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

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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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
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• http://mi.mathnet.ru/eng/tvp2449
• https://doi.org/10.4213/tvp2449
• http://mi.mathnet.ru/eng/tvp/v53/i3/p557

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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.
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
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
4. Polunchenko A.S., “On the quasi-stationary distribution of the Shiryaev–Roberts diffusion”, Seq. Anal., 36:1 (2017), 126–149
5. Theory Probab. Appl., 62:4 (2018), 617–631
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
7. Theory Probab. Appl., 63:3 (2019), 464–478
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
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
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
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