Klüppelberg, C.; Kyprianou, A. E. On extreme ruinous behaviour of Lévy insurance risk processes. (English) Zbl 1118.60071 J. Appl. Probab. 43, No. 2, 594-598 (2006). Authors’ abstract: We show how new fluctuation identities and their associated asymptotics, given by V. Vigon [J. Lond. Math. Soc., II. Ser. 65, No. 1, 243–256 (2002; Zbl 1016.60054)], the authors and R.A. Maller [Ann. Appl. Probab. 14, No. 4, 1766–1801 (2004; Zbl 1066.60049)] and R. A. Doney and A. E. Kyprianou [ibid. 16, No. 1, 91–106 (2006; Zbl 1101.60029)], provide the basis for establishing, in an elementary way, asymptotic overshoot and undershoot distributions for a general class of Lévy insurance risk processes. The results bring the earlier conclusions of S. Asmussen and C. Klüppelberg [Stochastic Processes Appl. 64, No. 1, 103–125 (1996; Zbl 0879.60020)] for the Cramér-Lundberg process into greater generality. Reviewer: Bero Roos (Hamburg) Cited in 20 Documents MSC: 60K05 Renewal theory 91B30 Risk theory, insurance (MSC2010) 60G70 Extreme value theory; extremal stochastic processes 60J55 Local time and additive functionals 60K15 Markov renewal processes, semi-Markov processes Keywords:Lévy process; insurance risk process; ruin; extreme value theory PDF BibTeX XML Cite \textit{C. Klüppelberg} and \textit{A. E. Kyprianou}, J. Appl. Probab. 43, No. 2, 594--598 (2006; Zbl 1118.60071) Full Text: DOI References: [1] Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64 , 103–125. · Zbl 0879.60020 · doi:10.1016/S0304-4149(96)00087-7 [2] Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16 , 91–106. · Zbl 1101.60029 · doi:10.1214/105051605000000647 [3] Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49 , 335–347. · Zbl 0397.60024 · doi:10.1007/BF00535504 [4] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. · Zbl 0873.62116 [5] Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails , eds R. Adler, R. Feldman and M. S. Taqqu, Birkhäuser, Boston, MA, pp. 435–459. · Zbl 0923.62021 [6] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004a). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14 , 1378–1397. · Zbl 1061.60075 · doi:10.1214/105051604000000332 [7] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004b). Ruin probabilities for competing claim processes. J. Appl. Prob. 41 , 679–690. · Zbl 1065.60100 · doi:10.1239/jap/1091543418 [8] Klüppelberg, C., Kyprianou, A. E. and Maller, R. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14 , 1766–1801. · Zbl 1066.60049 · doi:10.1214/105051604000000927 · euclid:aoap/1099674077 [9] Vigon, V. (2002). Votre Lévy rampe-t-il? J. London Math. Soc. 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.