# Talk:Itô calculus

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## 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)

## 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)
I just checked on MathSciNet. Pretty much all of his articles are signed "Kiyosi Itô", except for those written in the eighties that are signed "Kiyosi Itō". Some of them are signed "Kiyoshi Itô" but again, it is a minority. By that time, every editor would have been able to typeset "Itō" if this is what he wanted. I conclude that either Itô didn't have much of an opinion on how to romanize his name, or he actually preferred "Kiyosi Itô", whatever the standard romanization. Since "Kiyosi Itô" is also the romanization used in various prominent locations (see http://www.ams.org/notices/199808/comm-kyoto.pdf or http://www.kurims.kyoto-u.ac.jp/~kenkyubu/past-director/ito/ito-kiyosi.html), I propose to change the name back to "Itô". Hairer (talk) 13:06, 15 September 2010 (UTC)

The correct answer is "Itō," the macron being the most usual way of indicating the long Japanese "o"; others are "o-o," and "oh." There is no circumflex in any normally used transcription of Japanese, and I would guess that the peculiar intrusion here is the result of somebody having a French keyboard but no easily available macron, and then somebody else having no relevant knowledge or judgement.

The circumflex accent is known to most English-speakers through their exposure to French. In French the "ô," I was taught at Lycee, is the modern version of a "s" or "ss" that vanished in the Middle Ages, and indeed this seems to be reflected in modern phonology. There is no analogous effect, nor historical background, in Japanese-English transcription.

I am very chuffed that the Unicode Consortium introduced a version of Unicode which includes the Welsh y-circumflex at my request. Now you know why Unicode version-numbers run to four digits: there are many odds and ends in many languages. My Welsh, however, is so long gone and forgotten that I've forgotten that bit of pronunciation. I do, however, speak Japanese, and can say it has nothing to do with Welsh, circumflected or otherwise.

The Japanese word "Roido," "Lloyd," is sometimes blamed on me, but no, it's from Harold Lloyd, and described his thin-rimmed round glasses.

DavidLJ (talk) 10:56, 29 June 2014 (UTC)

I spoke to a number of people who knew Itô personally. They all agreed that the way he wanted his name to be spelled is "Kiyosi Itô". As a matter of fact, he was apparently quite upset when people spelled his name as "Kiyoshi Itô" or "Kiyoshi Itō". I believe that ultimately the bearer of a name gets to decide what the "correct" spelling of that name is, so I will revert to his preferred spelling if there is no objection. Hairer (talk) 01:39, 11 September 2015 (UTC)

## 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 (talkcontribs) 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)

This section strikes me as strongly misrepresenting the commonly held view of stochastic analysts for at least two reasons. First, the notion of stochastic derivative "introduced" by Allouba is nothing but the well-known martingale representation theorem (see for example Section V.3 of Revuz and Yor). There is absolutely nothing new in his construction (except maybe for the name), so I do not believe that Allouba is a suitable reference. Second, when stochastic analysts refer to the "stochastic derivative", they are usually thinking about the Malliavin derivative, with the associated "fundamental theorem" being given by the Clark-Ocone formula. The best course of action would be to rewrite this section in a more balanced way by explaining both possible notions of a stochastic derivative and by giving both versions of the associated "fundamental theorem". Hairer (talk) 11:21, 6 July 2010 (UTC)

I agree with the above. The Allouba deriavtive is not found in any textbook on stochastic analysis. The paper itself has only been cited once. It's just not relevant enough to the subject to be included in a short summary. 129.31.204.62 (talk) 12:12, 7 September 2010 (UTC)

Hairer's absolute statements gravely and unfortunately misrepresent Allouba's contribution and ignore facts giving a misleading picture that attempts to rewrite history, and they can't be left unanswered. As B. Oksendal (p.43 in the fifth edition of his book) unequivocally and correctly states when talking about Ito's calculus and formula "In this context, however, we have no differentiation theory, only integration theory". This IS A FACT, period! This quadratic covariation pathwise differentiation theory program, which CREATES for the first time the notion of a semimartingale pathwise derivative with respect to Brownian motion (BM) via their quadratic covariation as well as CREATES an associated systematic and complete differentiation theory for Ito's calculus, is undeniably Allouba's and it has not been done by ANYBODY for over 60 years before. What IS new here is Allouba's DEFINITION of the pathwise derivative of a semimartingale with respect to a BM in terms of their quadratic covariation derivative (QCD)---the "correct" measure of their nearly Holder-1/2 paths regularity---which yields the "right" DEFINITION; and then his building of a systematic complete pathwise differentiation theory counterpart to Ito's purely integral calculus, based on this quadratic covariation derivative (and INDEPENDENT of ANY representation theorems). The martingale representation theorem, without any definition of a derivative of semimartingale with respect to the BM integrator, is a purely integral theorem that is certainly NOT Allouba's differentiation theory; and to say that Allouba's approach is "nothing but the martingale representation theorem" is a gross misrepresentation of his work and is disingenuous at the very best! His theory includes

(1) a QCD fundamental theorem of stochastic calculus (FTSC) that relates HIS quadratic covariation DERIVATIVE of a semimartingale with respect to BM to integral with respect to BM, which is certainly NOT "nothing but the well-known martingale representation theorem" as it also characterizes the integrand as a QCD with respect to the BM integrator and is stated for semimartingales (general) with an independent proof (in fact, Allouba shows how to prove Ito's rule itself from the his QCD fundamental theorem of calculus and his differentiation theory in a separate preprint (upcoming article); it is also shown in a more recent upcoming article by Allouba et al. how to use the QCD FTSC to obtain representations of random variables that are not standard Malliavin differentiable like the Brownian indicator via a QCD variant of the Clark-Ocone formula, without the need for weak derivatives or Hida-Malliavin calculus);

(2) differential QCD chain rules (not Ito integral rule);

(3) a differential QCD mean value theorem; and

(4) different QCD differentiation rules.

