The Optional Sampling Theorem for Martingales Indexed by Directed Sets

Abstract

A natural generalization of the optional sampling theorem for martingales is given. For discrete valued stopping times the result holds for directed sets; for more general stopping times the result holds for lattices satisfying a type of separability condition. The discrete case improves a lemma of Chow. The general case depends upon a lemma showing that all martingales with respect to $\sigma$-algebras satisfying a "right continuity" condition have a modification which has a regularity property that is similar to, but weaker than, right continuity. A result of Wong and Zakai is obtained as a corollary.