On confidence intervals for Brownian motion change point times. (On confidence intervals for Brownian motion changepoint times.) (English. Russian original) Zbl 1369.62205

Zhitlukhin, Mikhail V.; Muravlev, Alexey A.; Shiryaev, Albert N.

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