This theory most certainly has not been given before Allouba ANYWHERE. This theory has also been generalized most recently to a very general theory, in an upcoming article by Allouba, that covers processes of different order and types variations, including many processes that are outside the traditional Gaussian, semimartingale, Markovian classes (certainly no martingale representation result there!).

It's one thing to say everybody knew about quadratic covariations and their simple facts; and quite another to say somebody thinking of a derivative of semimartingale with respect to BM then RECOGNIZING that the derivative of that quadratic covariation IS the right definition of a semimartingale derivative with respect to BM, giving an anti-Ito integral and leading to a systematic differentiation theory counterpart to Ito's integral calculus. One that remained a purely integral calculus for over 60 years, despite the many giants who wrote extensively about Ito's calculus without making Allouba's observation and/or seeing the resulting differentiation theory and despite their obvious knowledge of the martingale representation theorem (not a single one of them! and Allouba's approach doesn't appear in ANY of the standard references). This makes his results all the more (not less) notable. Others have recognized the novelty and significance of his approach, that's why they have added his contribution to the Wiki site. It is most certainly NOT nothing new, as is very inaccurately and sinisterly implied. Every new discovery/or approach becomes obvious and linked to prior results AFTER it is made, and one can argue that every theorem has its ingredients somewhere before it was discovered and put together. It can't be a political process whether to decide to accredit somebody with a discovery or approach or to simply claim that that result or part of it is included in some ambiguous way in a prior result, based on whether we like their name or not. Allouba's differentiation approach deserves to be highlighted in the Ito calculus section where it belongs, along with reference to his original article. Anything less is petty, motivated by nonscientific motives, and doesn't serve the scientific goal of timely dissemination of knowledge and proper accreditation of discoveries (not robbing people of their due credit!). This IS the way to be balanced. In fact, one can argue that NO CALCULUS is complete without BOTH an INTEGRATION AND A DIFFERENTIATION theories.

I do agree with Hairer about the name. It should be written under the name quadratic covariation derivative and not simply stochastic derivative to avoid confusion with Malliavin's derivative. But I think the Malliavin calculus with its tremendous applications needs a separate article altogether.

Changed the name. —Preceding unsigned comment added by AmericanProbabilist (talkcontribs) 18:49, 11 October 2010 (UTC)

If this is an important new concept, it will soon be introduced and cited in textbooks. Until then the article should not give undue weight to this new concept. Best regards Ulner (talk) 19:17, 11 October 2010 (UTC)
So far, according to MathSciNet and scholar.google.com the article by Allouba has no citations. Ulner (talk) 21:35, 11 October 2010 (UTC)
For these reasons I have deleted the section about this. Ulner (talk) 21:36, 11 October 2010 (UTC)

The article has been cited once already and is cited twice in 2 upcoming articles, with more to come. Returned the section. Best, —Preceding unsigned comment added by AmericanProbabilist (talkcontribs) 23:02, 11 October 2010 (UTC)

Three citations is a rather low number of citations if this is an important concept; by the way, which articles are citing the Allouba article? Ulner (talk) 20:40, 12 October 2010 (UTC)
I suggest that this section is deleted; one reason is that it is not widely cited by other articles or cited in textbooks and hence not of great importance. Ulner (talk) 20:17, 13 October 2010 (UTC)

I suppose that I shouldn't bother to try to reply to the long rant (kind of reminds me of the hate-posts that show up whenever someone blogs about El Naschie). Maybe I "gravely and profoundly" misrepresent Allouba's achievements and maybe he will be hailed as a genius for generations to come. For the moment, I believe that I faithfully represented the prevailing opinion of active researchers in the stochastic analysis community of which I am one. Actually, a much more useful and profound theory, which can be interpreted as a pathwise theory of differentiation with respect to a Brownian path, is given by Gubinelli's theory of controlled rough paths. However, this is in my opinion somewhat too technical to be presented here and of marginal interest for the relatively general audience targeted by this article. Hairer (talk) 22Here:52, 13 October 2010 (UTC)

Here are my two cents : I agree that this section should be deleted. The main reason is that Allouba's theory is not notable among probabilists. This can't be debated. The number of citations of the paper and the impact factor of the journal (Stochastic Analysis and Applications) are objective proofs, but any researcher in probability (not working directly with Allouba...) would not even need to look at that. The second reason is that I don't think his theory really brings anything to stochastic analysis. But this can be debated, as user AmericanProbabilist does. However, if this "stochastic differentiation" theory is really a "novel and significant approach" as claimed by AmericanProbabilist, then the paper will be cited and notability criterion will be meeted in the future. Until then, I suggest the section be deleted.

