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On the drawdown of completely asymmetric Levy processes

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  • Aleksandar Mijatovic
  • Martijn R. Pistorius

Abstract

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.

Suggested Citation

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

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    Cited by:

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    2. Salminen, Paavo & Vallois, Pierre, 2020. "On the maximum increase and decrease of one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5592-5604.
    3. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
    4. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, December.
    5. Zied Ben-Salah & H'el`ene Gu'erin & Manuel Morales & Hassan Omidi Firouzi, 2014. "On the Depletion Problem for an Insurance Risk Process: New Non-ruin Quantities in Collective Risk Theory," Papers 1406.6952, arXiv.org.
    6. Zhang, Gongqiu & Li, Lingfei, 2023. "A general method for analysis and valuation of drawdown risk," Journal of Economic Dynamics and Control, Elsevier, vol. 152(C).

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