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Published March 2004 | public
Journal Article

On the Maximum Drawdown of a Brownian Motion


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).

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.

Additional details

August 22, 2023
October 20, 2023