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On the drawdown of completely asymmetric Levy processes


  • Aleksandar Mijatovic
  • Martijn R. Pistorius


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 $\unl X_{\tau_a}$ and supremum $\ovl X_{\tau_a}$ 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 L\'{e}vy model.

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  • Aleksandar Mijatovic & Martijn R. Pistorius, 2011. "On the drawdown of completely asymmetric Levy processes," Papers 1103.1460,, revised Sep 2012.
  • Handle: RePEc:arx:papers:1103.1460

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    References listed on IDEAS

    1. Olympia Hadjiliadis & Jan Vecer, 2006. "Drawdowns preceding rallies in the Brownian motion model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 403-409.
    2. Pospisil, Libor & Vecer, Jan & Hadjiliadis, Olympia, 2009. "Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2563-2578, August.
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    Cited by:

    1. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786,

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