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December, 1981 Optional Sampling of Submartingales Indexed by Partially Ordered Sets
Robert B. Washburn Jr., Alan S. Willsky
Ann. Probab. 9(6): 957-970 (December, 1981). DOI: 10.1214/aop/1176994267

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 not true in general. 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 $R^2$ the optional sampling theorem is true for all uniformly bounded, right continuous 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 counterexample shows that the optional sampling result is false for $R^n$ when $n > 2$ even if the conditional independence property is satisfied.

Citation

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Robert B. Washburn Jr.. Alan S. Willsky. "Optional Sampling of Submartingales Indexed by Partially Ordered Sets." Ann. Probab. 9 (6) 957 - 970, December, 1981. https://doi.org/10.1214/aop/1176994267

Information

Published: December, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0477.60048
MathSciNet: MR632969
Digital Object Identifier: 10.1214/aop/1176994267

Subjects:
Primary: 60G40
Secondary: 60G45

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 6 • December, 1981
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