Frostig, Esther A Markov additive risk process with a dividend barrier. (English) Zbl 1301.60063 Adv. Appl. Probab. 45, No. 2, 451-489 (2013). Summary: 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. MSC: 60G51 Processes with independent increments; Lévy processes 60J27 Continuous-time Markov processes on discrete state spaces 60J75 Jump processes (MSC2010) 60G46 Martingales and classical analysis 62P05 Applications of statistics to actuarial sciences and financial mathematics Keywords:Markov arrival process; spectrally negative Lévy process; phase-type distribution; exit time; reflected process PDFBibTeX XMLCite \textit{E. Frostig}, Adv. Appl. Probab. 45, No. 2, 451--489 (2013; Zbl 1301.60063) Full Text: DOI Euclid References: [1] 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, 234-249. · Zbl 1193.60103 [2] Asmussen, S. (1989). Exponential families generated by phase-type distribution and other Markov lifetimes. Scand. J. Statist. 16, 319-334. · Zbl 0697.60076 [3] Asmussen, S. (2003). Applied Probability and Queues , 2nd edn. Springer, New York. · Zbl 1029.60001 [4] Avanzi, B. and Gerber, H. U. (2008). Optimal dividends in the dual model with diffusion. ASTIN Bull. 38, 653-667. · Zbl 1274.91463 [5] Avanzi, B., Gerber, H. U. and Shiu, E. S. W. (2007). Optimal dividends in the dual model. Insurance Math. Econom. 41, 111-123. · Zbl 1131.91026 [6] 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 [7] Bertoin, J. (1996). Lévy Processes. Cambridge University Press. · Zbl 0861.60003 [8] Breuer, L. (2008). First passage time for Markov additive processes with positive jumps of phase type. J. Appl. Prob. 45, 779-799. · Zbl 1156.60059 [9] Breuer, L. (2010) A quintuple law for Markov processes with phase-type jumps. J. Appl. Prob. 47, 441-458. · Zbl 1205.60095 [10] Cheung, E. C. K. (2011). On the class of stochastic models with two-sided jumps. Queueing Systems 69, 1-28. · Zbl 1235.60126 [11] Cheung, E. C. K. and Landriault, D. (2009). Perturbed MAP risk models with dividend barrier strategies. J. Appl. Prob. 46, 521-541. · Zbl 1180.60071 [12] 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. [13] Kyprianou, A. E. (2006), Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001 [14] 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. 121-135. · Zbl 1156.60060 [15] 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, 65-76. [16] 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, 53-71. · Zbl 1169.91390 [17] Lu, Y. and Li, S. (2009). The Markovian regime-switching risk model with threshold dividend strategy. Insurance Math. Econom. 44, 296-303. · Zbl 1163.91438 [18] 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, 136-149. [19] 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 [20] Zhu, J. and Yang, H. (2008). Ruin theory for a Markov regime-switching model under threshold dividend strategy. Insurance Math. Econom. 42, 311-318. \endharvreferences · Zbl 1141.91558 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.