The estimation problem of an unknown random parameter $\theta=\theta(\omega)$ is studied in the case when $\theta$ takes values 1 and 0 with probabilities $\pi_0$, $1-\pi_0$, respectively, and the observed process is $$ \xi_t(\omega)=r\theta(\omega)t+\sigma W_t(\omega),\qquad\sigma>0,\qquad r\ne 0,\qquad t\ge 0, $$ where $W$ is a standard Wiener process. Denote by $\tau=\tau(\omega)$ a Markov time with respect to the family of $\sigma$-algebras $\mathscr F_{\tau}^{\xi}=\sigma\{\xi_s,\,s\le t\}$, and by $d=d(\omega)$ a decision function which is $\mathscr F_{\tau+m}^{\xi}$-measurable, where $m\ge 0$ is the delay time. We find a pair $(\tau,d)$ which minimizes $$ \mathbf M[c\tau+a\chi_{(d=0,\theta=1)}+b\chi_{(d=1,\theta=0)}], $$ where $a$, $b$, $c$ are positive constants, $\chi_A$ is the characterictic function of a set $A$ .