The Free Boundary for Variational Inequalities with Nonlocal Operators

It is well known that the solution of a stationary optimal stopping time problem corresponding to a diffusion process is a solution of an elliptic variational inequality. In this paper we study the corresponding situation for some other Markov processes whose paths are right continuous and whose generator is a nonlocal operator. An existence and uniqueness of a solution of the variational inequality is proved. Special attention is then paid to the shape of the free boundary. It is shown that the free boundary is a curve $y = \varphi (x)$ and $\varphi (x)$ is piecewise monotone.