Journal
SCANDINAVIAN ACTUARIAL JOURNAL
Volume -, Issue 7, Pages 573-591Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/03461238.2013.865257
Keywords
heavy tails; phase-type distributions; ruin probability
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Funding
- ARO [W911NF-07-1-0078, W911NF-12-10385]
- NSF [DMS-1005903]
- NSA at Cornell University [H98230-11-1-0154]
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In this paper, we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional phase-type distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.
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