×

zbMATH — the first resource for mathematics

On a generalization of the risk model with Markovian claim arrivals. (English) Zbl 1237.91124
Summary: The class of risk models with Markovian arrival process (MAP, see e.g., [M. F. Neuts, J. Appl. Probab. 16, 764–774 (1979; Zbl 0422.60043)]) is generalized by allowing the waiting times between two successive events (which can be a change in the environmental state and/or a claim arrival) to have an arbitrary distribution. Using a probabilistic approach, we determine the solution for a class of Gerber-Shiu functions apart from some unknown constants when claim sizes have a mixed exponential distribution. Such constants are later determined using the more classic ruin-analytic approach. A numerical example is later considered to illustrate the tractability of the suggested methodology in the study of Gerber-Shiu functions.

MSC:
91B30 Risk theory, insurance (MSC2010)
60K15 Markov renewal processes, semi-Markov processes
60J75 Jump processes (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahn S., Stochastic Models 20 pp 71– (2004) · Zbl 1038.60086 · doi:10.1081/STM-120028392
[2] Ahn S., Queueing Systems 55 pp 207– (2007) · Zbl 1124.60067 · doi:10.1007/s11134-007-9017-x
[3] Albrecher H., Insurance: Mathematics and Economics 37 pp 650– (2005) · Zbl 1129.91023 · doi:10.1016/j.insmatheco.2005.06.007
[4] Asmussen S., Scandinavian Actuarial Journal 2 pp 69– (1989) · Zbl 0684.62073 · doi:10.1080/03461238.1989.10413858
[5] Asmussen S., Stochastic Models 11 pp 21– (1995) · Zbl 0817.60086 · doi:10.1080/15326349508807330
[6] Badescu A.L., Scandinavian Actuarial Journal 2 pp 127– (2005) · Zbl 1092.91037 · doi:10.1080/03461230410000565
[7] Cheung E.C.K., Journal of Applied Probability 46 pp 521– (2009) · Zbl 1180.60071 · doi:10.1239/jap/1245676104
[8] de Smit J.H.A., Advances in Queueing: Theory, Methods and Open Problems pp 293– (1995)
[9] Dufresne D., Applied Stochastic Models in Business and Industry 23 pp 23– (2007) · Zbl 1142.60321 · doi:10.1002/asmb.635
[10] Gerber H.U., Insurance: Mathematics and Economics 21 pp 129– (1997) · Zbl 0894.90047 · doi:10.1016/S0167-6687(97)00027-9
[11] Gerber H.U., North American Actuarial Journal 2 pp 48– (1998) · Zbl 1081.60550 · doi:10.1080/10920277.1998.10595671
[12] Landriault D., Insurance: Mathematics and Economics 42 pp 600– (2008) · Zbl 1152.91591 · doi:10.1016/j.insmatheco.2007.06.004
[13] Latouche G., Journal of Applied Probability 41 pp 746– (2004) · Zbl 1063.60127 · doi:10.1239/jap/1091543423
[14] Li S., North American Actuarial Journal 11 pp 65– (2007) · doi:10.1080/10920277.2007.10597448
[15] Neuts M.F., Journal of Applied Probability 16 pp 764– (1979) · Zbl 0422.60043 · doi:10.2307/3213143
[16] Ren J., North American Actuarial Journal 11 pp 128– (2007) · doi:10.1080/10920277.2007.10597471
[17] Rogers L.C.G., Annals of Applied Probability 4 pp 390– (1994) · Zbl 0806.60052 · doi:10.1214/aoap/1177005065
[18] Willmot G.E., Insurance: Mathematics and Economics 41 pp 17– (2007) · Zbl 1119.91058 · doi:10.1016/j.insmatheco.2006.08.005
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.