Regularity of the Value Function for a Two-Dimensional Singular Stochastic Control Problem

Abstract

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 $ - \nabla V$ direction.