Two footnotes : Firstly, I read Allouba's paper, and the comparison with El Naschie made by user Ulner seems unfair. Although as I said the interest of Allouba's paper should be debated, there is no question that the paper is clearly written and is mathematically rigorous. Secondly, and more important, AmericanProbabilists talks about "proper accreditation of discoveries". The paper is on arxiv, so I think the 'discovery' is already properly accreditated. And regarding 'dissemination of knowledge', I think talks and conferences are better suited to this purpose than wikipedia, when it comes to new results in stochastic analysis or probability in general. Actually, all this leaves me under the impression that the people restoring the section regularly are more concerned about Allouba's fame among the general public than accreditation of its 'discovery' or dissemination of knowledge, as suggested by Hairer. 82.232.50.199 (talk) 10:58, 14 October 2010 (UTC)

Sorry if my comment was misinterpreted. I didn't mean to suggest that Allouba was in any sense comparable to El Naschie. What I meant was that the passionate and lengthy rant written by one of his defenders reminded me of similar rants written by El Naschie's defenders as soon as he is criticised somewhere. I read Allouba's paper and although I stand by my opinion that it is no way sufficiently significant to be mentioned on this page, it is definitely mathematically correct and of publishable quality (unlike most of El Naschie's work....). Hairer (talk) 20:54, 14 October 2010 (UTC)

Just for the record, I have never compared the paper to El Naschie (as claimed above by 82.232.50.199). I have deleted the section for reasons explained earlier. At the same time this is not a judgement in positive or negative way about the paper by Allouba; it is not (at this stage at least) widely cited or used in textbooks - and for this reason inclusion in Wikipedia would give undue weight to the research explained in this paper. Ulner (talk) 22:28, 15 October 2010 (UTC)

I'm a probabilist & a stochastic analysis expert & I too agree with AmericanProbabilist that Hairer's opinion doesn't represent mine. The bottom line is that Allouba made connections that none on the impressive list of authors of books about Ito's calculus made for a very long time before him (over 60 years,) thereby creating an elegantly simple stochastic differentiation theory for Ito's calculus (with many interesting applications & far reaching generalizations to come.) A natural & fundamental question for the public, or starting mathematicians, or those outside the field is why doesn't this calculus come with a differentiation theory counterpart to Ito's purely integral calculus in Ito's setting? Allouba's work answers that elegantly & gives that theory, without requiring any extra settings from outside of Ito's calculus. His work needs to be highlighted to the public, since many people who are looking for an encyclopedic view of the subject should not be led to believe that his quadratic covariation differentiation theory for Ito's calculus doesn't exist when it actually does. Citations for Allouba's recent result are increasing. The section is balanced & well written for the Wikipedia, & I returned it to its rightful place. It shouldn't be deleted.

As for the journal, Stochastic Analysis & Applications is a respected journal with several top probabilists on the editorial board, & many top probabilists published & continue to publish there. Number of citations & specific journals' names are not always reliable metrics to judge a recent work (especially one that looks at the subject in a totally nontraditional way,) nor are they always objective. Without making any comparisons, Perelman's work, undeniably some of the finest & most impressive recorded mathematical work is not even published in any journal; only on the arXiv site. It is cited zero times as a result on Mathscinet, though many times outside it. It clearly is far superior to most articles in the highest standard journals.

The long & detailed response of AmericanProbabilist is not a rant & it is disrespectful of Hairer to characterize it as such. It was necessary to respond to his misrepresentation of Allouba's result & his overuse of the quotation marks & other less-than-flattering terms to try to detract from the value of that article. His invocation of the totally inappropriate & irrelevant EL Naschie afterwards doesn't serve any scientific purpose at all, & was totally out of bounds. —Preceding unsigned comment added by JRMATH (talkcontribs) 00:58, 20 October 2010 (UTC)

I am still not convinced. Regarding citations, even Google scholar finds no citation of Allouba's result, so I see no evidence that the number of citations to this result "is increasing". Regarding AmericanProbabilist's response, I qualified it as a rant because of its tone, its overuse of uppercase letters, and the fact that it made strong statements without real argumentation. The reason why I stated that Allouba's theory is nothing but the martingale representation theorem is that the latter precisely states that if M is a martingale with respect to the filtration generated by a Brownian motion B, then there exists a (unique) process X such that M_t = \int_0^t X_s dB_s. Furthermore, X can be recovered as the derivative of the quadratic covariation of M and B. (See for example the book by Revuz & Yor.) There is no new insight there at all, and this is reflected by the fact that the article has never been cited. Giving it this kind of prominence just because "it exists" seems misleading at best. As a sidenote, I am surprised by the fact that AmericanProbabilist has access to many "upcoming articles" by Allouba, none of which seems to be publicly available (neither from Allouba's homepage, nor from arXiv). Hairer (talk) 23:51, 24 October 2010 (UTC)

Anyone can choose to remain unconvinced despite the facts. I have explained in great details that the representation theorem is not (by a long shot) Allouba's differentiation theory. He and he alone made the connection (through his definition) between the derivative of the quadratic covariation and the derivative of semimartingale wrt Brownian motion, and developing that concept into a complete differentiation theory for Ito's calculus. Citations are increasing, papers can even be submitted without being publicly available. I also agree with all the others that it shouldn't be deleted. —Preceding unsigned comment added by AmericanProbabilist (talkcontribs) 02:12, 25 October 2010 (UTC)

