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In this paper, we introduce the concept of \emph{Poissonian occupation times}below level $0$ of spectrally negative L\'evy processes. In this case,occupation time is accumulated only when the process is observed to be negativeat arrival epochs of an independent Poisson process. Our results extend somewell known continuously observed quantities involving occupation times ofspectrally negative L\'evy processes. As an application, we establish a linkbetween Poissonian occupation times and insurance risk models with Parisianimplementation delays.

Poissonian occupation times of spectrally negative L\'evy processes with applications