Electronic Journal of Probability

Martingales on Random Sets and the Strong Martingale Property

Michael Sharpe

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Abstract

Let $X$ be a process defined on an optional random set. The paper develops two different conditions on $X$ guaranteeing that it is the restriction of a uniformly integrable martingale. In each case, it is supposed that $X$ is the restriction of some special semimartingale $Z$ with canonical decomposition $Z=M+A$. The first condition, which is both necessary and sufficient, is an absolute continuity condition on $A$. Under additional hypotheses, the existence of a martingale extension can be characterized by a strong martingale property of $X$. Uniqueness of the extension is also considered.

Article information

Source
Electron. J. Probab., Volume 5 (2000), paper no. 1, 17 pp.

Dates
Accepted: 16 December 1999
First available in Project Euclid: 7 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457376436

Digital Object Identifier
doi:10.1214/EJP.v5-57

Mathematical Reviews number (MathSciNet)
MR1741775

Zentralblatt MATH identifier
0953.60021

Subjects
Primary: 60J30

Keywords
Martingale random set strong martingale property

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Sharpe, Michael. Martingales on Random Sets and the Strong Martingale Property. Electron. J. Probab. 5 (2000), paper no. 1, 17 pp. doi:10.1214/EJP.v5-57. https://projecteuclid.org/euclid.ejp/1457376436


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