The monotone follower problem

The monotone follower problem is a stochastic control problem in which the state, a diffusion process, is controlled by a monotone nondecreasing process. For the 1-D case it has been shown that the optimal control is singular with respect to the Lebesgue measure as a function of time and is characterized by a region of inaction A and its complement, the free boundary ∂A being reduced to a point. The present authors identify the free boundary ∂A in the 2-D case under very mild conditions. Then, they assume that A is locally of finite perimeter (LFP) and show that A can be replaced by a new region of inaction A˜. They prove that the new free boundary, ∂A˜, is countably 1-rectifiable and give conditions under which LPF holds. They show that an optimal control exists under certain conditions