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# On Probability Characteristics of "Downfalls" in a Standard Brownian Motion

## Author

Listed:

() (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

• A.N. Shiryaev

(SMI - Steklov Mathematical Institute - RAS - Russian Academy of Sciences [Moscow])

• Marc Yor

() (IUF - Institut Universitaire de France - M.E.N.E.S.R. - Ministère de l'Éducation nationale, de l’Enseignement supérieur et de la Recherche, LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

## Abstract

For a Brownian motion $B=(B_t)_{t\le 1}$ with $B_0=0$, {\bf E}$B_t=0$, {\bf E}$B_t^2=t$ problems of probability distributions and their characteristics are considered for the variables $$\begin{array}{c} {\mathbb D} =\displaystyle\sup_{0\le t\le t'\le 1}(B_t-B_{t'}),\qquad {\mathbb D}_1=B_\sigma-\inf_{\sigma\le t'\le 1}B_{t'}, \\ {\mathbb D}_2=\displaystyle\sup_{0\le t\le\sigma'}B_{t}-B_{\sigma'}, \end{array}$$ where $\sigma$ and $\sigma'$ are times (non-Markov) of the absolute maximum and absolute minimum of the Brownian motion on $[0,1]$ (i.e., $B_\sigma=\sup_{0\le t\le 1}B_t$, $B_{\sigma'}=\inf_{0\le t'\le 1}B_{t'}$).

## Suggested Citation

• Raphaël Douady & A.N. Shiryaev & Marc Yor, 2000. "On Probability Characteristics of "Downfalls" in a Standard Brownian Motion," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01477104, HAL.
• Handle: RePEc:hal:cesptp:hal-01477104
DOI: 10.1137/S0040585X97977306
Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-01477104
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## Citations

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Cited by:

1. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
2. Ola Mahmoud, 2015. "The Temporal Dimension of Risk," Papers 1501.01573, arXiv.org, revised Jun 2016.
3. Kyo Yamamoto & Seisho Sato & Akihiko Takahashi, 2009. "Probability Distribution and Option Pricing for Drawdown in a Stochastic Volatility Environment," CIRJE F-Series CIRJE-F-625, CIRJE, Faculty of Economics, University of Tokyo.
4. Kyo Yamamoto & Seisho Sato & Akihiko Takahashi, 2008. "Probability Distribution and Option Pricing for Drawdown in a Stochastic Volatility Environment ( Revised in May 2009; Electronic version of an article will be published in "International Journal," CARF F-Series CARF-F-138, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
5. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
6. Zhang, Hongzhong & Leung, Tim & Hadjiliadis, Olympia, 2013. "Stochastic modeling and fair valuation of drawdown insurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 840-850.

### Keywords

Brownian motion; downfall;

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