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On extreme ruinous behaviour of Lévy insurance risk processes. (English) Zbl 1118.60071
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.

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
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