Parisian ruin probability for spectrally negative Lévy processes

Abstract

In this note we give, for a spectrally negative Lévy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period $r$. The formula involves only the scale function of the spectrally negative Lévy process and the distribution of the process at time $r$.