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On the Maximum Drawdown of a Brownian Motion

Magdon-Ismail, Malik and Atiya, Amir F. and Pratap, Amrit and Abu-Mostafa, Yaser S. (2004) On the Maximum Drawdown of a Brownian Motion. Journal of Applied Probability, 41 (1). pp. 147-161. ISSN 0021-9002. doi:10.1239/jap/1077134674.

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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).

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Additional Information:© 2004 Applied Probability Trust. Received 20 February 2003; revision received 18 July 2003. We thank Greg C. Bond at the Baker Investment Group for some useful comments and discussions. We thank Emmanuel Acar at Bank of America for useful discussion and pointers to existing results. Finally, we thank an anonymous referee for useful comments.
Subject Keywords:Random walk; asymptotic distribution; expected maximum drawdown
Issue or Number:1
Classification Code:2000 Mathematics Subject Classification: Primary 60G50; 60G51
Record Number:CaltechAUTHORS:20190702-143758047
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Official Citation:Magdon-Ismail, M., Atiya, A., Pratap, A., & Abu-Mostafa, Y. (2004). On the maximum drawdown of a Brownian motion. Journal of Applied Probability, 41(1), 147-161. doi:10.1239/jap/1077134674
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:96893
Deposited By: Tony Diaz
Deposited On:08 Jul 2019 17:17
Last Modified:16 Nov 2021 17:24

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