On the drawdown of completely asymmetric Levy processes

The {\em drawdown} process $Y$ of a completely asymmetric L'{e}vy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the L'{e}vy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the level $a>0$, the time ${\bar{G}}_{{\tau }_a}$ of the last supremum of $X$ prior to ${\tau }_a$, the infimum $\text{unl}{X}_{{\tau }_a}$ and supremum $\ovl X_{\tau_a}$of$X$at$\tau_a$andtheundershoot$a - Y_{\tau_a-}$ and overshoot $Y_{{\tau }_a}-a$ of $Y$ at ${\tau }_a$. As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential L'{e}vy model.