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 X ¯ : Y = X ¯ − X . 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 G ¯ τ a of the last supremum of X prior to τ a , the infimum X ¯ τ a and supremum X ¯ τ a of X at τ a and the undershoot a − Y τ a − and overshoot Y τ a − a of Y at τ a . 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.