The first passage event for sums of dependent L\'evy processes with applications to insurance risk

TL;DR

This paper derives a comprehensive law describing the first passage event for sums of dependent bivariate Lévy processes, with applications to insurance risk, highlighting the impact of dependence structures on ruin probabilities.

Contribution

It introduces a quintuple law for the first passage event of dependent Lévy processes and analyzes the influence of dependence via Lévy copulas, with applications to insurance risk.

Findings

Derived a quintuple law for dependent Lévy processes.

Analyzed the effect of dependence structures on first passage events.

Applied results to ruin probabilities in insurance risk models.

Abstract

For the sum process $X=X^1+X^2$ of a bivariate L\'evy process $(X^1,X^2)$ with possibly dependent components, we derive a quintuple law describing the first upwards passage event of $X$ over a fixed barrier, caused by a jump, by the joint distribution of five quantities: the time relative to the time of the previous maximum, the time of the previous maximum, the overshoot, the undershoot and the undershoot of the previous maximum. The dependence between the jumps of $X^1$ and $X^2$ is modeled by a L\'evy copula. We calculate these quantities for some examples, where we pay particular attention to the influence of the dependence structure. We apply our findings to the ruin event of an insurance risk process.