The Bivariate regular variation of randomly weighted sums revisited in the presence of interdependence

TL;DR

This paper extends the theory of bivariate regular variation to include dependent random weights, providing new results for stochastic modeling under economic instability, with applications to ruin probabilities and risk measures.

Contribution

It introduces a dependent bivariate extension of Breiman's theorem and applies it to randomly weighted sums with interdependence, enhancing modeling flexibility.

Findings

Extended Breiman's theorem for dependent variables

Derived ruin probability estimates under interdependence

Analyzed asymptotic behavior of joint expected shortfall

Abstract

We study the closure properties of the class of Bivariate Regular Variation, symbolically BRV , in standard and nonstandard cases, with respect to the randomly weighted sums. However, we take into consideration a weak dependence structure among the random weights and the main random variables, providing more flexibility in applications. In the first main result, under the condition that the main random variables belong to BRV , we provide a dependent bivariate extension of Breiman's theorem, under some sufficient conditions. Furthermore, we continue the extension into randomly weighted sums. The main target motivation of this work is stochastic modeling of regular variation under the condition of economic instability. For this reason we provide the ruin probability estimation over finite time horizon in a two dimensional discrete time risk model, with interdependence among the insurance and financial risks, as also the asymptotic behavior of a risk measure, known as joint expected shortfall.