Background risk model in presence of heavy tails under dependence

TL;DR

This paper studies dependence structures with heavy tails, establishing max-sum equivalence under a new dependence concept and analyzing ruin probabilities and risk measures in multivariate heavy-tailed models.

Contribution

It introduces Generalized Tail Asymptotic Independence and proves max-sum equivalence under broad dependence, with applications to ruin probabilities and risk measures in heavy-tailed models.

Findings

Max-sum equivalence holds under the new dependence structure.

Asymptotic ruin probabilities are characterized in bi-variate renewal models.

Tail Distortion Risk Measure behavior is analyzed using multivariate regular variation.

Abstract

In this paper, we examine two problems on applied probability, which are directly connected with the dependence in presence of heavy tails. The first problem, is related to max-sum equivalence of the randomly weighted sums in bi-variate set up. Introducing a new dependence, called Generalized Tail Asymptotic Independence, we establish the bi-variate max-sum equivalence, under a rather general dependence structure, when the primary random variables follow distributions from the intersection of the dominatedly varying and the long tailed distributions. On base of this max-sum equivalence, we provide a result about the asymptotic behavior of two kinds of ruin probabilities, over a finite time horizon, in a bi-variate renewal risk model, with constant interest rate. The second problem, is related to the asymptotic behavior of the Tail Distortion Risk Measure, in a static portfolio, called Background Risk Model. In opposite to other approaches on this topic, we use a general enough assumption, that is based on multivariate regular variation.