On the drawdown of completely asymmetric Levy processes

Abstract

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 ${\bar{G}}_{{\tau }_a}$ of the last supremum of $X$ prior to ${\tau }_a$, the infimum $\unl X_{τ_a}$ and supremum $\ovl X_{τ_a}$ of $X$ at ${\tau }_a$ and the undershoot $a-Y_{{\tau }_a-}$ 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 Lévy model.

arXiv commentapplications added, 26 pages, 3 figures, to appear in SPA