×

Ruin probabilities for Erlang (2) risk processes. (English) Zbl 0907.90097

Summary: We consider a risk process in which claim inter-arrival times have an Erlang(2) distribution. We consider the infinite time survival probability as a compound geometric random variable and give expressions from which both the survival probability from initial surplus zero and the ladder height distribution can be calculated. We consider explicit solutions for the survival ruin probability in the case where the individual claim amount distribution is phase-type, and show how the survival/ruin probability can be calculated for other individual claim amount distributions.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Asmussen, S., Applied Probability and Queues (1987), Wiley: Wiley New York · Zbl 0624.60098
[2] Asmussen, S., Phase-type representations in random walk and queueing problems, The Annals of Probability, 20, 772-789 (1992) · Zbl 0755.60049
[3] Asmussen, S.; Bladt, M., Renewal theory and queueing algorithms for matrix-exponential distributions, (Chakravarthy, S. R.; Alfa, A. S., Matrix-Analytic Methods in Stochastic Models (1996), Marcel Dekker: Marcel Dekker New York) · Zbl 0872.60064
[4] Asmussen, S.; Rolski, T., Computational methods in risk theory: A matrix-algorithmic approach, Insurance: Mathematics and Economic, 10, 259-274 (1991) · Zbl 0748.62058
[5] Dickson, D. C.M.; Egídio dos Reis, A. D.; Waters, H. R., Some stable algorithms in ruin theory and their applications, ASTIN Bulletin, 25, 153-175 (1995)
[6] Dickson, D. C.M., On a class of renewal risk process, North American Actuarial Journal (1998), to appear
[7] Dufresne, F.; Gerber, H. U., Three methods to calculate the probability of ruin, ASTIN Bulltein, 19, 71-90 (1989)
[8] Neuts, M. F., Probability distributions of phase-type, (Liber Amicorum Professor Emeritus H. Florin (1975), Department of Mathematics, University of Louvain: Department of Mathematics, University of Louvain Belgium), 173-206
[9] Neuts, M. F., Algorithms for the waiting time distribution under various queue disciplines in the M/G/1 queue with service time distribution of phase type, (Algorithmic Methods in Probability. Algorithmic Methods in Probability, TIMS Studies in the Management Sciences, Vol. 7 (1977)), 177-197
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.