Geodesic
The optimal stopping of a controlled diffusion
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 303-312 
L. G. Mikhaǐlovskaya. The optimal stopping of a controlled diffusion. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 303-312. http://geodesic.mathdoc.fr/item/TVP_1980_25_2_a5/
@article{TVP_1980_25_2_a5,
     author = {L. G. Mikhaǐlovskaya},
     title = {The optimal stopping of a controlled diffusion},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {303--312},
     year = {1980},
     volume = {25},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_2_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $$ v(s,x)=\sup_{\alpha\in\mathfrak A}\sup_{0\le\tau\le T-s}\mathbf M_{s,x}^\alpha \biggl\{\int_0^\tau f^{\alpha_t}(s+t,x_t)e^{-\varphi_t}\,dt+g(s+\tau,x_\tau)e^{-\varphi_\tau}\biggr\}. $$