On the maximum drawdown of a Brownian motion
Malik Magdon-Ismail, Amir F. Atiya, Amrit Pratap, and Yaser S. Abu-Mostafa
Source: J. Appl. Probab.
Volume 41, Number 1
(2004), 147-161.
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).
Primary Subjects: 60G50, 60G51
Keywords: Random walk; asymptotic distribution; expected maximum drawdown
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1077134674
Digital Object Identifier: doi:10.1239/jap/1077134674
Mathematical Reviews number (MathSciNet):
MR2036278
Zentralblatt MATH identifier:
1051.60083
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