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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: [email protected]
∗∗ Postal address: Department of Computer Engineering, Cairo University, Giza, Egypt. Email address: [email protected]
∗∗∗ Postal address: Department of Computer Science, Caltech, MC 136-93, Pasadena, CA 91125, USA. Email address: [email protected]
∗∗∗∗ Postal address: Department of Electrical Engineering and Department of Computer Science, Caltech, MC 136-93, Pasadena, CA 91125, USA. Email address: [email protected]
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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 

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