Abstract. In this paper, we investigate Parisian ruin for a Lévy surplus process with an adaptive premium rate, namely a refracted Lévy process. Our main contribution is a generalization of the result in [13] for the probability of Parisian ruin of a standard Lévy insurance risk process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Despite the more general setup considered here, our main result is as compact and has a similar structure. Examples are provided.

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.

In this short paper, we study a VaR-type risk measure introduced by Guérin and Renaud and which is based on cumulative Parisian ruin. We derive some properties of this risk measure and we compare it to the risk measures of Trufin et al. and Loisel and Trufin.

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