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A Lévy Insurance Risk Process with Tax

Part of:Markov processesStochastic processesMathematical economics

Published online by Cambridge University Press:  14 July 2016

Hansjörg Albrecher,
Jean-François Renaud and
Xiaowen Zhou
Hansjörg Albrecher*
Affiliation:
Austrian Academy of Sciences and University of Linz
Jean-François Renaud*
Affiliation:
Austrian Academy of Sciences
Xiaowen Zhou*
Affiliation:
Concordia University
*
∗ Postal address: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria.
∗ Postal address: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria.
∗∗∗∗ Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd W., Montréal, Québec, H3G 1M8, Canada. Email address: xzhou@mathstat.concordia.ca
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Abstract

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Using fluctuation theory, we solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Lévy risk process with tax payments of a loss-carry-forward type. We study arbitrary moments of the discounted total amount of tax payments and determine the surplus level to start taxation which maximises the expected discounted aggregate income for the tax authority in this model. The results considerably generalise those for the Cramér-Lundberg risk model with tax.

Keywords

Lévy processfluctuation theoryexcursion theoryscale functionsinsurance risk theoryruin probabilitytax payments

MSC classification

Secondary:60G51: Processes with independent increments; L'evy processes91B30: Risk theory, insurance60J75: Jump processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008, pp. 363 - 375
Copyright
Copyright © Applied Probability Trust 2008 

Footnotes

Supported by the Austrian Science Fund Project P18392.

Supported by an NSERC grant.

References

[1] Albrecher, H. and Hipp, C. (2007). Lundberg's risk process with tax. Blätter der DGVFM 28, 13–28.Google Scholar
[2] 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.Google Scholar
[3] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[4] Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.Google Scholar
[5] Chan, T. and Kyprianou, A. E. (2008). Smoothness of scale functions for spectrally negative Lévy processes. Submitted.Google Scholar
[6] Chiu, S. N. and Yin, C. (2005). Passage times for a spectrally negative Lévy process with applications to risk theory. Bernoulli 11, 511–522.Google Scholar
[7] Doney, R. A. (2005). Some excursion calculations for spectrally one-sided Lévy processes. In Séminaire de Probabilités XXXVIII, (Lecture Notes Math. 1857), Springer, Berlin, pp. 5–15.Google Scholar
[8] Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J. 1, 59–74.CrossRefGoogle Scholar
[9] Garrido, J. and Morales, M. (2006). On the expected discounted penalty function for Lévy risk processes. N. Amer. Actuarial J. 10, 196–218.CrossRefGoogle Scholar
[10] Gerber, H. U. and Shiu, E. S. W. (2004). Optimal dividends: analysis with Brownian motion. N. Amer. Actuarial J. 8, 1–20.Google Scholar
[11] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397.Google Scholar
[12] Klüppelberg, C. and Kyprianou, A. E. (2006). On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Prob. 43, 594–598.Google Scholar
[13] 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.Google Scholar
[14] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[15] 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.Google Scholar
[16] Loeffen, R. (2008). On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. To appear in Ann. Appl. Prob. Google Scholar
[17] 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.Google Scholar
[18] Yang, H. and Zhang, L. (2001). Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Prob. 33, 281–291.Google Scholar
[19] Zhou, X. (2006). Discussion on: On optimal dividend strategies in the compound Poisson model, by Gerber, H. and Shiu, E. (N. Amer. Actuar. J. 10, 76–93). N. Amer. Actuarial J. 10, 79–84.Google Scholar
[20] Zhou, X. (2007). Exit problems for spectrally negative Lévy processes reflected at either the supremum or the infimum. J. Appl. Prob. 44, 1012–1030.Google Scholar