Regularity of the value function for a two-dimensional singular stochastic control problem

It is desired to control a two-dimensional Brownian motion by adding a (possibly singularly) continuous process to it so as to minimize an expected infinite-horizon discounted running cost. The Hamilton-Jacobi-Bellman characterization of the value function V is a variational inequality which has a unique twice continuously differentiable solution. The optimal control process is constructed by solving the Skorokhod problem of reflecting the two-dimensional Brownian motion along a free boundary in the -▽ V direction.