  This is the first of a series of papers on the interior regularity of fully
nonlinear degenerate elliptic equations. We consider a stochastic optimal
control problem in which the diffusion coefficients, drift coefficients and
discount factor are independent of the spacial variables. Under suitable
assumptions, for $k=0,1$, when the terminal and running payoffs are globally
$C^{k,1}$, we obtain the $C^{k,1}$-smoothness of the value function, which
yields the existence and uniqueness of the solution to the associated Dirichlet
problem for the degenerate Bellman equation.
