Geodesic
Probability characteristics of downfalls of Brownian motion with drift
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 3, pp. 570-579

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This work studies the distribution of maximum downfall and downfall from maximum for Brownian motion with drift. Contrary to the usual measures of risk, these statistics do not use the value of process at time 0 as a reference point for measuring losses. Instead, these downfalls choose two points inside the interval. In particular maximum downfall chooses two consecutive moments to maximize the difference between the process values in these two points. The downfall from maximum uses the point of absolute maximum as an initial point and the consecutive local minimum as the final point. The main result of this work is the Laplace transform with respect to the length of the interval for the cumulative distribution function of downfalls. The Laplace transform turns out to be in this case more convenient for analytical work than explicit formulae. The distribution function is known only for the maximum downfall. It is expressed as an infinite sum with coefficients which are consecutive solutions of the equation which cannot be solved analytically. The inversion of the Laplace transform is not computationally more difficult than evaluation of the infinite sum which represents the explicit cumulative distribution function of the maximum downfall.
Keywords: downfalls of Brownian motion. 
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     author = {S. N. Lobanov},
     title = {Probability characteristics of downfalls of {Brownian} motion with drift},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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S. N. Lobanov. Probability characteristics of downfalls of Brownian motion with drift. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 3, pp. 570-579. http://geodesic.mathdoc.fr/item/TVP_2005_50_3_a10/