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Another look at the Picard–Lefèvre formula for finite-time ruin probabilities. (English) Zbl 1103.91048
In the compound Poisson risk model, with discrete claim size distribution, Picard and Lefèvre derived a formula to compute the finite-horizon ruin probability. Here, some alternatives to this formula are proposed: exact recursive formulas which provide the distribution of time to ruin at once and a Seal-type formula which only involve probabilistic quantities. Depending on the comparison between the initial reserve and the total premium up to the finite horizon, their different interests are discussed by comparing their performances. The numerical stability of the formulas is then investigated, and disagreements in the existing literature about the detection of critical values are explained.
Formal convolutions for pseudo-compound distributions are introduced, and a theorem is stated in order to switch between formulas based on Appell polynomials and Seal-type formulas. This also provides a derivation of the Picard-Lefèvre formula from sample path properties.

MSC:
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62E10 Characterization and structure theory of statistical distributions
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[1] De Vylder, F.E., La formule de picard – lefèvre pour la probabilité de ruine en temps fini, Bulletin français d’actuariat, 1-2, 31-40, (1997)
[2] De Vylder, F.E., Numerical finite-time ruin probabilities by the picard – lefèvre formula, Scand. actuarial J., 2, 97-105, (1999) · Zbl 0952.91042
[3] Gerber, H.-U., 1979. Introduction to Mathematical Risk Theory. Huebner Foundation Monograph
[4] Ignatov, G.; Kaishev, V.K.; Krachunov, R.S., An improved finite-time ruin probabilities formula and its Mathematica implementation, Insur.: math. econ., 29, 375-386, (2001) · Zbl 1074.62528
[5] Panjer, H., Recursive evaluation of a family of compound distributions, Astin bull., 12, 22-26, (1981)
[6] Panjer, H.; Wang, S., On the stability of recursive formulas, Astin bull., 23, 2, (1993)
[7] Picard, P.; Lefèvre, C., The probability of ruin in finite time with discrete claim size distribution, Scand. actuarial J., 1, 58-69, (1997) · Zbl 0926.62103
[8] Picard, P.; Lefèvre, C., The moments of ruin time in the classical risk model with discrete claim size distribution, Insur.: math. econ., 23, 157-172, (1998) · Zbl 0957.62089
[9] Picard, P.; Lefèvre, C.; Coulibaly, I., Problèmes de ruine en théorie du risque à temps discret avec horizon fini, J. appl. probab., 40, 3, 527-542, (2003) · Zbl 1049.62113
[10] Picard, P.; Lefèvre, C.; Coulibaly, I., Multirisks model and finite-time ruin probabilities, Methodol. comput. appl. probab., 5, 3, 337-353, (2003) · Zbl 1035.62109
[11] Rullière D. (2000). Mesure et contrôle du risque technique dans une compagnie d’assurance sur la vie, PhD dissertation, pp. 154-155.
[12] Seal, H.L., Stochastic theory of a risky business, (1969), Wiley New York · Zbl 0196.23501
[13] Takács, L., A generalization of the ballot problem and its application in the theory of queues, J. am. stat. assoc., 57, 327-337, (1962) · Zbl 0109.36702
[14] Takács, L., The time dependence of a single-server queue with Poisson input and general service times, Ann. math. stat., 33, 1340-1348, (1962) · Zbl 0117.36003
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