On Conditional-Extremal Problems of the Quickest Detection of Nonpredictable Times of the Observable Brownian Motion

We consider nonpredictable stopping times $\theta=\inf\{t\le 1:B_t=\max_{0\le s\le 1}B_s\}$, $g=\sup\{t\le 1:B_t=0\}$ for the Brownian motion $B=(B_t)_{0\le t\le 1}$. The main results of the paper concern solving the following conditional-extremal problems: in classes of Markov times $\mathfrak{M}_\alpha^B(\theta)=\{\tau\le 1:P\,\{\tau<\theta\}\le\alpha\}$, $\mathfrak{M}_\alpha^B(g)=\{\sigma\le 1:P\,\{\sigma<g\}\le\alpha\}$, where $0<\alpha<1$, to describe a structure of optimal stopping times $\tau_\alpha^*(\theta)$ and $\sigma_\alpha^*(g)$, for which $E\,[\tau_\alpha^*(\theta)-\theta]^+=\inf_{\tau\in\mathfrak{M}_\alpha^B(\theta)}E\,(\tau-\theta)^+$, $E\,[\sigma_\alpha^*(g)-g]^+=\inf_{\sigma\in\mathfrak{M}_\alpha^B(g)}E\,(\sigma-g)^+$. We also consider the problems of the structure of some special stopping times in the classes $\mathfrak{M}_\alpha^B(\theta^\mu)$ and $\mathfrak{M}_\alpha^B(g^\mu)$ for the case of Brownian motion with drift $B^\mu=(B_t^\mu)_{0\le t\le 1}$, where $B_t^\mu=\mu t+B_t$.