Heilpern, Stanisław Dependent discrete risk processes – calculation of the probability of ruin. (English) Zbl 1492.91082 Oper. Res. Decis. 20, No. 2, 59-76 (2010). Summary: This paper is devoted to discrete processes of dependent risks. The random variables describing the time between claims can be dependent in such processes, unlike under the classical approach. The ruin problem is investigated and the probably of ruin is computed. The relation between the degree of dependence and the probability of ruin is studied.Three cases are presented. Different methods of characterizing the dependency structure are examined. First, strictly dependent times between claims are investigated. Next, the dependency structure is described using an Archimedean copula or using Markov chains. In the last case, three situations in which the probability of ruin can be exactly computed are presented. Numerical examples in which the claims have a geometric distribution are investigated. A regular relation between the probability of ruin and the degree of dependence is only observed in the Markov chain case. MSC: 91B05 Risk models (general) 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 62H05 Characterization and structure theory for multivariate probability distributions; copulas Keywords:risk process; probability of ruin; dependence; copula; Markov chain PDFBibTeX XMLCite \textit{S. Heilpern}, Oper. Res. Decis. 20, No. 2, 59--76 (2010; Zbl 1492.91082) Full Text: Link References: [1] COSSETE H., LANDRIAULT D., MARCEAU E., Exact expressions and upper bound for ruin probabilities in the compound Markov binomial model, Insurance: Mathematics and Economics, 2004, 34, 449-466. · Zbl 1188.91086 [2] COSSETE H., LANDRIAULT D., MARCEAU E., Ruin probabilities in the compound Markov binomial model, Scandinavian Actuarial Journal, 2003, 4, 301-323. · Zbl 1092.91040 [3] DICKSON D.C.M., EGIDO DOS REIS A.D., WALTERS H.R., Some stable algorithms in ruin theory and their applications, ASTIN Bulletin, 1995, 25, 153-175. [4] FREES E.W., VALDEZ E.A., Understanding relationships using copulas, North Amer. Actuarial J., 1998, 2, 1-25. · Zbl 1081.62564 [5] GERBER E., Mathematical fun with the compound binomial process, ASTIN Bulletin, 1988, 18, 161-168. [6] HEILPERN S., Funkcje łączące, Wyd. AE Wrocław, Wrocław, 2007. [7] NELSEN R.B., An Introduction to copulas, Springer, New York, 1999. · Zbl 0909.62052 [8] OAKES D., Bivariate survival models induced by frailties, JASA, 1989, 84, 487-493. · Zbl 0677.62094 [9] SHIU E., The probability of eventual ruin in the compound binomial model, ASTIN Bulletin, 1989, 19, 179-190. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.