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.
Michael Sharpe. "Martingales on Random Sets and the Strong Martingale Property." Electron. J. Probab. 5 1 - 17, 2000. https://doi.org/10.1214/EJP.v5-57