An optimal stopping time problem with time average cost in a bounded interval (Q1085882)

This paper deals with an optimal stopping problem of one dimensional diffusion y, with the cost defined by \[ J(\tau)=\lim_{T\to \infty}\inf (E\int^{\tau \wedge \sigma \wedge T}_{0}f(y(t))dt+\psi (y(\tau \wedge T))\chi_{(\tau <\sigma)})/E(\tau \wedge \sigma \wedge T), \] where \(\sigma\) is the exit time of y from [a,b]. Using a variational inequality with Dirichlet boundary condition, the author characterized the value function and obtained an optimal stopping policy.