Talk:Martingale (probability theory)

message: 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)
thread_title: tau

message: 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)
replies: text: I added it to the disambig at the top --14:18, 8 March 2006 (UTC)
thread_title: Recent addition

message: "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)
replies: text: 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  <Small>(what?)</Small> 13:38, 21 May 2006 (UTC)
thread_title:  Gambler's Fallacy 

message: I think that we had better to put supmartingale, submartingale, semimartingale together in the article. Jackzhp 16:29, 9 September 2006 (UTC)
thread_title:  Compared to others 

message: 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)
replies: text: A coincidence?  The gambling system has an obvious connection to the mathematical concept.  I find that hard to believe that it would be coincidence.  The entire point of the gambling system is a claim that the optional sampling theorem can be violated.  —Preceding unsigned comment added by 132.228.195.207 (talk) 21:35, 31 March 2009 (UTC)
replies: text: You may wish to see [] Melcombe (talk) 13:31, 2 April 2009 (UTC)
thread_title: Origin of the name

message: I think that the R-programmed simulation of a Brownian motion is just a sequence of independent variables, whereas it should be the sum of independent random variables. I haven't changed it, because I'd like someone to provide a second opinion. Thanks.  —Preceding unsigned comment added by Philip Maton (talk • contribs) 16:46, 27 May 2009 (UTC)
replies: text: I agree. As it stands, the Excel formula and the R command/graphic are inconsistent. -- Chadhoward (talk) 22:18, 14 March 2010 (UTC)
thread_title: Demonstration of R software

message: Polya's urn redirects here, Wouldn't it be better if redirected to Urn problem? --Squidonius (talk) 23:20, 23 June 2010 (UTC)
thread_title:  Polya's urn redirect 

message: I think the term martingale was coined by Ville, somewhat before Lévy used them. About the Girsanov theorem, it only permits to construct a measure that makes SOME Ito processes a martingale (not any Ito process!). And Balsara and Kleinert seem out of place in the references. Why not link to Bachelier or some other financial applicant instead of referencing Balsara's book "for futures traders". Likewise, treatises of Doob, Meyer or Neveu are actually about martingales, whereas Kleinert's book seems to be about "a lot of other stuff".90.27.21.180 (talk) 13:50, 28 April 2010 (UTC)
thread_title: Coined by Ville 

message: Surely, there must be some theorems relating martingales to Markov chains? The definitions are quite similar; one speaks of probabilities, the other of expectations. Surely there must be overlap! Yet the current article breaths narry a word of this. linas (talk) 03:40, 20 November 2010 (UTC)
replies: text: They are only related in that they both describe stochastic processes. The Markov property (roughly) states that after observing the prior state of the system, there is no additional information that can be gained by observing earlier states. In a Markov chain, once we enter one link of the chain, the next transitions out of the chain only depend on that link; they do not depend on how we got to that link. However, in a Martingale (roughly), the conditional expectation of the next observation given this and all prior observations is equal to this observation. So Martingales need not have a Markov property as the next transition can depend on more than the present state of the system. —TedPavlic (talk/contrib/@) 02:15, 15 August 2011 (UTC)

text: I had the same confusion reading the article for the first time. It says that martingale is "a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters." This is just misleading, and sounds like a special case of Markov property. It should be changed to "a model in which the expected value of the future state is always equal to the current state."Sprlzrd (talk) 18:56, 20 January 2017 (UTC)
thread_title:  Relation to Markov chains ??? 

message: Something about the biased coin as an example of a sub- and supermartingale seems misleading. The expectation value of a biased coin doesn't depend on its history – it's always p. Even in the case where you have a coin and are told that it's biased (but not by how much), you wouldn't know whether it was biased up or down, so still it's not a sub- or supermartingale. And if you were told that it was biased in a particular direction, the initial expectation value probably isn't 0.5.
206.124.141.187 (talk) 19:56, 29 June 2013 (UTC)
replies: text: Never mind – I realized that the bankroll is the random variable, not the outcome of the coin flips. 206.124.141.187 (talk) 03:44, 2 July 2013 (UTC)
thread_title:  Biased Coin? 

message: I would like to suggest adding a section on applications for martingales. The abstract of The least variable phase type distribution is Erlang concludes stating that their "proof is simple and elegant and is a nice example of the power of martingales; it seems intractible without them." Could we collect together some examples of other insights given by the theory of martingales? Gareth Jones (talk) 16:50, 12 November 2013 (UTC)
thread_title:  Applications 

message: "To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time. However, knowledge of the prior outcomes (e.g., all prior cards drawn from a card deck) may be able to reduce the uncertainty of future outcomes. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning strategy is used. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games." [emphasis mine]
A martingale illustrates a fair game of chance, not skill. A fair game of skill simply requires that all parties have equal access to the same body of information, if they have the presence of mind to seek it out. That means that parties that are willfully ignorant (that is, "stupid" ["Boko Haram!"]) are not at a disadvantage in a game of chance, but are at a disadvantage in a game of skill. Life is arguably just as much a game of skill as it is of chance, or none of us would bother reading and writing an online encyclopedia. This may be a small difference in the nature of this article, but it's a ginormous difference in world-view. I'm not sure how to proceed from here, beyond pointing this out on a talk page, but the anomaly struck me rather hard at the time, sorry. TheLastWordSword (talk) 13:37, 20 June 2014 (UTC)
thread_title:  Chance, Not Skill 

message: The post of 20 June 2014 has been largely ignored. Currently the second paragraph in the lead says
:To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time. However, knowledge of the prior outcomes (e.g., all prior cards drawn from a card deck) may be able to reduce the uncertainty of future outcomes. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning strategy is used. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games.
This makes no sense. First, it refers to the expected rather than actual value of “the process at one time”. Second, it conflates the outcome of the stochastic process with the outcome of a bet chosen based on skill. And it talks about reducing the “uncertainty” of future outcomes, which is irrelevant to the issues at hand. So I’m removing this paragraph. Loraof (talk) 01:01, 17 November 2017 (UTC)
thread_title:  Reason for removal of paragraph 

message: English has been using the word "martingale" since 1580 at the latest. It originally meant a strap that could be added to a horse's harness to limit how far the horse could lift its head; this was used during training as a check against out-of-control behavior. TooManyFingers (talk) 22:45, 6 May 2024 (UTC)
replies: text: Please find a good RS reference and we can add it. Limit-theorem (talk) 06:06, 8 May 2024 (UTC)
replies: text: Actually, for the word origin we should start with the gambling use, as this is not about etymology but mathematics and probability. Limit-theorem (talk) 12:27, 8 May 2024 (UTC)
replies: text: The word does not come from the strap for horses, to my knowledge. If I remember correctly, there were other things that were also called a martingale in the past. The origin of the word in mathematics is to my knowledge unkown. There is also a French article about it: "Histoire de martingales" by Roger Mansuy (Roger Mansuy, “Histoire de martingales”, Mathématiques et sciences humaines, 169, Printemps 2005, DOI: 10.4000/msh.2945) Tensorproduct --talk 20:59, 8 May 2024 (UTC)
thread_title:  False statement about word origin