.60003
[9] Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7 , 156-169. · Zbl 0880.60077
[10] Cheung, E. C. K. and Drekic, S. (2008). Dividend moments in the dual risk model: exact and approximate approaches. ASTIN Bull. 38, 399-422. · Zbl 1256.91026
[11] Dickson, D. C. M. and Waters, H. R. (2004). Some optimal dividends problems. ASTIN Bull. 34, 49-74. · Zbl 1097.91040
[12] De Finetti, B. (1957). Su un’impostazione alternativa dell teoria collettiva del rischio. Trans. XVth Internat. Congr. Actuaries 2, 433-443.
[13] Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-prozess. Schweiz. Verein. Versicherungsmath. 69, 185-228. · Zbl 0193.20501
[14] Hubalek, F. and Kyprianou, A. E. (2011). Old and new examples of scale functions for spectrally negative Lévy processes. In Seminar on Stochastic Analysis, Random Fields and Applications VI (Progress Prob. 63 ), Birkhäuser, Basel, pp. 119-145. · Zbl 1274.60148
[15] Kulenko, N. and Schmidli, H. (2008). Optimal dividend strategies on a Cramér-Lundberg model with capital injections. Insurance Math. Econom. 43, 270-278. · Zbl 1189.91075
[16] Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061 ), Springer, Heidelberg, pp. 97-186. · Zbl 1261.60047
[17] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[18] Kyprianou, A. E. and Palmowski, Z. (2007). Distributional study of de Finetti’s dividend problem for a general Lévy insurance risk process. J. Appl. Prob. 44, 428-443. · Zbl 1137.60047
[19] Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 1669-1680. · Zbl 1152.60344
[20] Mijatović, A. and Pistorius, M. R. (2012). On the drawdown of completely asymmetric Lévy processes. Stoch. Process. Appl. 122, 3812-3836. · Zbl 1252.60046
[21] Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183-220. · Zbl 1049.60042
[22] Renaud, J-F. 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
[23] Schmidli, H. (2008). Stochastic Control in Insurance . Springer, London. · Zbl 1133.93002
[24] Suprun, V. N. (1976). Problem of destruction and resolvent of a terminating process with independent increments. Ukrainian Math. J. 28, 39-51. · Zbl 0349.60075
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.