I have been asked to look at this discussion as an uninvolved outsider with a mathematical background. I am a mathematician, but have no prior knowledge of this particular subject. However, with or without knowledge of the subject, it is not difficult to assess this issue in terms of Wikipedia's policies and guidelines. It is also clear that Allouba's contribution has not been widely cited, and there is a complete lack of the sort of substantial independent coverage which is required by Wikipedia's standards. The arguments for inclusion of the passage make heavy use of such expressions as "dissemination of knowledge", "Allouba's differentiation approach deserves to be highlighted", in other words the view is being expressed that the article should include coverage of Allouba's work in order to publicise it. Furthermore, the lack of existing coverage is used as an argument in favour of inclusion, as for example in "His theory doesn't appear in ANY of the standard references, including Protter's book, and it deserves to be highlighted", and again "... and Allouba's approach doesn't appear in ANY of the standard references". With all due respect for the user who has argued so passionately along these lines, a greater knowledge of Wikipedia's guidelines and policies would have made it clear that these are in fact reasons for exclusion, not for inclusion. Wikipedia does not include new or little known work in order to give it more publicity, no matter how important that work is or how deserving of being better known it is. We also have remarks such as "It can't be a political process whether to decide to accredit somebody with a discovery", and reference to excluding the passage as "robbing people of their due credit", suggesting that at least one purpose of including the material is to get accreditation for Allouba, which again is inconsistent with Wikipedia's standards. Wikipedia's basic criterion for inclusion of material is that it has received substantial coverage in independent reliable sources, not that it is important or deserving, or that its author deserves greater recognition.
Wikipedia works by consensus, and editors should be willing to back out gracefully if there is a consensus against their position. There is consensus against giving prominence to Allouba's contribution. The only dissent is from a small number of single purpose accounts and one single purpose IP (assuming we ignore a single vandalism edit so long ago as to be irrelevant, and probably from a different person).
It is also very unfortunate that we have numerous remarks imputing negative motivations to editors. (For example, "petty, motivated by nonscientific motives", "based on whether we like their name or not", "a political process whether to decide to accredit somebody", "Anyone can choose to remain unconvinced despite the facts".) I see no evidence at all that the arguments for exclusion were made in anything other than perfectly good faith, and I strongly urge all concerned to be willing to assume good faith in the absence of evidence to the contrary.
As I said at the beginning of this post, I have come here in response to a request, as an uninvolved outsider. I have looked at the matter from a completely external perspective, with a view to helping resolve a dispute, not with a view to being partisan. As will be clear from my comments above, it is perfectly clear that based purely on Wikipedia's policies and guidelines no case has been made for inclusion of the disputed passage. I am aware that, having said that, I am likely henceforth to be seen as an involved party, perhaps particularly by the editor responsible for the passage. Nevertheless, I prefer to remain as uninvolved as I can for the present, and I am not myself removing the passage from the article. I say that without prejudice as to any further steps I may make in the future. JamesBWatson (talk) 10:40, 3 November 2010 (UTC)
I've edited the controversial section to begin with, for want of a better term, "differentiation by formal interpretation of the martingale representation theorem". This can be seen as a motivator for Allouba's QCD — Allouba's definition is set up so that the fundamental theorem of calculus for the QCD is isomorphic to the martingale representation theorem. Hopefully without sparking yet another edit-revert war, this will shift some of the spotlight onto more conventional and well-known differentiation ideas while also preserving some mention of Allouba's work, which seems to matter so much to some contributors. Sullivan.t.j (talk) 00:47, 2 December 2010 (UTC)

I've attempted to remove the reference in question, though my changes have been reverted. I suggest we seek the opinion of an informed third party. Failing that, I will apply for formal dispute resolution. I'm new to editing wikipedia, so I apologise if I've breached etiquette along the way. In my opinion, the "Allouba derivative" is actively misleading. People who attempt to learn about stochastic calculus from this page will pick up a highly nonstandard viewpoint on the subject. Indeed, this happened to me when I first studied stochastic analysis in 2007. SimonL (talk) 22:02, 16 August 2011 (UTC)

Many informed independent experts have already decided to keep it, ending with Sullivan.t.j, who is also clearly an independent third party, and who has done a nice job re-writing the section in a balanced and clear way. Wikipedia articles are about describing as completely as possible facts that exist in the human body of knowledge in a given area. Removing Allouba's quadratic covariation differentiation theory gives an outdated, distorted, and fundamentaly incomplete (what you call standard) view of Ito's calculus as it currently stands. All's theory not only gives a complete differentiation theory counterpart to Ito's integral calculus, with a definition of derivative that's independent of the integral, but it gives new variants of famous results that compare well with Malliavin's derivative (which no one in his right mind can discount), see for example Allouba latest arXiv article together with Fontes (accepted for publication) for one of many applications of All's theory. Writing a section on differentiation in Ito's calculus without including All's theory (as you keep doing in your reverts) is simply distorting the facts. Pretending that All's theory is not a worthy addition to Ito's integral calculus is like saying only integrals are important in a given calculus, which doesn't pass the laughing test. Keeping All's QCD section as an integral part of differentiation in Ito's calculus setting is only fair and is the right thing to do to give a complete updated view of Ito's calculus to the wiki readers. I've reverted to the Sullivan.t.j, balanced version. RHarryd (talk) 02:46, 17 August 2011 (UTC)

As was mentioned earlier in this section: "Wikipedia's basic criterion for inclusion of material is that it has received substantial coverage in independent reliable sources". MathSciNet lists 0 citations of this work. google scholar lists five citations. Of these, the only one that Allouba was not clearly directly involved with is not written in English. As far as I can see, the theory has received neither substanstial nor reliable coverage. Whether or not multiple so-called experts on the wikipedia talk page agree or disagree is irrelevant. Anyone can claim to be an expert, and multiple accounts can come from one poster. I'm not claiming that this is what is happening, but I'm saying that the "majority vote" argument doesn't hold much weight (though I believe that in this case the "majority" are in favour of removal). What I propose is that we ask someone who is well qualified to comment on the relevance of differentiation in Ito calculus. This includes the martingale representation theorem section and the Allouba derivative. Would you be amenable to this idea RHarryd? It's futile to get stuck in an edit war. SimonL (talk) 11:44, 17 August 2011 (UTC).

