The first passage event for sums of dependent Lévy processes with applications to insurance risk

Abstract

For the sum process X=X¹+X² of a bivariate Lévy process (X¹, X²) 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 X¹ and X² is modeled by a Lévy 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.