On the drawdown of completely asymmetric Levy processes

The {\em drawdown} process $Y$ of a completely asymmetric Lévy 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 Lévy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the level $a>0$, the time $\bar{G}_{τ_a}$ of the last supremum of $X$ prior to $τ_a$, the infimum $\unl X_{τ_a}$ and supremum $\ovl X_{τ_a}$ 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.

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