Approximation of ruin probabilities via Erlangized scale mixtures

Abstract

In this paper, we extend an existing scheme for numerically calculating the probability of ruin of a classical Cramér-Lundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a dense class of distributions that we denominate Erlangized scale mixtures (ESM) and correspond to nonnegative and absolutely continuous distributions which can be written as a Mellin-Stieltjes convolution Π G of a nonnegative distribution Π with an Erlang distribution G. A distinctive feature of such a class is that it contains heavy-tailed distributions. We suggest a simple methodology for constructing a sequence of distributions having the form Π G to approximate the integrated tail distribution of the claim sizes. Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.

Asymptotic tail behaviour of phase-type scale mixture distributions

“…As an example of the later, Bladt et al (2015aBladt et al ( , 2015b provide renewal results that can be applied to obtain exact expressions for the ruin probability of a classical Cramér-Lundberg risk process having claim sizes distributed according to a PH scale mixture distribution with discrete scaling. This approach is further explored in Peralta et al (2016), where a systematic methodology for approximating arbitrary heavy-tailed distributions via PH scale mixtures is provided; such a formulation provides simplified formulas for approximating ruin probabilities with arbitrary claim size distributions. Furthermore, Bladt & Rojas-Nandayapa (2017) provide statistical inference procedures based on the expectation maximization algorithm to adjust PH scale mixtures to heavy-tailed data/distributions.…”

Ruin Probability Approximations in Sparre Andersen Models with Completely Monotone Claims

“…Proof. The result is a direct application of Theorem 4.1 of Peralta et al (2018) by (i) choosing the functionsF 1 andF 2 to be H andĤ, respectively; (ii) taking ρ = φ; and (iii) recognising that sup y<u H(y) −Ĥ(y) ≤ D(H,Ĥ).…”

Modelling heavy-tails with phase-type scale mixture distributions

“…Such is the case of ruin probabilities in the Crámer-Lundberg process having claims sizes distributed according to a phase-type scale mixture (cf. Peralta et al, 2018). Notice for instance, that such exact results are not available for the case of continuous scaling distributions.…”

“…This approach has been tested successfully in Peralta et al (2018), where they considered discretizing a Pareto distribution over a geometric progression and used the corresponding phase-type scale mixture distribution to approximate Pareto claim size distributions in ruin probability calculations. This selection of the scaling distribution is of critical importance in Bladt and Rojas-Nandayapa (2017) for estimating the parameters of a phase-type scale mixture distribution via the EM algorithm.…”

“…As an example of the later, provide renewal results that can be applied to obtain exact expressions for the ruin probability of a classical Cramér-Lundberg risk process having claim sizes distributed according to a phase-type scale mixture distribution with discrete scaling. This approach is further explored in Peralta et al (2018), where a systematic methodology for approximating arbitrary heavy-tailed distributions via phase-type scale mixtures is provided; such a formulation provides simplified formulas for approximating ruin probabilities with arbitrary claim size distributions. Furthermore, Bladt and Rojas-Nandayapa (2017) provide statistical inference procedures based on the EM algorithm to adjust phase-type scale mixtures to heavy-tailed data/distributions.…”