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Let X be a process dened on an optional random set. The paper develops two dier- ent 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 to of some special semimartingale Z with canonical decomposition Z = M + A. The rst condition, which is both necessary and sucient, 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 on . Uniqueness of the extension is also considered.

Martingales on Random Sets and the Strong Martingale Property