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Teor. Veroyatnost. i Primenen., 2008, Volume 53, Issue 4, Pages 751–768 (Mi tvp2463)  

This article is cited in 10 scientific papers (total in 10 papers)

On Conditional-Extremal Problems of the Quickest Detection of Nonpredictable Times of the Observable Brownian Motion

A. N. Shiryaev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider nonpredictable stopping times $\theta=\inf\{t\le 1:B_t=\max_{0\le s\le 1}B_s\}$, $g=\sup\{t\le 1:B_t=0\}$ for the Brownian motion $B=(B_t)_{0\le t\le 1}$. The main results of the paper concern solving the following conditional-extremal problems: in classes of Markov times $\mathfrak{M}_\alpha^B(\theta)=\{\tau\le 1:P \{\tau<\theta\}\le\alpha\}$, $\mathfrak{M}_\alpha^B(g)=\{\sigma\le 1:P \{\sigma<g\}\le\alpha\}$, where $0<\alpha<1$, to describe a structure of optimal stopping times $\tau_\alpha^*(\theta)$ and $\sigma_\alpha^*(g)$, for which $E [\tau_\alpha^*(\theta)-\theta]^+=\inf_{\tau\in\mathfrak{M}_\alpha^B(\theta)}E (\tau-\theta)^+$, $E [\sigma_\alpha^*(g)-g]^+=\inf_{\sigma\in\mathfrak{M}_\alpha^B(g)}E (\sigma-g)^+$. We also consider the problems of the structure of some special stopping times in the classes $\mathfrak{M}_\alpha^B(\theta^\mu)$ and $\mathfrak{M}_\alpha^B(g^\mu)$ for the case of Brownian motion with drift $B^\mu=(B_t^\mu)_{0\le t\le 1}$, where $B_t^\mu=\mu t+B_t$.

Keywords: conditional-extremal problems, nonpredictable time, quickest detection, Brownian motion.

DOI: https://doi.org/10.4213/tvp2463

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English version:
Theory of Probability and its Applications, 2009, 53:4, 663–678

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Received: 23.07.2007

Citation: A. N. Shiryaev, “On Conditional-Extremal Problems of the Quickest Detection of Nonpredictable Times of the Observable Brownian Motion”, Teor. Veroyatnost. i Primenen., 53:4 (2008), 751–768; Theory Probab. Appl., 53:4 (2009), 663–678

Citation in format AMSBIB
\by A.~N.~Shiryaev
\paper On Conditional-Extremal Problems of the Quickest Detection of Nonpredictable Times of the Observable Brownian Motion
\jour Teor. Veroyatnost. i Primenen.
\yr 2008
\vol 53
\issue 4
\pages 751--768
\jour Theory Probab. Appl.
\yr 2009
\vol 53
\issue 4
\pages 663--678

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    This publication is cited in the following articles:
    1. du Toit J., Peskir G., Shiryaev A.N., “Predicting the last zero of Brownian motion with drift”, Stochastics, 80:2-3 (2008), 229–245  crossref  mathscinet  zmath  isi  elib  scopus
    2. du Toit J., Peskir G., “Selling a stock at the ultimate maximum”, Ann. Appl. Probab., 19:3 (2009), 983–1014  crossref  mathscinet  zmath  isi  scopus
    3. S. S. Sinelnikov, “On optimal stopping for Brownian motion with a negative drift”, Theory Probab. Appl., 56:2 (2011), 343–350  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. Bernyk V., Dalang R.C., Peskir G., “Predicting the ultimate supremum of a stable Lévy process with no negative jumps”, Ann. Probab., 39:6 (2011), 2385–2423  crossref  mathscinet  zmath  isi  elib  scopus
    5. Peskir G., “Optimal detection of a hidden target: the median rule”, Stochastic Process. Appl., 122:5 (2012), 2249–2263  crossref  mathscinet  zmath  isi  elib  scopus
    6. Kozinov I.A., Maltsev G.N., “Modifitsirovannyi algoritm obnaruzheniya razladki sluchainogo protsessa i ego primenenie pri obrabotke mnogospektralnykh dannykh”, Informatsionno-upravlyayuschie sistemy, 3:58 (2012), 9–17  elib
    7. Glover K. Hulley H. Peskir G., “Three-Dimensional Brownian Motion and the Golden Ratio Rule”, Ann. Appl. Probab., 23:3 (2013), 895–922  crossref  mathscinet  zmath  isi  elib  scopus
    8. Peskir G., “Quickest Detection of a Hidden Target and Extremal Surfaces”, Ann. Appl. Probab., 24:6 (2014), 2340–2370  crossref  mathscinet  zmath  isi  scopus
    9. Ankirchner S., Kazi-Tani N., Klein M., Kruse T., “Stopping With Expectation Constraints: 3 Points Suffice”, Electron. J. Probab., 24 (2019), 66  crossref  isi
    10. Ilicheva V.V., Guda A.N., Shevchuk P.S., “Logical Approaches to Anomaly Detection in Industrial Dynamic Processes”, Proceedings of the Fourth International Scientific Conference Intelligent Information Technologies For Industry (Iiti'19), Advances in Intelligent Systems and Computing, 1156, eds. Kovalev S., Tarassov V., Snasel V., Sukhanov A., Springer International Publishing Ag, 2020, 352–361  crossref  isi
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