The drawdown process Y of a completely asymmetric Lévy process X is equal to X reflected at its running supremum X̄: Y=X̄-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 Ḡ τa of the last supremum of X prior to τa, the infimum X̄ τa and supremum X̄ τa 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. © 2012 Elsevier B.V. All rights reserved.
Mijatović, A., & Pistorius, M. R. (2012). On the drawdown of completely asymmetric Lévy processes. Stochastic Processes and Their Applications, 122(11), 3812–3836. https://doi.org/10.1016/j.spa.2012.06.012