Randomized Stopping Points and Optimal Stopping on the Plane

Abstract

We prove that in continuous time, the extremal elements of the set of adapted random measures on $\mathbb{R}^2_+$ are Dirac measures, assuming the underlying filtration satisfies the conditional qualitative independence property. This result is deduced from a theorem in discrete time which provides a correspondence between adapted random measures on $\mathbb{N}^2$ and two-parameter randomized stopping points in the sense of Baxter and Chacon. As an application we show the existence of optimal stopping points for upper semicontinuous two-parameter processes in continuous time.