Optional Sampling of Submartingales Indexed by Partially Ordered Sets, Optional Sampling of Submartingales.

Abstract

Abstract : The optional sampling theorem for martingales indexed by a partially ordered set is true if the index set is directed. However, the corresponding result for submartingales indexed by a partially ordered set is generally false. In this paper we completely characterize the class of stopping times for which the optional sampling theorem is true for all uniformly bounded submartingales indexed by countable partially ordered sets. By assuming a conditional independence property, we show that when the index set is R to the 2nd power the optional sampling theorem is true for all uniformly bounded submartingales and all stopping times. This conditional independence property is satisfied in cases where the submartingales and stopping times are measurable with respect to the two-parameter Wiener process. A counter example shows that the optional sampling result is false for R to the n power when n greater than 2 even if the conditional independence property is satisfied. (Author)