We study the partial differential equation
max{Lu − f, H(Du)} = 0
where u is the unknown
function, L is a second-order elliptic operator, f is a
given smooth function and H is a convex function. This is a model
equation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. We
establish the existence of a unique viscosity solution of the Dirichlet problem that has a
Hölder continuous gradient. We also show that if H is uniformly
convex, the gradient of this solution is Lipschitz continuous.