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As stated, the aricle is wrong : (Zt) won't be a (positive) local martingale if (Xt) itself is not a local martingale (take dXt = dt as the simplest counter-example). — Preceding unsigned comment added by 184.108.40.206 (talk) 00:14, 31 January 2012 (UTC)
Article is in need of attention
The statement of this difficult theorem is hard to follow and definitely wrong. For example, the article has the equation.
The left hand side is a Radon-Nikodym derivative. The notation on the right hand side is nonstandard but it is clear from what follows that it is denoting a geometric Brownian motion process. But the Radon-Nikodym derivative of one probability measure with respect to another is a random variable, not a process. Brian Tvedt 12:05, 25 January 2006 (UTC)
- Good point, but you can always turn it into a process (and a martingale in fact) by taking conditional expectation with respect to the probability measure P. --DudeOnTheStreet (talk) 18:18, 27 May 2011 (UTC)
Agreed. Lacks historical content as well. Also, there is very little mention on the Black-Scholes formula page regarding the importance of the Girsanov theorem, risk neutral measure, etc. Dmaher 00:48, 4 April 2006 (UTC)
No reference is made to quadratic variation -- the square brackets [X]_t -- which makes the article unreadable for non-specialists. Rogier 220.127.116.11 (talk) —Preceding undated comment was added at 13:42, 20 November 2008 (UTC).
I agree that quadratic variation should at least be referenced; I wasn't even sure that the square-bracket notation meant quadratic variation. However, I have a more substantial issue, it seems that the page, as-is, contains a significant error. In the section "Statement of the Theorem," the process X_t is only assumed to be measurable and adapted to the Wiener filter. With only these assumptions, I do not believe that the Doleans exponential is a local martingale, as stated. In fact, I'm about 99 positive this is wrong. I think a simple assumption on X (something like X being a local martingale) was omitted. — Preceding unsigned comment added by 18.104.22.168 (talk) 17:59, 16 July 2011 (UTC)