Talk:Martingale representation theorem

Untitled

I have proposed deletion for this article. The theorem stated in this way is substantially uncorrect. IMHO, it needs a major re-writing. So, before doing the eventual job, we should decide if this theorem should be on wikipedia or not. If yes, I could eventually work it out. gala.martin (what?) 23:31, 29 April 2006 (UTC)

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 AJR_1978

Please, note that discussion about deletion is going on this page (as stated in the template). gala.martin (what?) 17:01, 30 April 2006 (UTC)

Alternative version

Some books refer to the MRT as being the "all cts local martingales are integrals of B.m" version, rather than "all square integrable F_T-measurable random variables are integrals of B.m". For example, Oksendal's "Stochastic Differential Equations" takes this approach, as does Shreve's "Stochastic calculus for finance" and Karatzas and Shreve's "Brownian motion and stochastic calculus" (although K&S doesn't explicitly refer to it by name).

Definition required

Can someone provide a definition of the term "previsible"? In appears in a few other articles, seemingly without definition. Melcombe (talk) 15:06, 17 February 2009 (UTC)

Merger proposal

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

Since there has been no comment within two weeks, I'll go ahead with the merger. Hairer (talk) 13:29, 7 March 2021 (UTC)

---

I propose to merge Integral representation theorem for classical Wiener space into Martingale representation theorem. It mostly just duplicates the material here, but something from the second section may be rescued. Hairer (talk) 15:25, 21 February 2021 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Clarification of the topic

There is a confusion what is Martingale representation theorem and what is Ito representation theorem. The article should state the former one and not the latter.

For didactic reasons, it would be good to have a section about the Ito repr theorem, since they are very much connected. The difference is not huge, but making the differences clear can help to better understand the theorem.

I think it is okay that the bulk of the article is about the continuous version, though, it could include jump processes too. Zsocca (talk) 23:30, 11 July 2023 (UTC)