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A unified analysis of claim costs up to ruin in a Markovian arrival risk model. (English) Zbl 1284.91214

Summary: An insurance risk model where claims follow a Markovian arrival process (MArP) is considered in this paper. It is shown that the expected present value of total operating costs up to default \(H\), as a generalization of the classical Gerber-Shiu function, contains more non-trivial quantities than those covered in [J. Cai et al., Adv. Appl. Probab. 41, No. 2, 495–522 (2009; Zbl 1173.91023)], such as all moments of the discounted claim costs until ruin. However, it does not appear that the Gerber-Shiu function \(\phi\) with a generalized penalty function which additionally depends on the surplus level immediately after the second last claim before ruin [E. C. K. Cheung et al., Scand. Actuar. J. 2010, No. 3, 185–199 (2010; Zbl 1226.60123)] is contained in \(H\). This motivates us to investigate an even more general function \(Z\) from which both \(H\) and \(\phi\) can be retrieved as special cases. Using a matrix version of Dickson-Hipp operator [R. Feng, “A matrix operator approach to the analysis of ruin-related quantities in the phase-type renewal risk model”, Mitt., Schweiz. Aktuarver. 2009, No. 1–2, 71–87 (2009), http://math.uiuc.edu/~rfeng/BSAA2009.pdf], it is shown that \(Z\) satisfies a Markov renewal equation and hence admits a general solution. Applications to other related problems such as the matrix scale function, the minimum and maximum surplus levels before ruin are given as well.

MSC:

91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60J28 Applications of continuous-time Markov processes on discrete state spaces
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References:

