zbMATH — the first resource for mathematics

The first passage event for sums of dependent Lévy processes with applications to insurance risk. (English) Zbl 1209.60029
This paper deals with first passage events and ruin probabilities for a Lévy process which is the sum of dependent Lévy processes. The dependence structure between the two processes is modeled by a Lévy copula. In particular, the authors consider a \(2\)-dimensional Lévy process \((X^1,X^2)\) and derive a quintuple law describing the first upwards passage event of \(X^1+X^2\) over a fixed barrier, 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 authors first provide the general expression for this quintuple law, and then consider two examples where explicit computations are carried out: when the jump parts of \((X^1,X^2)\) are spectrally positive compound Poisson processes, and when they are subordinators with negative drift. They also consider different Lévy copulas for modeling the dependence structure, in particular Clayton and (non-homogeneous) Archimedean Lévy copulas. Finally, they use their results to obtain asymptotic expressions for the ruin event of \(X^1+X^2\).

60G51 Processes with independent increments; Lévy processes
60G50 Sums of independent random variables; random walks
60J75 Jump processes (MSC2010)
91B30 Risk theory, insurance (MSC2010)
Full Text: DOI arXiv
[1] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121 . Cambridge Univ. Press, Cambridge. · Zbl 0861.60003
[2] Billingsley, P. (1979). Probability and Measure . Wiley, New York. · Zbl 0411.60001
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27 . Cambridge Univ. Press, Cambridge. · Zbl 0617.26001
[4] Böcker, K. and Klüppelberg, C. (2006). Multivariate models for operational risk. Quant. Finance . · Zbl 1204.91059
[5] Bregman, Y. and Klüppelberg, C. (2005). Ruin estimation in multivariate models with Clayton dependence structure. Scand. Actuar. J. 6 462-480. · Zbl 1145.91031 · doi:10.1080/03461230500362065
[6] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes . Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1052.91043
[7] Doney, R. A. (2007). Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897 . Springer, Berlin. · Zbl 1128.60036
[8] Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 91-106. · Zbl 1101.60029 · doi:10.1214/105051605000000647
[9] Eder, I. and Klüppelberg, C. (2007). Pareto Lévy measures and multivariate regular variation. · Zbl 1248.60052
[10] Esmaeili, H. and Klüppelberg, C. (2008). Parameter estimation of a bivariate compound Poisson process. · Zbl 1231.62150
[11] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events : For Insurance and Finance. Applications of Mathematics ( New York ) 33 . Springer, Berlin. · Zbl 0873.62116
[12] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Probab. 14 1378-1397. · Zbl 1061.60075 · doi:10.1214/105051604000000332
[13] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities for competing claim processes. J. Appl. Probab. 41 679-690. · Zbl 1065.60100 · doi:10.1239/jap/1091543418
[14] Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73 . Chapman & Hall, London. · Zbl 0990.62517
[15] Kallsen, J. and Tankov, P. (2006). Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivariate Anal. 97 1551-1572. · Zbl 1099.62048 · doi:10.1016/j.jmva.2005.11.001
[16] Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Probab. 25 132-141. · Zbl 0651.60020 · doi:10.2307/3214240
[17] Klüppelberg, C. and Kyprianou, A. (2006). On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43 1-5. · Zbl 1118.60071 · doi:10.1239/jap/1152413744
[18] Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 1766-1801. · Zbl 1066.60049 · doi:10.1214/105051604000000927 · euclid:aoap/1099674077
[19] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[20] Nelsen, R. B. (2006). An Introduction to Copulas , 2nd ed. Springer, New York. · Zbl 1152.62030
[21] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions . Cambridge Univ. Press, Cambridge. · Zbl 0973.60001
[22] Vigon, V. (2002). Votre Lévy rampe-t-il? J. London Math. Soc. (2) 65 243-256. · Zbl 1016.60054 · doi:10.1112/S0024610701002885
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.