Gerber–Shiu Function at Draw-Down Parisian Ruin Time for the Spectrally Negative Lévy Risk Process

Motivated by Baurdoux et al. (J Appl Probab 53:572–584, 2016) andWang and Zhou (Appl Probab 52:1–33, 2020), in this paperweintroduce the general draw-down feature into the study of the Parisian ruin with exponential implementation delays. For the spectrally negative Lévy processeswith paths of bounded and unbounded variation, we study the exit problem and the Gerber–Shiu function involving the draw-down Parisian ruin time with exponential implementation delays, via an excursion approach. In our setting, the Gerber–Shiu function considers joint law of the time of drawdown Parisian ruin with exponential delays, the running maximum until the drawdown Parisian ruin with exponential delays, and the surplus at drawdown Parisian ruin with exponential delays. The solutions to the two-sided exit problem and theGerber–Shiu function at the draw-down Parisian ruin time are obtained in terms of the scale functions associated with the underlying spectrally negative Lévy process.