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
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