Discounted penalty function at Parisian ruin for Lévy insurance risk process

In the setting of a Lévy 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 generalizes 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 Lévy process.