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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)
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[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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