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

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

2009-12-10

arxiv.org/abs/0912.1925v1

Annals of Applied Probability 2009, Vol. 19, No. 6, 2047-2079

Mathematics

Probability

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

Scientific paper

10.1214/09-AAP601

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.

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