Wikipedia:Articles for deletion/Martingale representation theorem

The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.

The result of the debate was keep. Mailer Diablo 01:07, 5 May 2006 (UTC)[reply]

Martingale representation theorem[edit]

I think the theorem needs not its own article. On the other hand, merging cannot be considered since the article is substantially wrong. gala.martin (what?) 23:39, 29 April 2006 (UTC)[reply]

Comment In order to assert the theorem, you need the martingale to have finite quadratic variation (and of course, a.e. path should be continuous). This is not just a math technicality, but it is the heart of the theorem. The representation theorem tells you that a BM-adapted, continuous, L^2 martingale is a brownian motion at a random time (or, if you want to make things more complicated, it is a stochastic integral driven by brownian motion). Now, the random time you need (or the process you need to integrate), is defined exactly by the quadratic variation of the original martingale. If we want to go through the quadratic variation stuff for martingales, the representation theorem article is not the right place. On the other hand, IMHO the article does not meet the wikipedia standard right now. That's why I purposed deletion.

Also, I have some doubts about the way to solve this problem: we could create an article for continuous time martingales, since they are the most commonly used martingales, both in mathematics and applications. In this case, we could state the theorem there. Otherwise, we could create some articles in order to cover this subjects: add info on quadratic variations article, etc, and then improve this article. By the way, I am not sure at all that the representation theorem should be on wikipedia. I hope to receive further comments. gala.martin (what?) 00:02, 30 April 2006 (UTC)[reply]

Comment If it's substantially wrong as written, but a legitimate theorem, then the remedy is to correct the errors, not to delete the article. I think several theorems about martingales probably deserve their own articles. Michael Hardy 02:24, 30 April 2006 (UTC)[reply]
Keep, if it's wrong it should be fixed and not deleted, as per Michael Hardy --Deville (Talk) 05:57, 30 April 2006 (UTC)[reply]
- Comment Yes, the issue is not about the fact that the theorem is uncorrect. It is about if we want this theorem with its own article. And of course, it is better to decide that before correcting. A couple of months ago, I would surely thought we should keep the theorem, but I am changing my ideas about the best way to organize the math stuff. Anyway, I am convinced martingales' theorems are important and interesting. On the other hand, my nick here is just a transposition of the italian word for martingale :). gala.martin (what?) 16:56, 30 April 2006 (UTC)[reply]

Keep
- I agree that the statement of the theorem is poor, but it is a start. Why don't we just rewrite it? Perhaps we could restructure it by the following layout?
  - 1. the statement for continuous martingales and give a few references for the proof,
  - 2. the statement for general martingales, and few references for the proof
  - 3. then perhaps some motivation for it along the lines of saying that Brownian motion is the canonical continuous martingale, while compensated compound Poisson processes are the canonical processes for martingales with jumps and that all martingales are just integrals of a Brownian motion plus some sort of compensated compound Poisson process, and lastly
  - 4. some places where the theorem is used such as in finance where it is used to prove the existence of portfolios User_talk:AJR_1978.

Comment. Sorry for comments inflaction! I think I did not explain myself properly. I would like to receive comments about this: ok, we keep this material. I am the first one to say that this stuff is interesting. I just wonder if the best thing is to keep theorems separated, or to merge together, for instance, this one, this one, this one, some stuff about quadratic variation etc. gala.martin (what?) 19:36, 30 April 2006 (UTC)[reply]
Keep, relevant theorem in financial mathematics. Stifle (talk) 18:52, 1 May 2006 (UTC)[reply]

The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.