The Optional Sampling Theorem for Processes Indexed by a Partially Ordered Set

Abstract

The optimal sampling theorem (OST) is not necessarily true for supermartingales indexed by a partially ordered set. However, if the index set satisfies a mild separability condition, without necessarily being directed or countable, we prove the OS inequality for a class of supermartingales that extends the concept of $S$-processes defined by Cairoli on the plane $\mathbb{R}^2_+$. Under a further restriction on these processes we obtain the OS equation, thus extending the corresponding result for martingales to the case of nondirected index sets. We then introduce strong martingales and strong supermartingales for separable partially ordered index sets, and show that these processes again satisfy the OST. By defining stopping domains as well as the value of a process for a stopping domain, we show that the strong (super)martingales are precisely those processes which satisfy the OST for all bounded stopping domains. This extends a result of Cairoli-Walsh and Wong-Zakai on $\mathbb{R}^2_+$.