A probabilistic approach to interior regularity of fully nonlinear degenerate elliptic equations in smooth domains

We consider the value function of a stochastic optimal control of degenerate diffusion processes in a domain $D$. We study the smoothness of the value function, under the assumption of the non-degeneracy of the diffusion term along the normal to the boundary and an interior condition weaker than the non-degeneracy of the diffusion term. When the diffusion term, drift term, discount factor, running payoff and terminal payoff are all in the class of $C^{1,1}(\bar D)$, the value function turns out to be the unique solution in the class of $C_{loc}^{1,1}(D)\cap C^{0,1}(\bar D)$ to the associated degenerate Bellman equation with Dirichlet boundary data. Our approach is probabilistic.