Optimal stopping and cluster point processes

In some recent work it has been shown how to solve optimal stopping problems approximatively for independent sequences and also for some dependent sequences, when the associated embedded point processes converge to a Poisson process. In this paper we extend these results to the case where the limit is a Poisson cluster process with random or with deterministic cluster. We develop a new method of directly proving convergence of optimal stopping times, stopping curves, and values and to identify the limiting stopping curve by a unique solution of some first order differential equation. In the random cluster case one has to combine the optimal stopping curve of the underlying hidden Poisson process with a statistical prediction procedure for the maximal point in the cluster. We study in detail some finite and infinite moving average processes.