A Construction Scheme of the Minimal Dominating Supermartingale arising in the Discrete Parameter Optimal Stopping Problems

Abstract

General optimal stopping problem with discrete parameter reward process Z=(Z(n), K(n), n∈N), with N={0, 1, …}, is considered. The problem is described as follows: find the optimal stopping time τ* such that E[Z(τ*)]=sup {E[Z(τ)]: τ is K(n)-stopping time}, In order to characterize τ*, the minimal dominating supermartingale Γ=(γ(n), K(n), n∈N) such that γ(n)=ess sup {E[Z(τ) | K(n)]: τ>n, τ is K(n)-stoppiūg time} is needed. Here, a new construction scheme of Γ is developed. By this new scheme we can derive Γ and determine the optimal stopping time τ* for some problems. We apply our results to a job search type optimal stopping problem.