Gerber-Shiu functionals at Parisian ruin for L\'evy insurance risk processes

Inspired by works of Landriault et al. \cite{LRZ-0, LRZ}, we study discounted penalties at ruin for surplus dynamics driven by a spectrally negative L\'evy process with Parisian implementation delays. To be specific, we study the so-called Gerber-Shiu functional for a ruin model where at each time the surplus process goes negative, an independent exponential clock with rate $q>0$ is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative L\'evy processes and relies on the theory of the so-called scale functions. In particular, our results extend recent results of Landriault et al. \cite{LRZ-0, LRZ}.