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Finite time ruin problems for the Erlang\((2)\) risk model. (English) Zbl 1231.91176

Summary: We consider the Erlang\((2)\) risk model and derive expressions for the density of the time to ruin and the joint density of the time to ruin and the deficit at ruin when the individual claim amount distribution is (i) an exponential distribution and (ii) an Erlang\((2)\) distribution. We also consider the special case when the initial surplus is zero.

MSC:

91B30 Risk theory, insurance (MSC2010)
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[1] Borovkov, K. A.; Dickson, D. C.M., On the ruin time distribution for a Sparre Andersen process with exponential claim sizes, Insurance: Mathematics & Economics, 42, 1104-1108 (2008) · Zbl 1141.91486
[2] Cheung, E. C.K.; Dickson, D. C.M.; Drekic, S., Moments of discounted dividends for a threshold strategy in the compound Poisson risk model, North American Actuarial Journal, 12, 3, 299-318 (2008) · Zbl 1481.91166
[3] Dickson, D. C.M., Some explicit solutions for the joint density of the time of ruin and the deficit at ruin, ASTIN Bulletin, 38, 259-276 (2008) · Zbl 1169.91386
[4] Dickson, D. C.M.; Hipp, C., On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics & Economics, 29, 333-344 (2001) · Zbl 1074.91549
[5] Dickson, D. C.M.; Willmot, G. E., The density of the time to ruin in the classical Poisson risk model, ASTIN Bulletin, 35, 45-60 (2005) · Zbl 1097.62113
[6] Dickson, D. C.M.; Hughes, B. D.; Lianzeng, Z., The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims, Scandinavian Actuarial Journal, 2005, 5, 358-376 (2005) · Zbl 1144.91025
[7] Drekic, S.; Willmot, G. E., On the density and moments of the time to ruin with exponential claims, ASTIN Bulletin, 33, 11-21 (2003) · Zbl 1062.60007
[8] Garcia, J. M.A., Explicit solutions for survival probabilities in the classical risk model, ASTIN Bulletin, 35, 113-130 (2005) · Zbl 1101.62100
[9] 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
[10] Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics (1994), Addison-Wesley: Addison-Wesley Upper Saddle River, NJ · Zbl 0836.00001
[11] Li, S.; Garrido, J., On ruin for the Erlang \((n)\) risk process, Insurance: Mathematics & Economics, 34, 391-408 (2004) · Zbl 1188.91089
[12] Mazza, C.; Rullière, C., A link between wave governed random motions and ruin processes, Insurance: Mathematics & Economics, 35, 205-222 (2004) · Zbl 1103.91045
[13] Sun, L.-J., The expected discounted penalty at ruin in the Erlang(2) risk process, Statistics & Probability Letters, 72, 205-217 (2005) · Zbl 1104.91046
[14] Willmot, G. E., On the discounted penalty function in the renewal risk model with general interclaim times, Insurance: Mathematics & Economics, 41, 17-31 (2007) · Zbl 1119.91058
[15] 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) · Zbl 1480.91253
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