Statement

Let $(X_t)_{t\ge 0}$ be a continuous-time martingale with continuous paths, and let $T$ be a stopping time with $\mathbb P(T<\infty)=1$.

If

$$ \mathbb E[|X_T|]<\infty $$
$$ \lim_{t\to\infty}\mathbb E[|X_t|\mathbf 1(T>t)]=0 $$

then

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

Practical criteria

As in discrete time, the theorem is easy to apply when:

  • the process is bounded before stopping, or
  • the martingale is uniformly integrable.