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
In Optional Sampling Theorem there is a reference to undefined condition (c): "a gambler with a finite lifetime (which gives conditions (a) and (b)) and a house limit on bets (condition (c))"
[edit] Martingale systems in investment, and the value of stating the obvious
It seems that more snake oil is being sold, and more ignorance spread, in the form of martingale systems promoted as a sound investment strategy, for stocks or foreign exchange. Shouldn't it perhaps be mentioned that martingale systems have been used, not just by casino gamblers, but by investors who perceive some kind of scientifically proven advantage to the use of such systems?
On the subject of stating the obvious, perhaps a plain language explanation of the fallacy of martingale systems would be suitable. Simply put, if you you're betting on a flip of a fair coin, then your expectation is zero (that is, your current status +/- 0), and by the very definition of random outcomes there is no need to refer to previous results to draw this conclusion. When betting in a "supermartingale" situation, such as a bet on black on a roulette wheel with two green spots, your expectation is determined by the odds of one trial, again without reference to previous trials, and this is largely self-evident without lengthy mathematical proof (I'm not saying we shouldn't show the proof! Only that it's fair to make a plain language statement of the obvious fact as well).
A similar plain language explanation might be to point out that, "if it makes sense, after six consecutive losses, to bet 2^6 times your original bet, and if results of each bet are independent, then it makes just as much sense to bet that amount right now. If it ever makes sense to bet $64.00 (or $64,000.00), then it always makes sense to do so. If it ever doesn't make sense to bet that amount, then it never makes sense to, unless somehow the conditions of the game itself are changed... or unless you take the rational approach and constrain your bets, not by tying them to a chain of random past events according to an irrational system, but rather by the limits of one's own tolerance for risk.
Maybe this doesn't sound scientific enough, but sometimes common sense should be expressed in a common way.zadignose 04:56, 22 August 2007 (UTC)
[edit] Origin of the name
There is no citation of the etymology of Martingale. In fact, I am reasonably confident that the stochastic process called a Martingale had its origin from the horse collar, not the gambling system. It is to give the imagery that there is a constraint on where the horse's head can move in the next time instant. It was a coincidence that this also happened to be a name of a gambling system. —Preceding unsigned comment added by 124.171.59.106 (talk) 11:37, 24 April 2008 (UTC)
