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The optimal stopping of a controlled diffusion
L. G. Mikhaĭlovskaya Moscow
Abstract:
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
$$
v(s,x)=\sup_{\alpha\in\mathfrak A}\sup_{0\le\tau\le T-s}\mathbf M_{s,x}^\alpha
\biggl\{\int_0^\tau f^{\alpha_t}(s+t,x_t)e^{-\varphi_t}\,dt+g(s+\tau,x_\tau)e^{-\varphi_\tau}\biggr\}.
$$
Received: 23.08.1978