Martingales on Random Sets and the Strong Martingale Property

Michael J. Sharpe (University of California, San Diego)


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.

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Pages: 1-17

Publication Date: December 16, 1999

DOI: 10.1214/EJP.v5-57


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