Dependent Discrete Risk Processes - Calculation of the Probability of Ruin
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 probability 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. (original abstract)
-  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.
-  COSSETE H., LANDRIAULT D., MARCEAU E., Ruin probabilities in the compound Markov binomial model, Scandinavian Actuarial Journal, 2003, 4, 301-323.
-  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.
-  FREES E.W., VALDEZ E.A., Understanding relationships using copulas, North Amer. Actuarial J., 1998, 2, 1-25.
-  GERBER E., Mathematical fun with the compound binomial process, ASTIN Bulletin, 1988, 18, 161-168.
-  HEILPERN S., Funkcje łączące, Wyd. AE Wrocław, Wrocław, 2007.
-  NELSEN R.B., An Introduction to copulas, Springer, New York, 1999.
-  OAKES D., Bivariate survival models induced by frailties, JASA, 1989, 84, 487-493.
-  SHIU E., The probability of eventual ruin in the compound binomial model, ASTIN Bulletin, 1989, 19, 179-190.