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On the maximum drawdown of a Brownian motion
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).
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“It highlights the worst loss an investor could have faced and is commonly used to assess risk and volatility (Magdon-Ismail et al., 2004; Chekhlov et al., 2005).”
From StockBench: Can LLM Agents Trade Stocks Profitably In Real-world Markets? (2025) · cites this paper as Magdon-Ismail et al., 2004
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StockBench: Can LLM Agents Trade Stocks Profitably In Real-world Markets?
Chen, Y., Yao, Z., Liu, Y., Xin, A., Ye, J., Yu, J., Hou, L., & Li, J. (2025) · arXiv · Cited by 62
Metadata
- Authors
- Magdon-Ismail, M., Atiya, A. F., Pratap, A., & Abu-Mostafa, Y. S.
- Affiliations
- Cairo University (EG)
- California Institute of Technology (US)
- Corresponding author
- Atiya, Amir F.
- Pratap, Amrit
- ORCID
- Atiya, Amir F.
- Venue
- Journal of Applied Probability
- Publisher
- Cambridge University Press (CUP)
- Published
- 2004-03
- Volume
- 41
- Issue
- 1
- Pages
- 147-161
- ISSN
- 0021-9002 (ISSN-L)
- 1475-6072
- OpenAlex
- W2119657026
- Cited by
- 180
- References
- 0
- Citations by year
- 2026: 7 · 2025: 12 · 2024: 7 · 2023: 11 · 2022: 4 · 2021: 7 · 2020: 11 · 2019: 16 · 2018: 7 · 2017: 7 · 2016: 12 · 2015: 10 · 2014: 8 · 2013: 14 · 2012: 7
- FWCI
- 5.21
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Applied Probability (publishedVersion)
- CaltechAUTHORS (California Institute of Technology) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Social Sciences › Economics, Econometrics and Finance › Finance › Stochastic processes and financial applications
- Social Sciences › Decision Sciences › Management Science and Operations Research › Probability and Risk Models
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Keywords
- maximum drawdown, Brownian motion with drift, infinite series representation, asymptotic behavior, extreme value statistics, random walks
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