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I have been reading a book called Stochastic Calculus by Steele. Inside, they have a theorem they state as the "Optional Stopping Time Theorem":

If $M_n$ is a martingale with respect to $\mathcal{F}_n$, then the stopped process $M_{n \wedge \tau}$ is also a martingale with respect to $\mathcal{F}_n$.

However, on Wikipedia, Optional Stopping Theorem - Wikipedia they state the optional stopping theorem in terms of three separate sufficient statements. At first glance, these two don't look the same. Is there a chance they are talking about the same thing?

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That is not the Optional Stopping Theorem (OST). That is the Optional Stopping Time Theorem. In Williams' Probability with Martingales, such statement (which doesn't have the cool name) comes before the OST:


p. 99

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p. 100

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