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A Markov Additive Risk Process with a Dividend Barrier

Published online by Cambridge University Press:  22 February 2016

Esther Frostig*
Affiliation:
University of Haifa
*
Postal address: Department of Statistics, University of Haifa, Haifa 31905, Israel. Email address: frostig@stat.haifa.ac.il
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Abstract

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We study a risk process with dividend barrier b where the claims arrive according to a Markovian additive process (MAP). For spectrally negative MAPs, we present linear equations for the expected discounted dividends and the expected discounted penalty function. We apply results for the first exit times of spectrally negative Lévy processes and change-of-measure techniques. Explicit expressions are given when there are positive and negative claims, with phase-type distribution.

Type
General Applied Probability
Copyright
© Applied Probability Trust 

Footnotes

Research supported by the Israel Science Foundation, under grant 606/09.

References

Ahn, S. and Badescu, A. L. (2007). On the analysis of the Gerber–Shiu discounted penalty function for risk processes with Markovian arrivals. Insurance Math. Econom. 41, 234249.CrossRefGoogle Scholar
Asmussen, S. (1989). Exponential families generated by phase-type distribution and other Markov lifetimes. Scand. J. Statist. 16, 319334.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Avanzi, B. and Gerber, H. U. (2008). Optimal dividends in the dual model with diffusion. ASTIN Bull. 38, 653667.Google Scholar
Avanzi, B., Gerber, H. U. and Shiu, E. S. W. (2007). Optimal dividends in the dual model. Insurance Math. Econom. 41, 111123.Google Scholar
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, 215238.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Breuer, L. (2008). First passage time for Markov additive processes with positive Jumps of phase type. J. Appl. Prob. 45, 779799.Google Scholar
Breuer, L. (2010) A quintuple law for Markov processes with phase-type Jumps. J. Appl. Prob. 47, 441458.CrossRefGoogle Scholar
Cheung, E. C. K. (2011). On the class of stochastic models with two-sided Jumps. Queueing Systems 69, 128.CrossRefGoogle Scholar
Cheung, E. C. K. and Landriault, D. (2009). Perturbed MAP risk models with dividend barrier strategies. J. Appl. Prob. 46, 521541.Google Scholar
Egami, M. and Yamazaki, K. (2012). Phase-type fitting of scale functions for spectrally negative Lévy processes. Preprint. Available at http://arxiv.org/abs/1005.0064v6.Google Scholar
Kyprianou, A. E. (2006), Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Kyprianou, A. E. and Palmowski, Z. (2008). Fluctuations of spectrally negative Markov additive processes. In Séminaire de Probabiltés XLI (Lecture Notes Math. 1934), Springer, Berlin, pp. 121135.Google Scholar
Li, S. and Lu, Y. (2007). Moments of the dividend payments and related problems in a Markov-modulated risk model. N. Amer. Actuarial J. 11, 6576.CrossRefGoogle Scholar
Li, S. and Lu, Y. (2008). The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model. ASTIN Bull. 38, 5371.Google Scholar
Lu, Y. and Li, S. (2009). The Markovian regime-switching risk model with threshold dividend strategy. Insurance Math. Econom. 44, 296303.Google Scholar
Lu, Y. and Tsai, C. C.-L. (2007). The expected discounted penalty at ruin for Markov-modulated risk process perturbed by diffusion. N. Amer. Actuarial J. 11, 136149.Google Scholar
Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183220.Google Scholar
Zhu, J. and Yang, H. (2008). Ruin theory for a Markov regime-switching model under threshold dividend strategy. Insurance Math. Econom. 42, 311318.Google Scholar