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

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  • Irmingard Eder
  • Claudia Kluppelberg

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.

Suggested Citation

  • Irmingard Eder & Claudia Kluppelberg, 2009. "The first passage event for sums of dependent L\'evy processes with applications to insurance risk," Papers 0912.1925, arXiv.org.
    • Handle: RePEc:arx:papers:0912.1925
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    Cited by:

    1. Chi, Zhiyi, 2016. "On exact sampling of the first passage event of a Lévy process with infinite Lévy measure and bounded variation," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1124-1144.
    2. Esmaeili, Habib & Klüppelberg, Claudia, 2010. "Parameter estimation of a bivariate compound Poisson process," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 224-233, October.
    3. Oliver Kley & Claudia Klüppelberg & Gesine Reinert, 2016. "Risk in a Large Claims Insurance Market with Bipartite Graph Structure," Operations Research, INFORMS, vol. 64(5), pages 1159-1176, October.
    4. Dimitrina S. Dimitrova & Zvetan G. Ignatov & Vladimir K. Kaishev, 2019. "Ruin and Deficit Under Claim Arrivals with the Order Statistics Property," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 511-530, June.

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