Electronic Journal of Probability
- Electron. J. Probab.
- Volume 5 (2000), paper no. 1, 17 pp.
Martingales on Random Sets and the Strong Martingale Property
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.
Electron. J. Probab., Volume 5 (2000), paper no. 1, 17 pp.
Accepted: 16 December 1999
First available in Project Euclid: 7 March 2016
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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