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We study the partial differential equation max{Lu - f, H(Du)}=0 where u isthe unknown function, L is a second-order elliptic operator, f is a givensmooth function and H is a convex function. This is a model equation forHamilton-Jacobi-Bellman equations arising in stochastic singular control. Weestablish the existence of a unique viscosity solution of the Dirichlet problemthat has a Holder continuous gradient. We also show that if H is uniformlyconvex, the gradient of this solution is Lipschitz continuous.

Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control