Martingales on Random Sets and the Strong Martingale Property
Michael J. Sharpe, University of California, San Diego
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.
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