# The first passage event for sums of dependent Lévy processes with applications to insurance risk - Mathematics > Probability

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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 thefirst upwards passage event of $X$ over a fixed barrier, caused by a jump, bythe joint distribution of five quantities: the time relative to the time of theprevious maximum, the time of the previous maximum, the overshoot, theundershoot and the undershoot of the previous maximum. The dependence betweenthe jumps of $X^1$ and $X^2$ is modeled by a L\-evy copula. We calculate thesequantities for some examples, where we pay particular attention to theinfluence of the dependence structure. We apply our findings to the ruin eventof an insurance risk process.

Author: ** Irmingard Eder, Claudia Klüppelberg**

Source: https://arxiv.org/