On a constrained optimal stopping problem

Abstract

For a sequence of random variables {Xn, n ≧ 0}, optimal stopping is considered over stopping times T constrained so that ET ≦ α, for some fixed α > 0. It is shown that under certain circumstances a Lagrangian approach may be used to reduce the problem to an unconstrained optimal stopping problem of a conventional type. The optimal value of the natural dual problem is shown to be equal to the optimal value of the original (primal) problem when certain randomised stopping times are permitted. Two examples are considered in detail.