The optimal stopping of a controlled diffusion

We prove that a stopping time
$$ \tau=\inf\{t:(s+t,x_t)\notin Q_0\}, $$
where $Q_0=\{(t,x):v(t,x)-g(t,x)>0\}$ is the optimal stopping time for the controlled diffusion
$$ x_t=x+\int_0^t\sigma(\alpha_r,s+r,x_r)\,dw_r+\int_0^tb(\alpha_r,s+r,x_r)\,dr $$
with gain
$$ v(s,x)=\sup_{\alpha\in\mathfrak A}\sup_{0\le\tau\le T-s}\mathbf M_{s,x}^\alpha \biggl\{\int_0^\tau f^{\alpha_t}(s+t,x_t)e^{-\varphi_t}\,dt+g(s+\tau,x_\tau)e^{-\varphi_\tau}\biggr\}. $$