How do I prove that a martingale has a constant expected value?

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I can´t prove that a martingale has constant expected value. $$ \mathbf{E}[M_t]=\mathbf{E}[M_0] $$

Thanks people.

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1 Answer

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It holds for any sigma-algebra $\mathcal{F}$ that

$$\mathbb{E}[ \mathbb{E}(X \mid \mathcal{F}) ] = \mathbb{E}(X).$$

Note that this does not require that $\mathcal{F}$ and $X$ are independent. Since a martingale satisfies

$$\mathbb{E}(M_{n+1} \mid \mathcal{F}_n) = M_n,$$

we get

$$\mathbb{E}(M_{n+1}) = \mathbb{E}[\mathbb{E}(M_{n+1} \mid \mathcal{F}_n)] = \mathbb{E}(M_n)$$ for all $n \in \mathbb{N}$. Hence, $\mathbb{E}(M_n) = \mathbb{E}(M_0)$.

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