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On the maximum increase and decrease of one-dimensional diffusions

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  • Salminen, Paavo
  • Vallois, Pierre

Abstract

In this paper the joint distribution of the maximum increase and the maximum decrease up to a first hitting time is calculated for a regular one-dimensional diffusion. Moreover, it is shown that the process given by the maximum decrease when the hitting level is the “time” parameter is a pure jump Markov process and its generator is found. As examples, Brownian motion and three dimensional Bessel process are analyzed more in detail.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:9:p:5592-5604
    DOI: 10.1016/j.spa.2020.04.001
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    References listed on IDEAS

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    1. Alexei Chekhlov & Stanislav Uryasev & Michael Zabarankin, 2005. "Drawdown Measure In Portfolio Optimization," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 13-58.
    2. Aleksandar Mijatovic & Martijn R. Pistorius, 2011. "On the drawdown of completely asymmetric Levy processes," Papers 1103.1460, arXiv.org, revised Sep 2012.
    3. 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.
    4. Mijatović, Aleksandar & Pistorius, Martijn R., 2012. "On the drawdown of completely asymmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3812-3836.
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