The optimal stopping of an integral with respect to the Wiener process

In this paper an optimal bound is found in the stopping problem, for the Wiener process $W_t$, $0\le t\le1$, with gain
$$ V(0,x)=\sup_{0\le\tau\le1}\mathbf M\int_0^\tau(W_s+x)\,ds, $$
where $\tau$ is a Markov time with respect to the family of $\sigma$-algebras $\mathscr F_t^W=\sigma\{W_s,s\le t\}$.