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Let $M_n$ be a martingale in discrete time with respect to the filtration $(\mathcal{F}_n)_{n \in \mathbb{N}}$, and $T$ a stopping time with respect to the same filtration, a.s. bounded by $k$. For $n \geq k$, is $\mathbb{E}[M_n |\mathcal{F}_T] = M_T$?

If not, is something similar true?

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Yes, this is an immediate consequence of what is commonly referred to as Doob's Optional Sampling Theorem.

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