An excursion theoretic approach to Parisian ruin problem (Q6607483)

For a surplus process modelled by \(X\), the Parisian ruin time with duration \(\gamma>0\) is defined as the first time when the process has continuously stayed below \(0\) for a period of length \(\gamma\), i.e. when an excursion below \(0\) of the surplus process has first reached length \(\gamma.\) Applying excursion theory, the authors re-express several well studied fluctuation quantities associated to Parisian ruin for Lévy risk processes in terms of integrals with respect to the corresponding excursion measure. \newline It is shown that these new expressions reconcile with the previous results on the Parisian ruin problem.