Talk:Martingale representation theorem
|WikiProject Statistics||(Rated Start-class, Low-importance)|
|WikiProject Mathematics||(Rated Start-class, Low-importance)|
||This article may be too technical for most readers to understand. (September 2010)|
|This page was nominated for deletion on 2006 March 29. The result of the discussion was keep.|
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
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).