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Poissonian potential measures for Lévy risk models. (English) Zbl 1416.91198

Summary: This paper studies the potential (or resolvent) measures of spectrally negative Lévy processes killed on exiting (bounded or unbounded) intervals, when the underlying process is observed at the arrival epochs of an independent Poisson process. Explicit representations of these so-called Poissonian potential measures are established in terms of newly defined Poissonian scale functions. Moreover, Poissonian exit measures are explicitly solved by finding a direct relation with Poissonian potential measures. Our results generalize [H. Albrecher et al., Bernoulli 22, No. 3, 1364–1382 (2016; Zbl 1338.60125)] in which Poissonian exit identities are solved. As an application of Poissonian potential measures, we extend the Gerber-Shiu analysis in [E. J. Baurdoux et al., J. Appl. Probab. 53, No. 2, 572–584 (2016; Zbl 1344.60046)] to a (more general) Parisian risk model subject to Poissonian observations.

MSC:

91B30 Risk theory, insurance (MSC2010)
60G51 Processes with independent increments; Lévy processes
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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