UI, L1 Convergence, and OST

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 $L^1$.

Under the same assumptions,

\mathbb E[X_0]. $$

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 $X_\infty$.