Definition

Let $X$ be an integrable random variable and let $(\mathcal F_n)$ be a filtration. Define

$$ M_n=\mathbb E[X\mid \mathcal F_n]. $$

Then $(M_n)$ is called Doob’s martingale.

Why it is a martingale

Using the tower property,

$$ \mathbb E[M_{n+1}\mid \mathcal F_n]

\mathbb E[\mathbb E[X\mid \mathcal F_{n+1}]\mid \mathcal F_n]

\mathbb E[X\mid \mathcal F_n]

M_n. $$

Important consequence

Doob’s martingale is automatically uniformly integrable. Therefore, for any stopping time $T$,

$$ \mathbb E[M_T]=\mathbb E[M_0]. $$

Interpretation

$M_n$ is the best prediction of $X$ based on the information available at time $n$.