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On a Sparre Andersen risk model perturbed by a spectrally negative Lévy process. (English) Zbl 1408.91109

Summary: In this paper, we consider a Sparre Andersen risk model perturbed by a spectrally negative Lévy process (SNLP). Assuming that the interclaim times follow a Coxian distribution, we show that the Laplace transforms and defective renewal equations for the Gerber-Shiu functions can be obtained by employing the roots of a generalized Lundberg equation. When the SNLP is a combination of a Brownian motion and a compound Poisson process with exponential jumps, explicit expressions and asymptotic formulas for the Gerber-Shiu functions are obtained for exponential claim size distribution and heavy-tailed claim size distribution, respectively.

MSC:

91B30 Risk theory, insurance (MSC2010)
91B70 Stochastic models in economics
60G51 Processes with independent increments; Lévy processes
60K05 Renewal theory
60G70 Extreme value theory; extremal stochastic processes
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