On the drawdown of completely asymmetric Lévy processes

The drawdown   process YY of a completely asymmetric Lévy process XX is equal to XX reflected at its running supremum X¯: Y=X¯−X. In this paper we explicitly express in terms of the scale function and the Lévy measure of XX the law of the sextuple of the first-passage time of YY over the level a>0a>0, the time G¯τa of the last supremum of XX prior to τaτa, the infimum X¯τa and supremum X¯τa of XX at τaτa and the undershoot a−Yτa−a−Yτa− and overshoot Yτa−aYτa−a of YY at τaτa. As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential Lévy model.