Geodesic
On~control of a~diffusion process up to the time of first exit from a~region
Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 297-313

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The problem of controlling a diffusion process with smooth coefficients up to the time of first exit from a region is reduced to the problem of controlling a diffusion process with smooth coefficients in a whole space. Three examples are considered which show the usefulness of such a reduction in deriving the Bellman equation for the original problem. Bibliography: 12 titles. 
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     author = {N. V. Krylov},
     title = {On~control of a~diffusion process up to the time of first exit from a~region},
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N. V. Krylov. On~control of a~diffusion process up to the time of first exit from a~region. Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 297-313. http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a5/