Approximation of Ruin Probabilities via Erlangized Scale Mixtures

TL;DR

This paper introduces a new numerical method using Erlangized scale mixtures to approximate ruin probabilities in risk processes with complex claim size distributions, including heavy-tailed cases.

Contribution

It extends existing ruin probability calculations by employing Erlangized scale mixtures, enabling approximation for a broader class of claim size distributions.

Findings

Provides explicit formulas for ruin probabilities with Erlangized scale mixtures.

Constructs bounds for approximation errors in ruin probability calculations.

Demonstrates effectiveness with a heavy-tailed claim size example.

Abstract

In this paper, we extend an existing scheme for numerically calculating the probability of ruin of a classical Cram\'er--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 $\Pi\star G$ of a nonnegative distribution $\Pi$ 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 $\Pi\star G$ 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.