Sullivan.t.j already did that in a balanced way as explained earlier. Again, as I said, many respected experts have already, regardless of your opinion, decided in favor of keeping the Allouba differentiation section. Besides, and more importantly, math is about facts. The relevance of Allouba differentiation theory to Ito's calculus is a fact and not a matter of opinion: his derivative is an anti-Ito integral which leads to a complete differentiation theory counterpart to Ito's integral calculus. It was proven and appeared in a peer reviewed respected journal so it is definitely reliable. It (and a subsequent important application of his theory that's accepted and is to appear) is widely disseminated on respected archival sites all over the world including arXiv. Also important, number of papers and citations are snapshots in time and are not always a reliable measure for how great a mathematician or work is. With no comparisons being made here, consider the Fields medalists, and undoubtedly two of the truly great mathematicians of our times, Bao Châu Ngô and Laurent Lafforgue. Ngo has "only" 19 papers since 1997 and Lafforgue has "only" 17 papers since 1996. The highest cited Ngo paper is currently "only" cited 20 times since 2002 and one of his important papers (in which he proves a famous Frenkel-Gaitsgory-Kazhdan-Vilonen conjecture) is only cited 5 times since 2000 (of which 2 are his citations). In addition, two of Lafforgue important papers (published in the Journal of the AMS and inventiones) are cited "only" 7 and 13 times since 1998 and 1999, respectively, in mathscinet. There are definitely many examples of lesser mathematicians with many more papers and citations in the same time span. I can go on and on, the point is made. As I said earlier, Wikipedia articles are about describing as completely as possible facts that exist in the human body of knowledge in a given area. All's QCD theory is a proven differentiation theory in Ito's calculus setting, so it is most definitely relevant since his derivative is the anti-Ito integral which is undeniably important, that's a fact. No amount of "opinion" or name dropping is going to change that. The unbiased written section on differentiation in Ito's calculus by Sullivan.t.j occupies one of 11 sections in Ito's calculus, and it doesn't even contain Allouba et al. latest accepted and very relevant results that express the integrand in the mart rep theorem in terms of his QCD of a conditional expectation, which as I said above compares very well with Malliavin famous derivative version. So, if anything, the section is not big enough, but it relays enough information so as to cause the interested wiki reader to go explore deeper and further. It therefore is reasonable and should not be removed at all. I am very firmly convinced of that.RHarryd (talk) 23:13, 17 August 2011 (UTC)

Which "experts" are they? If the user "Harier" is Martin Harier, then we already have one expert who disputes the relevance of this content. I attempted to build consensus by offering to consult a third party. It appears you are not happy to do this. Is that correct? How do you propose we resolve this dispute? SimonL (talk) 23:50, 17 August 2011 (UTC)

I'm simultaneously flattered and rather annoyed to see that my earlier edits are being held up as evidence of notability for the Allouba derivative. To avoid any conflict of interest, I will refrain from commenting on whether or not I think my particular version of the article should stay. Instead, I will offer some comments on the general situation and the reasoning behind my edits.
The Wikipedia standard for "notability of X" is how much traction X gets outside of Wikipedia, not the volume of argument that X generates inside Wikipedia. As the author of my own edits, I can say what their motivations were and how they should be interpreted. Quite simply, as a function of time, the Allouba derivative section was a sequence of inelegant messes, trapped in a bitter and acrimonious delete-revert cycle worthy of articles about politics and religion. (This, it should be clear, is not a compliment.) I offered my so-called "balanced" description of differentiation for three reasons: (1) it covered the conventional Martingale Representation Theorem way of doing things; (2) it mentioned the Allouba derivative, which has been peer-reviewed and published even if it has not yet received great acclaim; but most importantly, (3) as an attempt at a compromise that would cause everyone to shut up and stop their edit-warring, which it succeeded in doing for several months.
(While we're on the topic of history and singular contributions, there's no One True Way to Define the Derivative. I'm told that Gauss himself gave six different derivations of the Divergence Theorem.)
I do hold a PhD in mathematics; I am not an expert in stochastic analysis (although Martin Hairer is), and I was attempting to confer neither additional legitimacy and acclaim nor scorn and doubt on the Allouba derivative — this last point should be clear to all. Yes, the Allouba derivative is peer-reviewed and published work, and is relevant to Itō calculus. Wikipedia is not the right forum for attempting to elevate it beyond that humble status. (There is also, sadly, a lot of peer-reviewed and published dross in this world.) If/when MathSciNet, Zentralblatt Math and the mathematical community in general start to publish reviews that praise the Allouba derivative, then Wikipedians will be right to make edits reflecting that. In the meantime, I suggest that everyone keep calm, try to stay non-partisan, and (a personal plea) don't take my earlier edits out of context. Sullivan.t.j (talk) 04:15, 18 August 2011 (UTC)
• Both of you are edit warring. You may not revert someone more than three times within a 24 hour period. This issue is whether or not this section [1] is important. How many college level textbooks about calculus mention this guy and his formula? Dream Focus 10:15, 19 August 2011 (UTC)
I'm sorry if I broke etiquette. As I mentioned earlier, I'm new to editing, so I'm still on the learning curve. In answer to your question, no textbooks mention the result or the author. SimonL (talk) 10:41, 19 August 2011 (UTC)
Then I say remove it. I checked and didn't find any textbook mention of it either. Just now, searching this page for the word "textbook" I see others have brought this up as well. Some claimed a year ago it'd soon be in textbooks, but that apparently has not yet happen. If it gets enough coverage in scientific journals or other reliable sources, it can have its own article, but it doesn't really belong here. Dream Focus 10:47, 19 August 2011 (UTC)

