UI upgrades almost sure convergence

If $(X_n)$ is a martingale, $X_n\to X_\infty$ almost surely, and $(X_n)$ is uniformly integrable, then

$$ \mathbb E[|X_n-X_\infty|]\to 0. $$

So the convergence is not only almost sure but also in $L^1$.

Consequence for expectations

Under the same assumptions,

$$ \mathbb E[X_\infty]

\lim_{n\to\infty}\mathbb E[X_n]

\mathbb E[X_0]. $$

UI version of optional stopping

If $(X_n)$ is a uniformly integrable martingale and $T$ is any stopping time, then

$$ \mathbb E[X_T]=\mathbb E[X_0]. $$

This is the strongest version of OST and even allows $T=\infty$, with $X_T$ interpreted as $X_\infty$.

Source Links

• STA447
• Phase 5.2-5.5 Atomic Reading Order
• Phase 5.4 — Martingale Convergence Theorem + Uniform Integrability
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