Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-13T08:49:04.374Z Has data issue: false hasContentIssue false
  • English
  • Français

On the maximum drawdown of a Brownian motion

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Malik Magdon-Ismail ,
Amir F. Atiya ,
Amrit Pratap  and
Yaser S. Abu-Mostafa
Malik Magdon-Ismail*
Affiliation:
Rensselaer Polytechnic Institute
Amir F. Atiya*
Affiliation:
Cairo University
Amrit Pratap*
Affiliation:
Caltech, Pasadena
Yaser S. Abu-Mostafa*
Affiliation:
Caltech, Pasadena
*
∗ Postal address: Department of Computer Science, Rensselaer Polytechnic Institute, Room 207 Lally Building, 110 8th Street, Troy, NY 12180, USA. Email address: magdon@rpi.edu
∗∗ Postal address: Department of Computer Engineering, Cairo University, Giza, Egypt. Email address: amir@alumni.caltech.edu
∗∗∗ Postal address: Department of Computer Science, Caltech, MC 136-93, Pasadena, CA 91125, USA. Email address: apratap@caltech.edu
∗∗∗∗ Postal address: Department of Electrical Engineering and Department of Computer Science, Caltech, MC 136-93, Pasadena, CA 91125, USA. Email address: yaser@caltech.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The maximum drawdown at time T of a random process on [0,T] can be defined informally as the largest drop from a peak to a trough. In this paper, we investigate the behaviour of this statistic for a Brownian motion with drift. In particular, we give an infinite series representation of its distribution and consider its expected value. When the drift is zero, we give an analytic expression for the expected value, and for nonzero drift, we give an infinite series representation. For all cases, we compute the limiting (T → ∞) behaviour, which can be logarithmic (for positive drift), square root (for zero drift) or linear (for negative drift).

Keywords

Random walkasymptotic distributionexpected maximum drawdown

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; L'evy processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 147 - 161
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berger, A. W., and Whitt, W. (1995). Maximum values in queueing processes. Prob. Eng. Inf. Sci. 9, 375–409.CrossRefGoogle Scholar
Blasi, A. (1976). On a random walk between a reflecting and an absorbing barrier. Ann. Prob. 4, 695–696.Google Scholar
Bond, G. C. (2003). Green's function for a zero drift reflected Brownian motion and the expected maximum drawdown. Submitted.Google Scholar
Dominé, M. (1996). First passage time distribution of a Wiener process with drift concerning two elastic barriers. J. Appl. Prob. 33, 164–175.Google Scholar
Douady, R., Shiryaev, A., and Yor, M. (2000). On probability characteristics of downfalls in a standard Brownian motion. Theory Prob. Appl. 44, 29–38.Google Scholar
Feller, W. (1951). The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Statist. 22, 427–432.Google Scholar
Gradshteyn, I. S., and Ryzhik, I. M. (1980). Table of Integrals, Series and Products. Corrected and Enlarged Edition. Academic Press, San Diego, CA.Google Scholar
Graversen, S. E., and Shiryaev, A. (2000). An extension of P. Lévy's distributional properties to the case of a Brownian motion with drift. Bernoulli 6, 615–620.Google Scholar
Weesakul, B. (1961). The random walk between a reflecting and an absorbing barrier. Ann. Math. Statist. 32, 765–769.CrossRefGoogle Scholar