As mentioned above a couple times by users Mattrach and AmericanProbabilist, the standard book of Oksendal, which predates Allouba's theory and which obviously contains the classical martingale representation theorem, says explicitly "In this context, however, we have no differentiation theory, only integration theory", the context of course being Ito's setting. So, keeping the differentiation section, pretending that the classical mart rep theorem "formal" interpretation is all there is in the context of differentiation in Ito's setting, while at the same time purposely deleting any contribution by Allouba (which is both complete with fundamental elements of differentiation and rigorous and not merely "formal") or even reference to Allouba's paper is a deliberate and unacceptable distortion to the clear history and scientific facts. As to Dream Focus' comment, I've read the discussion above, what JRMATH said precisely was "Citations for Allouba's recent result are increasing", and he/she is absolutely correct we have already 5 Google scholar citations so far for his recent article (I have addressed number of citations and relevance above). AmericanProbabilist also said "Citations are increasing", also correct. Neither mentioned textbooks, so your comment is imprecise. Textbooks tend to take longer to include even big results, and facts are the relevant criteria here since All's QCD theory is fundamentally relevant to this section, we can't pretend it doesn't exist. Reverted to the historically accurate and balanced account.RHarryd (talk) 16:12, 19 August 2011 (UTC)

You *are* Mattrach and AmericanProbabilist, no? William M. Connolley (talk) 16:20, 19 August 2011 (UTC)
Special:Contributions/AmericanProbabilist, Special:Contributions/JRMATH, and Special:Contributions/RHarryd all have no edits outside of this article and its talk page. Anyway, post on the Wikiprojects for math and science, and get additional feedback from them. Dream Focus 16:42, 19 August 2011 (UTC)
You keep asserting that it's relevant and that this view is held by many people. You're not actually giving any evidence for this assertion. Three of the articles on google scholar are written by Allouba or his students. The other two are duplicates of a Serbian article on modelling central heating systems. So yes, you're right. The number of non-self citations has increased from 0 to 1 since 2006. If that's the threshold for notability, then this will turn out to be a very long article. I suggest you stop reverting the edits RHarryd. It's getting to the stage that your actions may actively hurt this theory and damage Allouba's reputation. SimonL (talk) 16:51, 19 August 2011 (UTC)

William, obviously you're way off bounds with your erroneous suggestion. SimonL, it's you who should stop reverting the edits, which have been stable for months since Sullivan.t.j. modifications. Your discussions are very unconvincing. The QCD subsection is simply stating a fact about differentiation in Ito's calculus, I, and many other see that deliberately omitting the QCD aspect and its proper attribution to its author in a section that is devoted to differentiation in Ito's setting is simply distorting facts by omitions. That's a no brainer. No one is hurting Allouba's reputation here. A threatening tone doesn't work well in these scientific discussions, so adjust your tone accordingly if you are really after a sincere conversation. The question is do we omit a mathematical work that's relevant? a differentiation theory counterpart to an integration theory is always relevant. You keep intentionally ignoring the other people who have clearly articulated that opinion. I just talked about # of citations and their relevance in the context of two super mathematicians above, very clearly before, I'm not repeating myself. Repeating ourselves is not going to change my convictions nor, apparently, yours. And that's perfectly fine with me.RHarryd (talk) 20:14, 19 August 2011 (UTC)

Perhaps I should introduce myself -- my name is Simon Lyons. If you're a seasoned probabilist, maybe you'd be open enough to share your real name? SimonL (talk) 19:05, 22 August 2011 (UTC)

Since I got somehow drawn into this "controversy", let me make a few points:

• Academic credentials. This seems a somewhat silly route to go down. However, anyone performing edits on the grounds of having some academic credentials should disclose their identity, otherwise this is completely meaningless. (In the interest of full disclosure, my username is indeed my real name and there is only one probabilist with this name.)
• Citation counts. Yes, the citation count of an article is not necessarily a reflection if its importance, especially in mathematics. There are for example excellent articles that close a subject area by proving the main result for which the area was created. Such articles are extremely important and will usually receive low citation counts. In the current situation, the article in question is meant to open a subject area so that one would be entitled to expect a high citation count if it had indeed been picked up by the probability community.
• Comparison to Fields medallists. The two works by Bao Châu Ngô and Laurent Lafforgue mentioned by RHarryD do not seem to be mentioned in any Wikipedia article, so I do not quite see the point. If anything, this confirms the fact that if a given piece of work does not gain sufficient notoriety, then it is not appropriate for an encyclopaedia, whether the said piece of work was created by a Fields medallist or not. In any case, alleged future notoriety (citation count will increase) does not count and Wikipedia is not meant to be a vehicle for gaining notoriety.
• The comment by Øksendal. If every paper that addresses a comment by a notorious probabilist was mentioned on Wikipedia, this would become completely unmanageable. In this particular case for example, a theory that addresses the exact same comment, but with a more "pathwise" perspective is Gubinelli's theory of "controlled rough paths". See also Section 2.3 in this article for a summary of the theory which I hope is slightly easier to read.
• Single purpose accounts. The account AmericanProbabilist was created on October 11, 2010 and did his first edit on this page on the same day. The exact same thing is true for JRMATH (October 20, 2010), Profezeuss (November 19, 2010), RHarryD (August 15, 2011) and JoshLev247 (August 21, 2011). With the exception of one minor edit to the "Quadratic Variation" page by JoshLev247, none of these accounts performed any edit outside of this page as of today, so one can certainly consider them as single-purpose accounts.

