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The optional sampling theorem for processes indexed by a partially ordered set. (English) Zbl 0576.60037

Summary: The optional sampling theorem (OST) is not necessarily true for super- martingales indexed by a partially ordered set. However, if the index set satisfies a mild separability condition, without necessarily being directed or countable, we prove that OS inequality for a class of supermartingales that extends the concept of S-processes defined by R. Cairoli [Décomposition de processus à indices doubles. Sém. Probab. V, Univ. Strasbourg 1969/1970, Lect. Notes Math. 191, 37-57 (1971)] 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 R. Cairoli and J. B. Walsh [Z. Wahrscheinlichkeitstheor. Verw. Geb. 44, 279-306 (1978; Zbl 0369.60043)] and E. Wong and M. Zakai [Ann. Probab. 4, 570-586 (1977; Zbl 0359.60053)] on \({\mathbb{R}}^ 2_+\).

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G48 Generalizations of martingales
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