In this paper, we introduce the concept of Poissonian occupation times below level 0 of a spectrally negative Lévy process. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known continuously observed quantities involving occupation times of spectrally negative Lévy processes. As an application, we establish a link between Poissonian occupation times and insurance risk models with Parisian implementation delays.

中文翻译：

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带应用的谱负列维过程的泊松占用时间

在本文中，我们介绍了谱负 Lévy 过程的 0 级以下的泊松占用时间的概念。在这种情况下，仅当在独立泊松过程的到达时期观察到该过程为负时，才会累积占用时间。我们的结果扩展了一些众所周知的连续观察到的数量，这些数量涉及光谱负列维过程的占用时间。作为一个应用程序，我们在泊松占用时间和保险风险模型与巴黎实施延迟之间建立了联系。