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

Abstract

For the sum process $X=X^1+X^2$ of a bivariate Lévy 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 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.

Published in at http://dx.doi.org/10.1214/09-AAP601 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Ruin Probabilities and Overshoots for General Levy Insurance Risk Processes
• Overshoots and undershoots of Lévy processes

Cited by in corpus (3)

• Exact joint laws associated with spectrally negative Levy processes and applications to insurance risk theory
• Sample path behavior of a Lévy insurance risk process approaching ruin, under the Cramér-Lundberg and convolution equivalent conditions
• On exact sampling of the first passage event of Levy process with infinite Levy measure and bounded variation