In this paper, we complement the existing literature on the occupation time in the red (below level 0) of a spectrally negative L\'evy process, and later extend the analysis to the refracted spectrally negative L\'evy process. For both classes of processes, we derive an explicit expression for the distribution of such occupation timeup to an independent exponential time. As an application, we consider theinverse occupation time (also known as the time of cumulative Parisian ruin in \mbox{%DIFAUXCMD guerinrenaud20150pt%DIFAUXCMD ), where ruin is deemed to occur at the earliest time the risk process cumulatively stays below a critical level over a pre-determined time-threshold. Some particular examples of spectrally negative L\'evy processes are also examined in more detail} vy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running maximum. By a further Laplace inversion to our results, we obtain the distribution of the occupation time up to a finite time horizon for refracted Brownian motion risk process and refracted Cram\'{e}r-Lundberg risk model with exponential claims} .
In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level 0) up to an (independent) exponential horizon for spectrally negative L\'{e refracted spectrally negative 0pt%DIFAUXCMD ), where ruin is deemed to occur at the earliest time the risk process cumulatively stays below a critical level over a pre-determined time-threshold. Some particular examples of spectrally negative L\'evy processes are also examined in more detail} L\'{evy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running maximum. By a further Laplace inversion to our results, we obtain the distribution of the occupation time up to a finite time horizon for refracted Brownian motion risk process and refracted Cram\'{e}r-Lundberg risk model with exponential claims} .
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complement the existing literature on
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L\'evy process. For both classes of processes, we derive an explicit expression for the distribution of such occupation timeup to an independent exponential time. As an application, we consider the
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