Continuous-Time Optional Stopping Theorem

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.

Source Links

• STA447
• Phase 6 Atomic Reading Order
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