×

zbMATH — the first resource for mathematics

The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model. (English) Zbl 1237.91125
Summary: We use probabilistic arguments to derive an expression for the joint density of the time to ruin and the number of claims until ruin in the classical risk model. From this we obtain a general expression for the probability function of the number of claims until ruin. We also consider the moments of the number of claims until ruin and illustrate our results in the case of exponentially distributed individual claims. Finally, we briefly discuss joint distributions involving the surplus prior to ruin and deficit at ruin.

MSC:
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albrecher, H.; Boxma, O.J., On the discounted penalty function in a Markov-dependent risk model, Insurance: mathematics & economics, 37, 650-672, (2005) · Zbl 1129.91023
[2] Beard, R.E., On the calculation of the ruin probability for a finite time interval, ASTIN bulletin, 6, 129-133, (1971)
[3] Dickson, D.C.M., Insurance risk and ruin, (2005), Cambridge University Press Cambridge · Zbl 1060.91078
[4] Dickson, D.C.M., Some finite time ruin problems, Annals of actuarial science, 2, 217-232, (2007)
[5] Dickson, D.C.M.; Li, S., Erlang risk models and finite time ruin problems, Scandinavian actuarial journal, (2010)
[6] Dickson, D.C.M.; Waters, H.R., Optimal dynamic reinsurance, ASTIN bulletin, 36, 415-432, (2006) · Zbl 1162.91408
[7] 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
[8] Egídio dos Reis, A.D., How many claims does it take to get ruined and recovered?, Insurance: mathematics & economics, 31, 235-248, (2002) · Zbl 1074.91550
[9] Gerber, H.U., Mathematical fun with ruin theory, Insurance: mathematics & economics, 7, 15-23, (1988) · Zbl 0657.62121
[10] 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
[11] Graham, R.L.; Knuth, D.E.; Patashnik, O., Concrete mathematics, (1994), Addison-Wesley Upper Saddle River, NJ · Zbl 0836.00001
[12] Landriault, D.; Shi, T.; Willmot, G.E., Joint density involving the time to ruin in the sparre Andersen risk model under exponential assumptions, Insurance: mathematics & economics, 49, 371-379, (2011) · Zbl 1229.91161
[13] Lee, J.M.L., 2011. The probability function of the number of claims until ruin in some risk models. Honours Research Essay, The University of Melbourne.
[14] Lin, X.; Willmot, G.E., The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: mathematics & economics, 27, 19-44, (2000) · Zbl 0971.91031
[15] Prabhu, N.U., On the ruin problem of collective risk theory, Annals of mathematical statistics, 32, 757-764, (1961) · Zbl 0103.13302
[16] Stanford, D.A.; Stroiński, K.J., Recursive methods for computing finite-time ruin probabilities for phase-distributed claims, ASTIN bulletin, 24, 235-254, (1994)
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.