On the Dirichlet problem for Bellman's equation in a plane domain

The Dirichlet problem for Bellman's equation in a plane domain is considered. It is shown that, under certain assumptions about the type of smoothness and the “weak” nondegeneracy of the corresponding controlled diffusion process, the solution of this Dirichlet problem is the pay-off function of an optimal control problem. Under supplementary assumptions about smoothness, the pay-off function is also smooth.
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