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Formulas for the Laplace Transform of Stopping Times based on Drawdowns and Drawups

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  • Hongzhong Zhang
  • Olympia Hadjiliadis

Abstract

In this work we study drawdowns and drawups of general diffusion processes. The drawdown process is defined as the current drop of the process from its running maximum, while the drawup process is defined as the current increase over its running minimum. The drawdown and the drawup are the first hitting times of the drawdown and the drawup processes respectively. In particular, we derive a closed-form formula for the Laplace transform of the probability density of the drawdown of a units when it precedes the drawup of b units. We then separately consider the special case of drifted Brownian motion, for which we derive a closed form formula for the above-mentioned density by inverting the Laplace transform. Finally, we apply the results to a problem of interest in financial risk-management and to the problem of transient signal detection and identification of two-sided changes in the drift of general diffusion processes.

Suggested Citation

  • Hongzhong Zhang & Olympia Hadjiliadis, 2009. "Formulas for the Laplace Transform of Stopping Times based on Drawdowns and Drawups," Papers 0911.1575, arXiv.org.
    • Handle: RePEc:arx:papers:0911.1575
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    Cited by:

    1. Zbigniew Palmowski & Joanna Tumilewicz, 2017. "Fair valuation of L\'evy-type drawdown-drawup contracts with general insured and penalty functions," Papers 1712.04418, arXiv.org, revised Feb 2018.
    2. Lochowski, Rafal Marcin, 2011. "Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift -- Their characteristics and applications," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 378-393, February.

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