Testing of two simple hypotheses in the presence of delayed observations

We consider the problem of testing hypotheses $H_0\colon \theta=0$ and $H_1\colon \theta=1$ for the process $\xi_t=\theta t+w_t$, $\xi_0=0$ where $w_t$ is a Wiener process, by means of a stopping rule $\delta=(\tau,d)\colon \tau$ is a Markov moment, $d$ ($d=0$ or $d=1$) is a decision function depending on the behaviour of $\xi_t$ on $[0,\tau+m]$. In a some class of rules $\Delta(\alpha,\beta)$ we find a rule $\delta\in\Delta(\alpha,\beta)$ which minimizes the functional
$$ \lambda\mathbf M_0\tau+(1-\lambda)\mathbf M_1\tau $$
for a fixed $\lambda\in[0,1]$ (here $\mathbf M_i$ is the expectation corresponding $H_i$).