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

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.

Keywords:

• heavy tails
• phase-type distributions
• ruin probability

Acknowledgments

We are grateful to the anonymous referee for useful comments that helped us to improve the paper. Samorodnitsky’s research was partially supported by the ARO grants W911NF-07-1-0078 and W911NF-12-10385, NSF grant DMS-1005903 and NSA grant H98230-11-1-0154 at Cornell University.