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The moments of ruin time in the classical risk model with discrete claim size distribution. (English) Zbl 0957.62089
The authors consider a classical ruin model with discrete claim sizes $$W_i$$, constant claim intensity $$c$$ and constant premium rate $$\lambda$$. Exact simple solutions are provided for the moments $$k$$ of ruin time for the case where the initial reserve $$u=0$$. Also, for the case of $$u$$ natural, an analytic expression is derived. The calculation involves recursive calculations in terms of $$k$$ and $$u$$ which can be rolled out into finite sums over $$r,\dots,u$$. The calculations are built on a generalized Appell structure of polynomials. Both cases $$c>$$ and $$c\leq\lambda E[W_i]$$ are treated.

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B30 Risk theory, insurance (MSC2010)
##### Keywords:
ruin theory; risk theory
Full Text:
##### References:
 [1] Delbaen, F., A remark on the moments of ruin time in classical risk theory, Insurance: mathematics and economics, 9, 121-126, (1990) · Zbl 0733.62108 [2] Delbaen, F.; Haezendonck, J., Martingales in Markov processes applied to risk theory, Insurance: mathematics and economics, 5, 201-215, (1986) · Zbl 0629.62100 [3] De Vylder, F.E., () [4] De Vylder, F.E., Numerical finite-time ruin probabilities by the Picard-lefèvre formula, (1997), submitted for publication · Zbl 0952.91042 [5] De Vylder, F.E., La formule de Picard et lefèvre pour la probabilité de ruine en temps fini, Bulletin français d’actuariat, 1, 2, 31-40, (1997) [6] Gerber, H.U., An introduction to mathematical risk theory, (1979), S.S. Huebner foundation, University of Pennsylvania Philadelphia, PA · Zbl 0431.62066 [7] Gerber, H.U., When does the surplus reach a given target?, Insurance: mathematics and economics, 9, 115-119, (1990) · Zbl 0731.62153 [8] Picard, Ph.; Lefèvre, C., On the first crossing of the surplus process with a given upper barrier, Insurance: mathematics and economics, 14, 163-179, (1994) · Zbl 0806.62089 [9] Picard, Ph.; Lefèvre, C., The probability of ruin in finite time with discrete claim size distribution, Scandinavian actuarial journal, 1, 58-69, (1997) · Zbl 0926.62103 [10] Prabhu, N.U., On the ruin problem of collective risk theory, Ann. math. statist., 32, 757-764, (1961) · Zbl 0103.13302 [11] Seal, H.L., Stochastic theory of a risky business, (1969), Wiley New York · Zbl 0196.23501 [12] Segerdahl, C.O., On homogeneous random processes and collective risk theory, Thesis, (1939), Stockholm · JFM 65.1371.01 [13] Segerdahl, C.O., When does ruin occur in the collective theory of risk, Skand. aktu. tidskr., 38, 22-36, (1955) · Zbl 0067.12105
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