The optimal control of the cheap monotone follower

Abstract

We construct the unique optimal control for the two-dimensional monotone follower problem in the cheap-control setting. It is obtained as the solution of a fixed point problem whose existence is guaranteed under the assumption that a Lipschitz conditions be satisfied in a small neighbourhood of a single “corner point” of the boundary of the region of inaction. Conditions under which this holds are given.

Keywords:

• Singular stochastic control
• variational inequality
• free boundary
• locally finite perimeter
• reflected diffusion