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Teor. Veroyatnost. i Primenen., 2008, Volume 53, Issue 3, Pages 557–575 (Mi tvp2449)  

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

On Asymptotic Optimality of the Second Order in the Minimax Quickest Detection Problem of Drift Change for Brownian Motion

E. V. Burnaeva, E. A. Feinbergb, A. N. Shiryaeva

a Steklov Mathematical Institute, Russian Academy of Sciences
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This paper deals with the minimax quickest detection problem of a drift change for the Brownian motion. The following minimax risks are studied: $C(T)=\inf_{\tau\in{\mathfrak{M}}_{T}}\sup_\thetaE_\theta(\tau-\theta | \tau\ge\theta)$ and $\overline{C}(T)=\inf_{\overline{\tau}\in\overline{\mathfrak{M}}_T}\sup_\thetaE_\theta(\overline{\tau}-\theta | \overline{\tau}\ge\theta)$, where ${\mathfrak{M}}_T$ is the set of stopping times $\tau$ such that $E_\infty\tau=T$ and ${\overline{\mathfrak{M}}}_T$ is the set of randomized stopping times ${\overline{\tau}}$ such that $E_\infty{\overline{\tau}}=T$. The goal of this paper is to obtain for these risks estimates from above and from below. Using these estimates we prove the existence of stopping times, which are asymptotically optimal of the first and second orders as $T\to\infty$ (for $C(T)$ and $\overline{C}(T)$, respectively).

Keywords: disorder problem, Brownian motion, minimax risk, asymptotical optimality of the first and second orders.

DOI: https://doi.org/10.4213/tvp2449

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English version:
Theory of Probability and its Applications, 2009, 53:3, 519–536

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Received: 08.11.2007

Citation: E. V. Burnaev, E. A. Feinberg, A. N. Shiryaev, “On Asymptotic Optimality of the Second Order in the Minimax Quickest Detection Problem of Drift Change for Brownian Motion”, Teor. Veroyatnost. i Primenen., 53:3 (2008), 557–575; Theory Probab. Appl., 53:3 (2009), 519–536

Citation in format AMSBIB
\by E.~V.~Burnaev, E.~A.~Feinberg, A.~N.~Shiryaev
\paper On Asymptotic Optimality of the Second Order in the Minimax Quickest Detection Problem of Drift Change for Brownian Motion
\jour Teor. Veroyatnost. i Primenen.
\yr 2008
\vol 53
\issue 3
\pages 557--575
\jour Theory Probab. Appl.
\yr 2009
\vol 53
\issue 3
\pages 519--536

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    This publication is cited in the following articles:
    1. Li K., Polunchenko A.S., Pepelyshev A., “Analytic Evaluation of the Fractional Moments For the Quasi-Stationary Distribution of the Shiryaev Martingale on An Interval”, Commun. Stat.-Simul. Comput.  crossref  isi
    2. Polunchenko A.S., Sokolov G., “An Analytic Expression for the Distribution of the Generalized Shiryaev–Roberts Diffusion”, Methodol. Comput. Appl. Probab., 18:4, SI (2016), 1153–1195  crossref  mathscinet  zmath  isi  elib  scopus
    3. Polunchenko A.S., “Exact distribution of the Generalized Shiryaev–Roberts stopping time under the minimax Brownian motion setup”, Seq. Anal., 35:1, SI (2016), 108–143  crossref  mathscinet  zmath  isi  elib  scopus
    4. Polunchenko A.S., “On the quasi-stationary distribution of the Shiryaev–Roberts diffusion”, Seq. Anal., 36:1 (2017), 126–149  crossref  mathscinet  zmath  isi  scopus
    5. Theory Probab. Appl., 62:4 (2018), 617–631  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. Polunchenko A.S., “Asymptotic Exponentiality of the First Exit Time of the Shiryaev-Roberts Diffusion With Constant Positive Drift”, Seq. Anal., 36:3 (2017), 370–383  crossref  mathscinet  zmath  isi  scopus
    7. Theory Probab. Appl., 63:3 (2019), 464–478  mathnet  crossref  crossref  isi  elib
    8. Polunchenko A.S., Pepelyshev A., “Analytic Moment and Laplace Transform Formulae For the Quasi-Stationary Distribution of the Shiryaev Diffusion on An Interval”, Stat. Pap., 59:4, SI (2018), 1351–1377  crossref  mathscinet  zmath  isi  scopus
    9. Li K., Polunchenko A.S., “On the Convergence Rate of the Quasi- to Stationary Distribution For the Shiryaev-Roberts Diffusion”, Seq. Anal., 39:2 (2020), 214–229  crossref  mathscinet  isi
    10. Romanenkova E., Zaytsev A., Klyuchnikov N., Gruzdev A., Antipova K., Ismailova L., Burnaev E., Semenikhin A., Koryabkin V., Simon I., Koroteev D., “Real-Time Data-Driven Detection of the Rock-Type Alteration During a Directional Drilling”, IEEE Geosci. Remote Sens. Lett., 17:11 (2020), 1861–1865  crossref  isi
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