On optimal stopping of Wiener process with incomplete data

We consider the optimal stopping problem for a Wiener process $W$ with reward $g(t,x)=x/(1+t)$ under the assumption that only the process
$$ \xi_t^{\varepsilon}=\int_0^t W_s\,ds+\varepsilon\widetilde W_t $$
is observed, where $\varepsilon>0$ and $\widetilde W$ is a Wiener process independent of $W$.
The convergence rate of the optimal mean reward $s^{\varepsilon}$ in this «$\varepsilon$-problem» to the optimal mean reward $s^0$ in the «0-problem» when $\varepsilon\to 0$ turns out to be of order $\sqrt{\varepsilon}$. It is shown that the observation domain is limited by a function for which an equation is derived.