We prove that in continuous time, the extremal elements of the set of adapted random measures on R2+ 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 N2 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.
This is the published version, also available here: http://dx.doi.org/10.1214/aop/1176989810.
Nualart, David. Randomized Stopping Points and Optimal Stopping on the Plane. Ann. Probab. 20 (1992), no. 2, 883--900. http://dx.doi.org/10.1214/aop/1176989810.