Talk:Itō's lemma
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Is anyone who knows what they are doing, re: Wiki, still monitoring this page? I note that Pcb21/Pete seems to have taken a break from Wiki.
In any case, (a) it is incorrect to state Itō's Lemma in "differential form" as is done in this article (and just about everywhere else that the Lemma is "stated") -- the equality holds for the integral, but not for the differential "equivalent;" (b) I can provide a formal proof (my own), if someone is willing to verify it, but I am completely unfamiliar with the math editor here. --Marsden 23:52, 23 August 2005 (UTC)
The differential and integral form are exactly equivalent. What makes you think they differ?
I've recently written up a formal proof to Ito's lemma if anyone wants it.
- Wikipedia itself would greatly benefit from such a proof. Would you be willing to release under our free licence? If you don't want to spend the time converting from what format you have it in to wiki-markup you could send it to me and I could do it. Let me know! Thanks for your interest Pete/Pcb21 (talk) 08:52, 11 Feb 2004 (UTC)
Shouldn't it be "Itô's lemma" with the appropriate accent mark? I think "Ito's lemma" (without the accent) was the cause of O.J. Simpson's acquittal or something? --Christofurio 15:36, Apr 12, 2004 (UTC)
- moved, kept redirect. If something is written about the OJ case, we would need to re-think the names to minimize confusion. Pete/Pcb21 (talk) 20:41, 12 Apr 2004 (UTC)
- What is with the ô? It's a Japanese name and in Romaji, o is o; there is no ô. Is it an Ainu name?
What does the formal proof require? We say it needs different things in two different places... can we rationalize/improve on this? Pete/Pcb21 (talk) 23:32, 15 Apr 2004 (UTC)
I think this "proof" should not really appear on the page, since the Taylor serie does not always exist, that we can't really reorder the terms of the expansion, and finally that the error term may not be negligeable. And finally, this proof does not give the intuition of why the Ito's lemma is true.. 140.247.43.68 20:23, 19 March 2006 (UTC)
i'm interested in this and other stochastic calc stuff for financial applications, but my background is only differential and integral calculus (called I and II usually), and that was years ago. can someone, please, explain what ito's lemma does or means just in english, without any mathematical symbols? that would be very helpful. thanks guys.
this probably applies to darn near all the math pages, i think. it makes it usable for more people if it has an english-only description (could be even 1 sentence) somewhere in there, preferably at the beginning.
i know you guys are very good with math but some of us plebeians are lost by this sort of notation. :) thanks guys.
- The differential of a function is a linear approximation of the change in the function. That is, the graph of the function is approximated by a straight line or a plane. If f is a function of t and W, then its differential df is a linear function of dt and dW that is approximately equal to the change in f. The differential df is calculated from the actual change by ignoring higher-order terms, e.g. squares and cubes. However, in stochastic calculus dW^2 is not really a higher-order term, and so it cannot be ignored. The order of dW is really one half so that the order of dW^2 is one; in fact dW^2 is equal to dt. The usual formula for the differential of a function has to be modified to take this into account, and this is what Ito's lemma does.—Zophar 15:59, 20 November 2006 (UTC)
- If you have a function f(x), the change in f due to a small change in x is df = (df/dx) dx. For a function of 2 variables g(x,t), a change in g due to small changes in x and t is dg = (dg/dx)dx + (dg/dt)dt.
- But that's for "sure variables". Variables that have a known value at a known time. At time 2, the variable t has a value of 2. We know that for sure. At time 5, the variable x has a value of x(5), with 100% certainty.
- There are things called random, or stochastic, variables (meaning a variable we don't know the value of at a given time until that time has occured) we only know the statistics of that variable. For example, w(3) is 10% likely to be between 9 and 10. Until t=3 happens and w(3) is "realized". They are also called "processes", but they can also be called "stochastic variables" too. Brownian motion is an example of such a process.
- We can have functions of these Brownian motion variables, like f=W^2(t) or f = Log(Sin(gd(arcTanh(1/W(t))))). For such a function, our differential scheme that I labeled above doesn't work, and there's no reason to expect that it does because we're working with a completely different type of function now.
- To get the change in this type of f, due to small changes of these stochastic variables, you need to use Ito's Lemma. That's all it is.
- Your goal is to get the change in f due to small changes in the variables f depends on. For "sure variables", we uses Newton's differential formula (dunno if it has a name). When f depends on stochastic variables, we use Ito's Lemma.
- Sliver 19:32, 3 May 2007 (UTC)
[edit] Entry Does Not Deliver Promise
The sentence "This is not Ito's Lemma, and is in fact just a specialization of the Lemma." is an odd one. I know this as Ito's Lemma. If it really isn't Ito's Lemma, then either this page should be renamed or Ito's full lemma needs to be stated. We can't have an encyclopedia that purports to have an entry named "Foo" and not describe "Foo". Either this page describes Ito's Lemma or it shouldn't be the Ito's Lemma page.
Can someone please tell me what Ito's Lemma is if this isn't it? Otherwise, I intend to delete this sentence since this is the Ito's lemma I've come to know and love.
If there are no objections, I also intend to change some of the wording so that terms don't "disappear" or get "deleted". Instead, we should state that the calculation is performed to O(delta t).
Sliver 19:32, 3 May 2007 (UTC)
Additional to all previously mentioned, does anyone have the correct form for when Ito's Lemma applies to more than one stochastically defined process? I think it's something in the order of basically doubling up on all of the terms and also including a d^2x/(dydz) value within the dt component. It appears to be coming up frequently in quant finance courses. DC34. —Preceding unsigned comment added by Dc34 (talk • contribs) 11:52, 20 October 2007 (UTC)
I removed the line "This is not Ito's Lemma", as it is a confusing, and very strange thing to say right after the definition of Ito's Lemma. This is clearly a case of Ito's lemma, even if it is not in the most general form. I think the general statement for multidimensional semimartingales should be added. If no-one else does this, then I will do this when I have the time.
Sorry, forgot to sign my previous comment.Roboquant (talk) 14:58, 3 March 2008 (UTC)
