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Last exit before an exponential time for spectrally negative Lévy processes. (English) Zbl 1170.60020
Summary: S. N. Chiu and C. Yin [Bernoulli 11, No. 3, 511–522 (2005; Zbl 1076.60038)] found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to \(\infty\), is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to \(\infty \), is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in [R. A. Doney, J. Lond. Math. Soc., II. Ser. 44, No. 3, 566–576 (1991; Zbl 0699.60061)].

MSC:
60G51 Processes with independent increments; Lévy processes
91B30 Risk theory, insurance (MSC2010)
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