Hairer (talk) 10:12, 23 August 2011 (UTC)

## 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)

## 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)

## 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)

## 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 (talkcontribs) 22:22, 22 March 2008 (UTC)

## 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)

Itô calculus gives many more hits than Ito calculus (but Google is smart enough to know that it's basically the same character anyway, so this is not reliable). If one wants to change the spelling, I suggest Itô instead of Itō, since this is what is being used consistently throughout most of the scientific literature. Hairer (talk) 08:44, 14 October 2010 (UTC)

As it is, this article is nicely written for mathematicians but hardly readable for physicists. This is really a sad state of affairs, since the same is true for most math textbooks on stochastic calculus. I would love to read here something in more familiar notation, e.g.

In physics, usually stochastic differential equations are used instead of stochastic integrals. A physicist would formulate an Ito stochastic differential equation (sde) as

${\displaystyle {\dot {x}}_{k}=h_{k}+g_{kl}\xi _{l}}$

where ${\displaystyle \xi _{j}}$ is Gaussian white noise with ${\displaystyle \langle \xi _{k}(t_{1})\,\xi _{l}(t_{2})\rangle =\delta _{kl}\delta (t_{1}-t_{2})}$ and Einstein summation convention is used.

If ${\displaystyle y=y(x_{k})}$ is a function of the ${\displaystyle x_{k}}$, then the Ito chain rule has to be used

${\displaystyle {\dot {y}}={\frac {\partial y}{\partial x_{j}}}{\dot {x}}_{j}+{\frac {1}{2}}{\frac {\partial ^{2}y}{\partial x_{k}\partial x_{l}}}g_{km}g_{ml}}$

An Ito sde as above corresponds to a Stratonovich sde which reads

${\displaystyle {\dot {x}}_{k}=h_{k}+g_{kl}\xi _{l}-{\frac {1}{2}}{\frac {\partial g_{kl}}{x_{m}}}g_{ml}}$

Stratonovich sde frequently occur in physics as the limit of a stochastic differential equation with colored noise if the correlation time of the noise term approaches zero. For a recent treatment of different interpretations of stochastic differential equations see for example Lau, Lubensky: State-dependent diffusion, Phys. Rev. E, 2007.

--Benjamin.friedrich (talk) 09:45, 28 August 2008 (UTC)

## Errors

I might be mistaken, but does not the infinite variation only hold almost surely ? —Preceding unsigned comment added by 129.240.176.119 (talk) 12:11, 5 April 2009 (UTC)

The article is full of such imprecisions. Commentor (talk) 14:03, 9 July 2010 (UTC)

I'm a professional mathematician (albeit an algebraist), and I find this article incomprehensible. It suffers from the same shortcomings as many other technical articles on Wikipedia: It doesn't explain its terminology (which certainly does not belong to the mathematical mainstream), it doesn't motivate, it provides too few examples. I'm not competent to improve it, but it would be worth for someone who is to rewrite it. —Preceding unsigned comment added by 128.240.229.65 (talk) 17:06, 27 October 2008 (UTC)

• Wikipedia is not about teaching things. It's about describing things that exist in the human body of knowlegde. If you want a textbook about Ito calculus, go to Wikibooks. That said, I have a complain: the article could improve if Ito integral was a _separate_ article, with less theoretical and more practical aspects of it, like a table of Ito integrals. Albmont (talk) 20:05, 28 July 2009 (UTC)

No, no, I agree. The article is incomprehensible. I have a Ph.D. in pure mathematics and I work in finance. The article is incomprehensible even though I know what it is about. I will see what can be done. Commentor (talk) 14:04, 9 July 2010 (UTC)

## Semimartingales as integrators

This chapter suddenly introduces a new notation In general, the stochastic integral H  ·  X can be defined even in cases where the predictable process H is not locally bounded., which is used in the next chapters, without giving a proper definition. IS this the same as ${\displaystyle \int _{0}^{t}H\,dX}$? Albmont (talk) 14:55, 12 August 2009 (UTC)

xvvxvx —Preceding unsigned comment added by 132.181.52.55 (talk) 01:53, 10 February 2010 (UTC)

## Further extensions section

I have added a clarify tag (put it back again) because the notation here is still unexplained. If the article quadratic variation is supposed to be supplying this inforamtion it fails to do so as it uses a different notation (square brackets rather than angle). Melcombe (talk) 09:51, 18 February 2010 (UTC)

## Formula missing descriptions

The 5-tuple characterizing a filtered probability space has missing or inadequate descriptions, but the underlying article also does not provide these descriptions. Vonkje (talk) 19:04, 18 November 2010 (UTC)

