# Formulas for the Laplace Transform of Stopping Times based on Drawdowns and Drawups - Mathematics > Probability

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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 itsrunning maximum, while the drawup process is defined as the current increaseover its running minimum. The drawdown and the drawup are the first hittingtimes of the drawdown and the drawup processes respectively. In particular, wederive a closed-form formula for the Laplace transform of the probabilitydensity of the drawdown of a units when it precedes the drawup of b units. Wethen separately consider the special case of drifted Brownian motion, for whichwe derive a closed form formula for the above-mentioned density by invertingthe Laplace transform. Finally, we apply the results to a problem of interestin financial risk-management and to the problem of transient signal detectionand identification of two-sided changes in the drift of general diffusionprocesses.

Author: ** Hongzhong Zhang, Olympia Hadjiliadis**

Source: https://arxiv.org/