Smoothness of the value function for a controlled diffusion process in a domain

The author gives conditions on the behavior of a controlled diffusion process within and on the boundary of a domain that are sufficient for the value function to have two bounded generalized derivatives and to satisfy the Bellman equation. These conditions are almost necessary even for uncontrolled diffusion processes, and at the same time they encompass, for example, the heat equation in a disc and the Monge舑AmpȲre equation in a convex domain.