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

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