Calculation of ruin probabilities for a dense class of heavy tailed distributions

Mogens Bladt, Bo Friis Nielsen, Gennady Samorodnitsky

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Abstract

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.
Original languageEnglish
JournalScandinavian Actuarial Journal
Volume2015
Issue number7
Pages (from-to)573-591
ISSN0346-1238
DOIs
Publication statusPublished - 2015

Keywords

  • Heavy tails
  • Phase-type distributions
  • Ruin probability

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