Talk:Itō calculus
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[edit] Who is Ito?
This should say who Ito is. Michael Hardy 22:31, 15 Mar 2005 (UTC)
- Why? If you want to know, read the link to his bio. --Prosfilaes 03:08, 30 July 2005 (UTC)
[edit] Itō or Itô
Should it be Itō Calculus or Itô Calculus? The article should be consistent in the math terminology, which I'm sure most math articles are, and not necessarily consistent to the name of the author. It's more important that it be consistent to modern math usage, then one arbitrary translation of the creator's name, IMO. --Prosfilaes 03:08, 30 July 2005 (UTC)
- The proper way I have seen his name written has been Itô. I am uncertain where this 'ō' came from, but I have only seen it used in the Wiki text. Mdbrack 07:36, 20 April 2007 (UTC)
- Neither is "proper", and neither is "wrong". They are simply two different romanizations of the same Japanese name, 伊藤 清. This is treated both on Itō's biographical page and pages on the Hepburn romanization, from which the following line might be helpful to you: "Circumflexes are how long vowels are indicated by the alternative Nihon-shiki and Kunrei-shiki romanizations [as opposed to macrons in standard Hepburn]. Circumflexes are often used when a word processor does not allow macrons. With the spread of Unicode, this is becoming rare." Sullivan.t.j 15:06, 20 April 2007 (UTC)
[edit] Stochastic derivative
The section on the stochastic derivative seems much to long in relation to its importance in Ito calculus. I suggest cutting it down to at most a couple of sentences. OliAtlason (talk) 16:48, 14 February 2008 (UTC)
Although I'm not happy with the state of this article at all the section on the stochastic derivative is the worst. Contrary to how it is stated here, these formulas are nothing new and are simple consequences of the properties of the quadratic varation. I have searched for the citation by Hassan Allouba, which is not freely available online. The only references I have found to this are by Hassan Allouba himself and this page. Also, Hassan Allouba links this wikipedia page from his own webpage (it looks suspiciously like he just added this section himself, refering to his own paper). I suggest this section should be deleted. Roboquant (talk) 15:11, 3 March 2008 (UTC)
Removed.Roboquant (talk) 01:40, 8 March 2008 (UTC)
First, Allouba did not add this section, and it's inappropriate to make unsubstantiated personal accusations. Second, his paper is readily available in the well known journal of Stochastic Analysis and Applications. It's a bit disingenuous to imply otherwise. Third, when talking about Ito's formula, B. Oksendal (p.43 in the fifth edition of his book) states clearly "In this context, however, we have no differentiation theory, only integration theory"; and what IS new here is Allouba's observation and his definition in terms of quadratic variation which yields the "right" definition for the pathwise stochastic derivative. This results in a differentiation theory---complete with the fundamental theorem of stochastic calculus and other crucial differentiation theorems that make the theory useful---which is the counterpart to Ito's integration theory. His theory doesn't appear in ANY of the standard references, including Protter's book, and it deserves to be highlighted. I agree though that the section needs to be shortened.
Section shortened. —Preceding unsigned comment added by Mattrach (talk • contribs) 06:17, 10 March 2008 (UTC)
Apologies for that comment. And, the section does look much better now. Roboquant (talk) 20:50, 20 March 2008 (UTC)
[edit] Causality
The causality of Itō Calculus should be emphasized linguistically, not just stated mathematically. I'd do it, but Stochastic calculus was my worst subject in my twenty-odd years of schooling., so perhaps someone less likely to introduce imperfections could do this. Calbaer 20:57, 10 March 2006 (UTC)
I agree, the need of the Ito integral should be motivated - even from the mathematical point of view: for most processes (Brownian motion and other diffusions, Levy Processes, etc.) one can not simply define the integral pathwise (in the ordinary Riemann-Stieltjes manner), as (almost all) sample paths do not have finite variation. Nevertheless, if the integrand is suitably adapted to the process, the Riemann sums do converge (at least in L^2) to a limit. If I find time, I will do the adding. --Uli.loewe 11:44, 12 April 2007 (UTC)
[edit] Merging
Should not this article be merged in Stochastic calculus? Gala.martin 22:28, 15 February 2006 (UTC)
- I believe that it is better not to merge them. Instead, some one please put all its properties in this page. and the properties of Stratonovich integral on its own page.
Ito integral's properties:
- ito isometry
- there exists t continous version
- its extension
- n-dimension
70.53.188.62 00:50, 9 April 2007 (UTC)
- No, the two should not be merged, for the simple reason that the Itō calculus is only one possible stochastic calculus, the one that arises from the consistent choice of values at left-hand end points in the Riemann-Stieltjes sum. The Stratonovich calculus arises from the consistent choice of values at mid-points in the Riemann-Stieltjes sum. The Paley-Wiener integral is yet another stochastic integral. All are stochastic integrals, although the Itō integral does have a strong claim to being "the" stochastic integral, at least by common abuse of notation. Sullivan.t.j 03:29, 9 April 2007 (UTC)
Should definitely not be merged. Ito calculus is a special subfield of stochastic calculus that deserves its own page given its special applications in ballistics and finance that other stochastic processes fail to describe. It is also a major intellectual breakthrough that deserves separate treatment. It should be integrated with Ito's Lemma which, while being an important argument in Ito calculus, is ultimately a way to increase the applications of the Ito integral. —Preceding unsigned comment added by 130.91.119.95 (talk) 19:17, 18 December 2007 (UTC)
In my opinion, this page should be deleted and Ito calculus redirect to stochastic calculus. Either that, or this page should be re-written from scratch. As it stands, the page Stochastic Calculus is a much better description of the Ito Integral than this page. I'm not even sure what "Ito Calculus" is supposed to mean, and it isn't explained here. Is it just the Ito integral, or is it Ito integral + Ito's Lemma + Ito processes? Roboquant (talk) 15:04, 3 March 2008 (UTC) Changed my mind here, we should keep this page. It needs major improvements though. Roboquant (talk) 00:49, 6 March 2008 (UTC)
[edit] Integration With Respect to Semimartingales
I added this section, and removed the old section "Generalization: integration with respect to a martingale" as it didn't make much sense, was full of mistakes, and is covered by the new section now.Roboquant (talk) 02:04, 7 March 2008 (UTC)
[edit] Rating
I added the maths rating template, rating it as High importance. Ito calculus is certainly very important to probability and statistics, but is also very important outside of maths. It is widely used in finance, and is fundamental to the theory of option pricing (e.g. Black-Scholes). It is also important to stochastic differential equations, areas of physics and in engineering (eg filtering). I propose increasing it to Top. Any comments? —Preceding unsigned comment added by Roboquant (talk • contribs) 22:22, 22 March 2008 (UTC)
[edit] Page move request to "Ito calculus"
The term is popularly referred to as Ito. As of 2008-04-24, Google returns 32,600 hits for "Ito calculus" -wikipedia. For reasons of simplicity, I recommend that this page be moved to Ito calculus. There is no good reason to use a non-standard character; it needlessly fragments search results. Please vote in favor of or against the move, along with your reasons. --AB (talk) 21:16, 24 April 2008 (UTC)
