Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T04:49:41.588Z Has data issue: false hasContentIssue false
  • English
  • Français

General tax Structures and the Lévy Insurance Risk Model

Part of: Markov processes Special processes Mathematical economics Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Andreas E. Kyprianou  and
Xiaowen Zhou
Andreas E. Kyprianou*
Affiliation:
The University of Bath
Xiaowen Zhou*
Affiliation:
Concordia University
*
∗Postal address: Department of Mathematical Sciences, The University of Bath, Claverton Down, Bath BA2 7AY, UK. Email address: a.kyprianou@bath.ac.uk
∗∗Postal address: Department of Mathematics and Statistics, Concordia University, 1455 De Maisonneuve Blvd. West, Montreal Quebec, H3G 1M8, Canada. Email address: xzhou@mathstat.concordia.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the spirit of Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) we consider a Lévy insurance risk model with tax payments of a more general structure than in the aforementioned papers, which was also considered in Albrecher, Borst, Boxma, and Resing (2009). In terms of scale functions, we establish three fundamental identities of interest which have stimulated a large volume of actuarial research in recent years. That is to say, the two-sided exit problem, the net present value of tax paid until ruin, as well as a generalized version of the Gerber–Shiu function. The method we appeal to differs from Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) in that we appeal predominantly to excursion theory.

Keywords

Reflected Lévy processpassage problemintegrated exponential Lévy processinsurance risk processruinexcursion theory

MSC classification

Secondary: 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes 91B30: Risk theory, insurance 60G70: Extreme value theory; extremal processes 60J55: Local time and additive functionals
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1146 - 1156
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Albrecher, H. and Hipp, C. (2007). Lundberg's risk process with tax. Blätter der DGVFM 28, 13–28.Google Scholar
[2] Albrecher, H., Renaud, J.-F. and Zhou, X. (2008). A Lévy insurance risk process with tax. J. Appl. Prob. 45, 363–375.Google Scholar
[3] Albrecher, H., Borst, S., Boxma, O. and Resing, J. (2009). The tax identity in risk theory—a simple proof and an extension. Insurance Math. Econom. 44, 304–306.Google Scholar
[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. Prob. 14, 215–238.CrossRefGoogle Scholar
[5] 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.CrossRefGoogle Scholar
[6] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[7] Chan, T., Kyprianou, A. E. and Savov, M. (2009). Smoothness properties of scale functions for spectrally negative Lévy processes. Preprint.Google Scholar
[8] Chaumont, L., Kyprianou, A. E. and Pardo, J. C. (2009). Some explicit identities associated with positive self-similar Markov processes. Stoch. Process. Appl. 119, 980–1000.Google Scholar
[9] Hubalek, F. and Kyprianou, A. E. (2008). Old and new examples of scale functions for spectrally negative Lévy processes. Preprint. Available at http://arxiv.org/abs/0801.0393.Google Scholar
[10] 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
[11] 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.CrossRefGoogle Scholar
[12] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[13] Kyprianou, A. E. and Loeffen, R. (2009). Refracted Lévy processes. To appear in Ann. Inst. H. Poincaré Prob. Statist.Google Scholar
[14] 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
[15] Kyprianou, A. E. and Rivero, V. (2008). Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Prob. 13, 1672–1701.Google Scholar
[16] Millar, P. W. (1977). Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226, 365–391.CrossRefGoogle Scholar
[17] Patie, P. (2009). Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration. Bull. Sci. Math. 133, 355–382.Google Scholar
[18] Pistorius, M. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183–220.Google Scholar
[19] Pistorius, M. R. (2007). An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. In Séminaire de Probabilités XL (Lecture Notes Math. 1899), Springer, Berlin, pp. 287–307.Google Scholar
[20] Renaud, J.-F. (2009). The distribution of tax payments in a Lévy insurance risk model with a surplus-dependent taxation structure. Insurance Math. Econom. 45, 242–246.Google Scholar
[21] 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
[22] Surya, B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135–149.Google Scholar