Zhitlukhin, Mikhail V.; Muravlev, Alexey A.; Shiryaev, Albert N. On confidence intervals for Brownian motion change point times. (On confidence intervals for Brownian motion changepoint times.) (English. Russian original) Zbl 1369.62205 Russ. Math. Surv. 71, No. 1, 159-160 (2016); translation from Usp. Mat. Nauk 71, No. 1, 171-172 (2016). From the text: We consider a sequential problem of finding the best confidence interval for a changepoint time of a Brownian motion. Namely, let \(B=(B_t)_{t\geqslant 0}\) be a standard Brownian motion defined on a probability space \((\Omega,\mathscr{F},\mathsf{P})\) and let \(\theta\) be an unobservable non-negative random variable which does not depend on \(B\) and has a known distribution. Observable is the process\[ X_t = \mu(t-\theta)^+ +B_t, \quad t\geqslant 0, \] where \(\mu\neq0\) is a known parameter. Thus, at time \(\theta\) a ‘disorder’ occurs, which is manifested through the change of the drift coefficient from zero to \(\mu\).Let \(\mathfrak{M}\) denote the class of stopping times \(\tau\) of the filtration \(\mathscr{F}_t^X=\sigma(X_s;s\leqslant t)\) generated by the observable process. The changepoint detection problem consists in finding a stopping time \(\tau^*\in\mathfrak{M}\) which is the closest to \(\theta\) in some sense. This paper deals with the proximity criterion in the form of a confidence interval of a given length \(h\), that is, in finding a \(\tau^*\) which maximizes the probability \(\mathsf{P}(|\tau-\theta|\leqslant h)\). MSC: 62L10 Sequential statistical analysis 60G40 Stopping times; optimal stopping problems; gambling theory 60J65 Brownian motion Keywords:Brownian motion; confidence intervals PDF BibTeX XML Cite \textit{M. V. Zhitlukhin} et al., Russ. Math. Surv. 71, No. 1, 159--160 (2016; Zbl 1369.62205); translation from Usp. Mat. Nauk 71, No. 1, 171--172 (2016) Full Text: DOI OpenURL