On the drawdown of completely asymmetric Lévy processes

The drawdown process Y of a completely asymmetric Lévy process X is equal to X reflected at its running supremum View the MathML source: View the MathML source. In this paper we explicitly express in terms of the scale function and the Lévy measure of X the law of the sextuple of the first-passage time of Y over the level a>0, the time View the MathML source of the last supremum of X prior to τa, the infimum View the MathML source and supremum View the MathML source of X at τa and the undershoot a−Yτa− and overshoot Yτa−a of Y at τa. 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.