A free boundary problem related to singular stochastic control

Abstract: "It is desired to control a multi-dimensional Brownian motion by adding a (possibly singularly) continuous process to its n[superscript th] components so as to minimize an expected infinite-horizon discounted running cost. The Hamilton-Jacobi-Bellman characterization of the value function is a variational inequality which has a unique twice continuously differentiable solution. The optimal process is constructed by solving the Skorokhod problem of reflecting the Brownian motion along a free boundary in the (0,0,..., -1) direction."