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Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions. (English) Zbl 1229.91161
Summary: Recent research into the nature of the distribution of the time of ruin in some Sparre Andersen risk models has resulted in series expansions for the associated density function. Examples include D. C. M. Dickson and G. E. Willmot [Astin Bull. 35, No. 1, 45–60 (2005; Zbl 1097.62113)] in the classical Poisson model with exponential interclaim times, and K. A. Borovkov and D. C. M. Dickson [Insur. Math. Econ. 42, No. 3, 1104–1108 (2008; Zbl 1141.91486)], who used a duality argument in the case with exponential claim amounts. The aim of this paper is not only to unify previous methodology through the use of Lagrange’s expansion theorem, but also to provide insight into the nature of the series expansions by identifying the probabilistic contribution of each term in the expansion through analysis involving the distribution of the number of claims until ruin. The (defective) distribution of the number of claims until ruin is then further examined. Interestingly, a connection to the well-known extended truncated negative binomial (ETNB) distribution is also established. Finally, a closed-form expression for the joint density of the time to ruin, the surplus prior to ruin, and the number of claims until ruin is derived. In the last section, the formula of Dickson and Willmot [loc. cit.] for the density of the time to ruin in the classical risk model is re-examined to identify its individual contributions based on the number of claims until ruin.

MSC:
91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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[1] Asmussen, S., Ruin probabilities, (2000), World Scientific Singapore
[2] Borovkov, K.A.; Dickson, D.C.M., On the ruin time distribution for a sparre Andersen risk process with exponential claim sizes, Insurance: mathematics and economics, 42, 1104-1108, (2008) · Zbl 1141.91486
[3] Chan, W.; Zhang, L., Direct derivation of finite-time ruin probabilities in the discrete risk model with exponential or geometric claims, North American actuarial journal, 10, 4, 269-279, (2006)
[4] 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
[5] Cohen, J.W., The single server queue, (1969), North-Holland, Publication Company Amsterdam · Zbl 0183.49204
[6] Dickson, D.C.M.; Hughes, B.D.; Zhang, L., The density of the time to ruin for a sparre Andersen process with Erlang arrivals and exponential claims, Scandinavian actuarial journal, 358-376, (2005) · Zbl 1144.91025
[7] Dickson, D.C.M., Li, S., 2011. Erlang risk models and finite time ruin problems. Scandinavian Actuarial Journal (forthcoming). · Zbl 1277.91081
[8] Dickson, D.C.M.; Willmot, G.E., The density of the time to ruin in the classical Poisson risk model, Astin bulletin, 35, 1, 45-60, (2005) · Zbl 1097.62113
[9] Drekic, S.; Willmot, G.E., On the density and moments of the time of ruin with exponential claims, Astin bulletin, 33, 11-21, (2003) · Zbl 1062.60007
[10] 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
[11] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998) · Zbl 1081.60550
[12] Landriault, D.; Willmot, G.E., On the joint distributions of the time to ruin, the surplus prior to ruin and the deficit at ruin in the classical risk model, North American actuarial journal, 13, 2, 252-270, (2009)
[13] Stanford, D.A.; Stroinski, K.J., Recursive methods for computing finite-time ruin probabilities for phase distributed claim sizes, Astin bulletin, 24, 235-254, (1994)
[14] Stanford, D.A.; Stroinski, K.J.; Lee, K., Ruin probabilities based at claim instants for some non-Poisson claim processes, Insurance: mathematics and economics, 26, 251-267, (2000) · Zbl 1013.91068
[15] Willmot, G.E., On the discounted penalty function in the renewal risk model with general interclaim times, Insurance: mathematics and economics, 41, 17-31, (2007) · Zbl 1119.91058
[16] Willmot, G.E., Sundt and jewell’s family of discrete distributions, Astin bulletin, 18, 1, 17-29, (1988)
[17] Willmot, G.E.; Lin, X.S., Risk modelling with the mixed Erlang distribution, Applied stochastic models in business and industry, 27, 1, 2-16, (2011)
[18] Willmot, G.E.; Woo, J.K., On the class of Erlang mixtures with risk theoretic applications, North American actuarial journal, 11, 2, 99-115, (2007)
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