Parametric divisibility of stochastic losses

TL;DR

This paper explores the property of n-divisibility in probability distributions, focusing on Gamma distributions, and proposes methods to approximate the distribution of convolution roots for modeling dependence in insurance losses.

Contribution

It demonstrates how the Gamma distribution's parametric divisibility can be used to efficiently approximate convolution roots of other distributions, aiding dependence modeling.

Findings

Gamma distributions allow exact parametric extraction of convolution roots.

The proposed algorithms enable efficient approximation of distribution pieces.

Applications include modeling dependence in insurance loss data.

Abstract

A probability distribution is n-divisible if its nth convolution root exists. While modeling the dependence structure between several (re)insurance losses by an additive risk factor model, the infinite divisibility, that is the $n$-divisibility for all $n \in\mathbb N$, is a very desirable property. Moreover, the capacity to compute the distribution of a piece (i.e., a convolution root) is also desirable. Unfortunately, if many useful distributions are infinitely divisible, computing the distributions of their pieces is usually a challenging task that requires heavy numerical computations. However, in a few selected cases, particularly the Gamma case, the extraction of the distribution of the pieces can be performed fully parametrically, that is with negligible numerical cost and zero error. We show how this neat property of Gamma distributions can be leveraged to approximate the pieces of other distributions, and we provide several illustrations of the resulting algorithms.