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The time to ruin and the number of claims until ruin for phase-type claims. (English) Zbl 1284.91232
Summary: We consider a renewal risk model with phase-type claims, and obtain an explicit expression for the joint transform of the time to ruin and the number of claims until ruin, with a penalty function applied to the deficit at ruin. The approach is via the duality between a risk model with phase-type claims and a particular single server queueing model with phase-type customer interarrival times; see [E. Frostig, Adv. Appl. Probab. 36, No. 2, 377–397 (2004; Zbl 1123.91335)]. This result specializes to one for the probability generating function of the number of claims until ruin. We obtain explicit expressions for the distribution of the number of claims until ruin for exponentially distributed claims when the inter-claim times have an Erlang-nn distribution.

91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
Full Text: DOI
[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, 234-249, (2007) · Zbl 1193.60103
[2] Asmussen, S., Ruin probabilities, (2000), World Scientific Singapore
[3] Asmussen, S.; Rolski, T., Computational methods in risk theory: a matrix – algorithmic approach, Insurance: mathematics and economics, 10, 259-274, (1991) · Zbl 0748.62058
[4] Badescu, A.; Breuer, L.; Da Silva Soars, A.; Latouche, G.; Remiche, M.A.; Stanford, D., Risk processes analyzed as fluid queues, Scandinavian actuarial journal, 2, 127-141, (2005) · Zbl 1092.91037
[5] Borovkov, K.A.; Dickson, D.C.M., On the ruin time for a sparre Andersen process with exponential claim sizes, Insurance: mathematics and economics, 42, 1104-1108, (2008) · Zbl 1141.91486
[6] Dickson, D.C.M.; Li, S., Finite time ruin problems for the Erlang(2) risk model, Insurance: mathematics and economics, 46, 12-18, (2010) · Zbl 1231.91176
[7] Drekic, D.; Dickson, D.C.M.; Stanford, D.A.; Willmot, G.E., On the distribution of the deficit at ruin when the claims are phase-type, Scandinavian actuarial journal, 105-120, (2004) · Zbl 1142.62088
[8] Egídio dos Reis, A.D., How many claims does it take to get ruined and recovered?, Insurance: mathematics and economics, 31, 2, 235-248, (2002) · Zbl 1074.91550
[9] Feller, W., An introduction to probability theory and its applications, vol. I, (1968), Wiley New York · Zbl 0155.23101
[10] Feller, W., An introduction to probability theory and its applications, vol. II, (1971), Wiley New York · Zbl 0219.60003
[11] Frostig, E., Upper bounds on the expected time to ruin and the expected recovery time, Advances in applied probability, 36, 377-397, (2004) · Zbl 1123.91335
[12] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 48-78, (1998) · Zbl 1081.60550
[13] Hesselager, O., A recursive procedure for calculation of some compound distributions, ASTIN bulletin, 24, 19-32, (1994)
[14] Klugman, S.A.; Panjer, H.H.; Willmot, G.E., Loss models: from data to decisions, (2008), Wiley New York · Zbl 1159.62070
[15] Landriault, D.; Shi, T.; Willmot, G.E., Joint densities involving the time to ruin in the sparre Andersen risk model under exponential assumptions, Insurance: mathematics and economics, 49, 371-379, (2011) · Zbl 1229.91161
[16] Landriault, D.; Willmot, G., On the gerber – shiu discounted penalty function in the sparre Andersen model with an arbitrary interclaim time distribution, Insurance: mathematics and economics, 42, 600-608, (2008) · Zbl 1152.91591
[17] Lucantoni, D.M., New results on the single server queue with a batch Markovian arrival process, Stochastic models, 7, 1-46, (1991) · Zbl 0733.60115
[18] Neuts, M.F., Matrix – geometric solutions in stochastic models: an algorithmic approach, (1981), The Johns Hopkins University Pess · Zbl 0469.60002
[19] Panjer, H.H.; Willmot, G.E., Recursions for compound distributions, ASTIN bulletin, 13, 1-11, (1982)
[20] Prabhu, N.U.; Zhu, Y., Markov-modulated queueing systems, Queueing systems, 5, 215-246, (1989) · Zbl 0694.60087
[21] Stanford, D.A.; Avram, F.; Badescu, A.B.; Breuer, L.; Da Silva Soars, A.; Latouche, G., Phase-type approximations in the sparre Andersen and stationary renewal risk models, ASTIN bulletin, 35, 131-144, (2005) · Zbl 1123.62078
[22] Willmot, G.E., Sundt and jewell’s family of discrete distributions, ASTIN bulletin, 18, 17-29, (1988)
[23] Zhu, Y.; Prabhu, N.U., Markov-modulated PH/G/1 queueing systems, Queueing systems, 9, 313-322, (1991) · Zbl 0734.60095
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