The classical solutions and the regularity of the free boundaries in multi-dimensional singular stochastic control

One traditional difficulty in stochastic singular control problem is to characterize the value function as a classical solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which involves free boundaries between the action and inaction regions. This difficulty is especially prominent in multi-dimensional singular control problems, where the HJB equations are elliptic partial differential equations (PDE) with free boundaries. In this paper, a type of multi-dimensional singular stochastic control problems is considered. Through a technique of Dynkin games (zero-sum games), it is shown that if the free boundaries have certain regularity properties such as Lipschitz continuity and smoothness, the classical solutions to the HJB equations exist. These regularities also enable us to characterize the boundary conditions of the PDEs. Then the verification theorem can be applied in order to show the optimality of the control.