[1] Ahn, S.; Badescu, A. L., On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals, Insurance: Mathematics and Economics, 41, 2, 234-249 (2007) · Zbl 1193.60103
[2] Ahn, S.; Badescu, A. L.; Ramaswami, V., Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier, Queueing Systems, 55, 4, 207-222 (2007) · Zbl 1124.60067
[3] Albrecher, H.; Boxma, O. J., On the discounted penalty function in a Markov-dependent risk model, Insurance: Mathematics and Economics, 37, 3, 650-672 (2005) · Zbl 1129.91023
[4] Asmussen, S., Applied Probability and Queues (2003), Springer: Springer New York · Zbl 1029.60001
[5] Badescu, A. L., Discussion of ‘The discounted joint distribution of the surplus prior to ruin and the deficit at ruin in a Sparre Andersen model’, North American Actuarial Journal, 12, 2, 210-212 (2008) · Zbl 1481.91047
[6] Badescu, A. L.; Breuer, L.; Da Silva Soares, A.; Latouche, G.; Remiche, M.-A.; Stanford, D., Risk processes analyzed as fluid queues, Scandinavian Actuarial Journal, 2005, 2, 127-141 (2005) · Zbl 1092.91037
[7] Badescu, A. L.; Drekic, S.; Landriault, D., Analysis of a threshold dividend strategy for a MAP risk model, Scandinavian Actuarial Journal, 2007, 4, 227-247 (2007) · Zbl 1164.91024
[8] Badescu, A. L.; Drekic, S.; Landriault, D., On the analysis of a multi-threshold Markovian risk model, Scandinavian Actuarial Journal, 2007, 4, 248-260 (2007) · Zbl 1164.91025
[9] Badescu, A. L.; Landriault, D., Recursive calculation of the dividend moments in a multi-threshold risk model, North American Actuarial Journal, 12, 1, 74-88 (2008) · Zbl 1481.91162
[10] Cai, J.; Feng, R.; Willmot, G. E., On the total discounted operating costs up to default and its applications, Advances in Applied Probability, 41, 2, 495-522 (2009) · Zbl 1173.91023
[11] Cheung, E. C.K., Discussion of ‘Moments of the dividend payments and related problems in a Markov-modulated risk model’, North American Actuarial Journal, 11, 4, 145-148 (2007) · Zbl 1480.91194
[12] Cheung, E. C.K., A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium, Insurance: Mathematics and Economics, 48, 3, 383-397 (2011) · Zbl 1229.91157
[13] Cheung, E. C.K.; Landriault, D., Perturbed MAP risk models with dividend barrier strategies, Journal of Applied Probability, 46, 2, 521-541 (2009) · Zbl 1180.60071
[14] Cheung, E. C.K.; Landriault, D., Analysis of a generalized penalty function in a semi-Markovian risk model, North American Actuarial Journal, 13, 4, 497-513 (2009) · Zbl 1483.91182
[15] Cheung, E. C.K.; Landriault, D., A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model, Insurance: Mathematics and Economics, 46, 1, 127-134 (2010) · Zbl 1231.91156
[16] Cheung, E. C.K.; Landriault, D.; Willmot, G. E.; Woo, J.-K., Gerber-Shiu analysis with a generalized penalty function, Scandinavian Actuarial Journal, 2010, 3, 185-199 (2010) · Zbl 1226.60123
[17] Cheung, E. C.K.; Landriault, D.; Willmot, G. E.; Woo, J.-K., Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models, Insurance: Mathematics and Economics, 46, 1, 117-126 (2010) · Zbl 1231.91157
[18] Cheung, E. C.K.; Landriault, D.; Willmot, G. E.; Woo, J.-K., On orderings and bounds in a generalized Sparre Andersen risk model, Applied Stochastic Models in Business and Industry, 27, 1, 51-60 (2011) · Zbl 1274.60050
[19] Çinlar, E., Markov renewal theory, Advances in Applied Probability, 1, 2, 123-187 (1969) · Zbl 0212.49601
[20] Dickson, D. C.M.; Hipp, C., On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics and Economics, 29, 3, 333-344 (2001) · Zbl 1074.91549
[21] Feng, R., On the total operating costs up to default in a renewal risk model, Insurance: Mathematics and Economics, 45, 2, 305-314 (2009) · Zbl 1231.91183
[22] Feng, R., A matrix operator approach to the analysis of ruin-related quantities in the phase-type renewal risk model, Schweizerische Aktuarvereinigung Mitteilungen, 2009, 1-2, 71-87 (2009), Correction Available at http://www.math.uiuc.edu/ rfeng · Zbl 1333.91025
[23] Feng, R., An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models, Insurance: Mathematics and Economics, 48, 2, 304-313 (2011) · Zbl 1218.91077
[26] Gerber, H. U.; Shiu, E. S.W., On the time value of ruin, North American Actuarial Journal, 2, 1, 48-72 (1998) · Zbl 1081.60550
[27] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Springer: Springer New York · Zbl 0734.60060
[28] Latouche, G.; Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic Modeling (1999), ASA SIAM: ASA SIAM Philadelphia · Zbl 0922.60001
[29] Li, S.; Lu, Y., Moments of the dividend payments and related problems in a Markov-modulated risk model, North American Actuarial Journal, 11, 2, 65-76 (2007) · Zbl 1480.91222
[30] Li, S.; Lu, Y., The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model, Astin Bulletin, 38, 1, 53-71 (2008) · Zbl 1169.91390
[31] Li, G.; Luo, J., Upper and lower bounds for the solutions of Markov renewal equations, Mathematical Methods of Operations Research, 62, 2, 243-253 (2005) · Zbl 1101.60066
[32] Miyazawa, M., A Markov renewal approach to the asymptotic decay of the tail probabilities in risk and queueing processes, Probability in the Engineering and Informational Sciences, 16, 2, 139-150 (2002) · Zbl 1005.60094
[33] Neuts, M. F., Structured Stochastic Matrices of M/G/1 Type and their Applications (1989), Marcel Dekker: Marcel Dekker New York · Zbl 0695.60088
[34] Ren, J., The discounted joint distribution of the surplus prior to ruin and the deficit at ruin in a Sparre Andersen model, North American Actuarial Journal, 11, 3, 128-136 (2007) · Zbl 1480.91079
[35] Woo, J.-K., Some remarks on delayed renewal risk models, Astin Bulletin, 40, 1, 199-219 (2010) · Zbl 1230.91083
[36] Woo, J.-K., A generalized penalty function for a class of discrete renewal processes, Scandinavian Actuarial Journal, 2012, 2, 130-152 (2012) · Zbl 1277.60146
[37] Wu, Y., Bounds for the ruin probability under a Markovian modulated risk model, Communications in Statistics-Stochastic Models, 15, 1, 125-136 (1999) · Zbl 0920.90042
[38] Zhu, J.; Yang, H., Ruin theory for a Markov regime-switching model under a threshold dividend strategy, Insurance: Mathematics and Economics, 42, 1, 311-318 (2008) · Zbl 1141.91558
[39] Zhu, J.; Yang, H., On differentiability of ruin functions under Markov-modulated models, Stochastic Processes and their Applications, 119, 5, 1673-1695 (2009) · Zbl 1168.91421
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