A note on joint occupation times of spectrally negative Lévy risk processes with tax

In this paper we consider the joint Laplace transform of occupation times over disjoint intervals for spectrally negative Lévy processes with a general loss-carry-forward taxation structure. This tax structure was first introduced by Albrecher and Hipp in their paper in 2007. We obtain representations of the joint Laplace transforms in terms of scale functions and the Lévy measure associated with the driven spectrally negative Lévy processes. Two numerical examples, i.e. a Brownian motion with drift and a compound Poisson model, are provided at the end of this paper and explicit results are presented with discussions.