Discounted Penalty Function at Parisian Ruin for L\'evy Insurance Risk Process

In the setting of a L\'evy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold $r$. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level $b$), which generalises known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the $q$-potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the $q$-scale function and the distribution of the L\'evy process.