  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.
