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What does it it mean to say a martingale is closed [in layman terms] ? Does it just mean the martingale converges ??

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Where did you encounter this? –  Stefan Hansen Nov 28 '12 at 20:17
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A martingale $(X_n,\mathcal{F}_n)_n$ is called (right-)closable if there exists a random variable $X$ such that $X_n = \mathbb{E}(X|\mathcal{F}_n)$. (A sufficient condition is that $(X_n)_n$ is uniformly integrable.)

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