Talk:Optional stopping theorem

Undefined symbol: $$\land$$
What does the symbol $$\land$$ mean in the condition (c)?
 * This is the meet. I added a brief description to the text. Zfeinst (talk) 21:05, 11 October 2015 (UTC)

Previous talk
One should add to the proof why E[X1]=E[Xttau], which is the only non-trivial thing in the proof in my opinion. Szepi (talk) 21:59, 24 October 2009 (UTC)

Reference to example section is problematic (as there is not example section). 83.248.213.36 (talk)

The theorem also applies to right-continuous martingales, not just martingales in discrete time.

Of course the proof is somewhat harder but it should at least be mentioned in the article. 77.1.167.9 (talk) 20:28, 17 November 2010 (UTC)

I could be missing something, but it seems like the proof given here is not really correct. In particular, we do not have

$$ \mathbb{E} \left(\sum_{i=1}^{\tau \land t-1}|X_{i+1}-X_i| \right) = \sum_{i=1}^{\tau \land t-1}\mathbb{E}|X_{i+1}-X_i|$$.

Also, the use of the dominated convergence theorem is a little confusing--it is not relevant that $$ \mathbb{E}|X_t^\tau|$$ is finite, but rather that $$\mathbb{E} \left( |X_1|+\sum_{i=1}^{\tau \land t-1}|X_{i+1}-X_i| \right)$$ is. The hypothesis $$\mathbb{E}(|X_{i + 1} - X_i|) \le c$$ may need to be replaced by $$\mathbb{E}(|X_{i + 1} - X_i| \mid X_i) \le c$$. (Arghbleargh (talk) 06:44, 18 November 2011 (UTC))

reference section
The link reference given is not accessible (access forbidden). A new source for the facts of the article is needed. Any text book providing an introduction to martingales should do. — Preceding unsigned comment added by Quarague (talk • contribs) 10:45, 2 August 2016 (UTC)

Condition (a) not stated properly
Condition (a) as it is now, is wrong. It should read: The stopping time $τ$ is almost surely bounded, i.e., $$\mathbb{P}[\tau<\infty]=1$$.

Otherwise the stopping times defined in the Applications section do not fulfill the requirements.

First application (gambler)
I don’t understand why the first application satisfies the conditions of the OST with only a house limit on bets. The issue is that the stopping time is not well-defined. If the stopping time is really whatever the gambler chooses, then why would the expectation be finite if the gambler’s life is infinite?

Then in the parenthesis it says the same conclusion would hold if the gambler had a debt limit. But again, if the stopping time is defined as hitting that debt limit, then its expectation could again be infinite depending on the strategy. Ofb (talk) 17:10, 13 May 2024 (UTC)