## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 6 (1981), 957-970.

### Optional Sampling of Submartingales Indexed by Partially Ordered Sets

Robert B. Washburn, Jr. and Alan S. Willsky

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 6 (1981), 957-970.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994267

**Digital Object Identifier**

doi:10.1214/aop/1176994267

**Mathematical Reviews number (MathSciNet)**

MR632969

**Zentralblatt MATH identifier**

0477.60048

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 60G45

**Keywords**

Optional sampling submartingale reachable stopping time optimal stopping problem partially ordered index set

#### Citation

Washburn, Robert B.; Willsky, Alan S. Optional Sampling of Submartingales Indexed by Partially Ordered Sets. Ann. Probab. 9 (1981), no. 6, 957--970. doi:10.1214/aop/1176994267. https://projecteuclid.org/euclid.aop/1176994267