On occupation times in the red of L\'evy risk models

David Landriault; Bin Li; Mohamed Amine Lkabous

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arXiv · 1903.03721

On occupation times in the red of L\'evy risk models

David Landriault·Bin Li·Mohamed Amine Lkabous

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Abstract

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}vy risk processes and refracted spectrally negative L\'{e}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.

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David Landriault, Bin Li, Mohamed Amine Lkabous. 2019-03-09. On occupation times in the red of L\'evy risk models. https://arxiv.org/abs/1903.03721

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