Talk:Martingale (probability theory)
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I don't think saying that [τ=t] is independent of Xt+1 etc. is quite what we want, but the issue is a little messy since I've been trying to avoid talking about filtrations. The definition of stopping time I'm used to is with respect to a filtration; the natural filtration to use (given that we aren't talking about filtrations) would seem to be the one generated by the Xi. In this case [τ=t] is a function of X1 through Xt, which is what I was aiming for in the definition given here.
I'm not sure how to make the "independent of the future" style definition work---for example, we could have a martingale where Xt+1 encodes in some subtle way (e.g. by minute perturbations of the lower-order bits) all of the preceding values. So for the moment I am pulling this paragraph out, since unless I am missing something it looks like the last claim is just plain false:
- To say that "the occurrence or non-occurrence of the event τ=t depends only on the values of X1, X2, ..., Xt" need not mean that the occurrence or non-occurrence of that event is completely determined by those values. It can mean rather that, although the occurrence or non-occurrence of the event τ=t may be correllated with the values of X1, X2, ..., Xt, that occurrence or non-occurrence is probabilistically independent of the values of Xt+1, Xt+2, Xt+3, ... .
Populus 03:06, 13 Nov 2003 (UTC)
Just added an article on filtrations, putting the link here in case anybody wants to link to it ... Bryan Barnard 22:40, 14 Jun 2004 (UTC)
"1 = E[Y1] = E[Yτ] = m² - E[τ]. We immediately get E[τ] = m²+1" Seems to me we get E[τ] = m2 − 1. Typo? - Gauge 00:49, 21 Aug 2004 (UTC)
Why on earth is a Martingale called a Martingale? One sentence on the historical reason would be interesting Torfason 18:52, 22 October 2005 (UTC)
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[edit] Recent addition
I added a general mathematical definition for martingales that take value in general topological vector spaces. They have quite a lot a applications in mathematical finance and in stochastic partial differential equations, so I think it is an important definition.
But I am not sure that the remark is placed in the right position (quite at the beginning). On one hand, it is a definition, and should be written in the beginning. On the other hand, it uses some abstract mathematical tools, and should be put at the end of the article in order to avoid to make it too hostile for non-mathematical readers. gala.martin
- Good points. For now I moved it down. See the article martingale and its history for my explanation. Oleg Alexandrov (talk) 01:51, 4 December 2005 (UTC)
I'm a wikipedia neophyte, so I won't try to sign this or anything like that. Nor am I an expert on sailing vessels, so I won't try to edit this page, either. However-- I do believe there is a type of sail called a martingale, which could benefit from its own page disambiguated from this one.
- I added it to the disambig at the top --14:18, 8 March 2006 (UTC)
[edit] Gambler's Fallacy
"Of course in reality the exponential growth of the bets would quickly bankrupt those foolish enough to use the martingale after even a moderately long run of bad luck."
I think this sentence qualifies as gambler's fallacy as it implies a long run of bad luck would affect future bets badly. --Kurulananfok 21:00, 20 May 2006 (UTC)
- Maybe yhe sentence should be changed (do we need to say that somebody is foolish?). Anyway, since nobody has infinite worth, after a long run of bad luck there are not future bets: the gambler bankrupts and cannot afford further betting. --gala.martin (what?) 13:38, 21 May 2006 (UTC)
[edit] Compared to others
I think that we had better to put supmartingale, submartingale, semimartingale together in the article. Jackzhp 16:29, 9 September 2006 (UTC)
[edit] suggestions
In optional sampling theorem condition (a) is redundant since you require (b). Second, the link of the word constant to mathematical_constant is irrelevant - it's not the same meaning/intention
