On occupation times in the red of Lévy risk models

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 Lévy risk processes and refracted spectrally negative Lévy 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 Cramér-Lundberg risk model with exponential claims. ∞ 0 1 (−∞,0) (X s ) ds. There exists a number of results on the occupation time O X t in the literature. For the standard Brownian motion, the distribution of O X t appeared in Lévy's [26] famous arc-sine law. This formula was generalized by Akahori [1] and Takács [36] to a Brownian motion with drift. For the classical compound Poisson process with some special jump distributions, Dos Reis [11] obtained the moment generating function of O X ∞ using a martingale approach. Zhang and Wu [37] further solved the Laplace transform of O X ∞ by considering a compound Poisson process perturbed by an independent Brownian motion.

Poissonian occupation times of spectrally negative Lévy processes with applications

“…Under continuous monitoring, we have the following expression for the distribution of [11] for more results on the distribution of occupations times). given by…”

“…For a more general treatment, Li and Palmowski [14] studied weighted occupation times. Recently, Landriault et al [11] obtained an analytical expression for the distribution of the occupation time below level 0 up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes.…”

“…Guérin and Renaud [8] studied the probability of cumulative Parisian ruin for the Cramér-Lundberg risk model with exponential claims by giving an explicit representation for the distribution of O X t . A general treatment for Lévy risk processes has been recently studied by Landriault et al [11]. On the other hand, the probability of Parisian ruin with exponential delays was first studied in [12] through the following connection between the occupation time O X ∞ and this type of Parisian ruin, for x ∈ R,…”

Sojourn Times of Gaussian Processes with Trend

“…The study of distributional properties of occupation-type functionals for Lévy processes is crucial for many applications in finance and insurance (e.g. the occupation time in red or the inverse occupation time -the time of cumulative Parisian ruin), see for instance [5,[7][8][9][10]. The number of papers dealing with occupation times (sojourn times) is huge; most of the articles discuss the derivation of Laplace transform, see the recent contributions [11][12][13][14] and references therein.…”

“…Recent paper [10] derives the density of occupation time for spectrally negative Lévy processes with exponential time horizon.…”