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 C1,1( D-bar ) , the value function turns out to be the unique solution in the class of Cloc1,1(D)∩C0,1( D-bar ) to the associated degenerate Bellman equation with Dirichlet boundary data. Our approach is probabilistic.