Actually, the underlying article provides an explanation in the "Measure Theory" section towards the end. The notation is basically the same with ${\displaystyle \Omega }$ denoting the underlying measure (probability) space, ${\displaystyle \{{\mathcal {F}}_{t}\}}$ the filtration, ${\displaystyle {\mathcal {F}}=\bigcup _{t}{\mathcal {F}}_{t}}$, and ${\displaystyle \mathbb {P} }$ the underlying probability measure.Hairer (talk) 18:09, 21 November 2010 (UTC)

## Differentiation in Itō calculus

This section is poorly-written and needs to be revamped. The differentiation theory has to be stated correctly with proper citation. AaronKauf (talk) 18:24, 29 August 2011 (UTC)

• Comment - There are couple of users here who are willing to fudge this section as long as it doesn't mention the work by author Allouba. Apparently they have personal issues with the author and they are settling a score without caring about mathematical facts. The Wiki is not a vehicle for this kind of behavior, and it is not tolerated. The revised version written by RHarryd should be put back and users and editors should be allowed to discuss it. The version was immediately removed by Mathsci without giving a chance for others to even read it. This domineering behavior is against Wiki policies. AaronKauf (talk) 18:16, 10 September 2011 (UTC)
I am actually in favour of having that whole section removed. As stated below a number of single-purpose accounts, of which RHarryd is the latest, have tried to add exactly the same material over the years. As explained on this talk page by regular watchers/editors of this page, the present version was a compromise to stop the constant addition of WP:UNDUE material on what is claimed to be "Allouba theory" and make the article stable. If AaronKauf, another recently arrived user, could tone down his language and assume good faith, that would be helpful. Mathsci (talk) 06:15, 11 September 2011 (UTC)
Your account has been single-purpose so far like the others described below; it is indistinguishable from those previous accounts. Again this content has not yet been recognized as notable as can be gauged from mathscinet and its non-inclusion in academic textbooks. There is a huge literature on the Ito calculus. The wikipedia article can only represent what has been assimilated and already recognized in the academic literature, in this case academic textbooks. All the material that has appeared in the section on differentiation, including the current content, is WP:UNDUE or WP:OR. There is a huge literature on the Ito calculus and it is not wikipedia's role to record all the content from individual papers. Wikipedia uses academic textbooks as the main sources in this particular context. On wikipedia personal opinions of editors on the merits of individual contributions are irrelevant, as we can only go by what secondary sources say, in this case academic textbooks. Personal opinions, as far as wikipedia is concerned, are just WP:OR and WP:SYNTH. Please see WP:NOTCRYSTALBALL. Making a comparison between Allouba and Fields medallists is also not particularly helpful. Thanks, Mathsci (talk) 04:53, 12 September 2011 (UTC)

RHarryd: could you please elaborate on what is "not rigorous" in the current version? Thanks. Hairer (talk) 12:22, 12 September 2011 (UTC)

Couple of points addressed to all parties:

• The quadratic covariation theory is a relevant part of Ito Calculus and should be included in this section. We are stating mathematical facts here. And even though the Malliavin Derivative is in the Gaussian setting, I would leave it in this section to satisfy all parties.
• On the textbook issue: There are several mathematical sections in the Wiki referencing research articles that didn't appear in textbooks. Advanced research articles often lead to the birth of a new book, however it is up to the author to decide what to include or exclude. The reference to the article of the quadratic covariation theory is peer-reviewed and should be cited in this section.
• On the “single-purpose” account issue: The Wiki is a vehicle for everyone to provide his or her own expertise in a certain field. That's why you will find what you call “single-purpose” accounts plastered all over the Wiki from the sports sections to the science sections where one provides the knowledge as a service for others outside the field. Not everybody has the luxury or time to sit all day contributing to the Wiki, especially that we're not getting paid for. My kids are participating in a project sponsored by Wiki, and they have those "single-purpose" accounts because all they care to contribute to is the mobile apps programming in which they are proficient.
• Now on the other hand, there are couple of accounts that pop up out of nowhere and they have one single mission: deleting and reverting parts of this section just because the word "Allouba" happens to appear in it, in some cases enticing an edit war. Ironically, those accounts are all from England, and probably were summoned by one big macher. Which begs the question, on what grounds they have the authority to veto? Contributing to the Wiki has to be done in a democratic environment.
• Finally on the point of notoriety: Notoriety is achieved based on one's work and accomplishments, not on a couple of lines in the Wiki! Common guys! As mathematicians, we are providing mathematical facts to the public. Mathematical researchers would turn to math engines like the arXiv or MathSciNet for their research. Usually, the ones who resort to the Wiki for reference are either students or non-mathematicians (e.g engineers) who don't like to bother with heavy mathematical articles. They are however provided with reference in case they need to investigate a subject further.

So, let me sum up by saying that we should all work together in the spirit of the Wiki, in a democratic friendly way, to write this section precisely and coherently, with all parties involved from both sides of the Atlantic. As I said, I would put the quadratic covariation theory along with the Malliavin derivative. The current section as it stands is not complete. And before anyone starts to spar, let's take a deep breath, watch some baseball or soccer, then come back here for a friendly discussion. Cheers! AaronKauf (talk) 21:12, 15 September 2011 (UTC)

I have restored the Allouba QCD section as it is inseparable from the differentiation section, and should never be removed. I have retained the Malliavin derivative section, even though it has its own page and even though the Malliavin calculus is a Gaussian calculus not Ito calculus. The section on the mart rep thm is not mathematics as it was: one cannot just say we can "formally" interpret something as a derivative without giving at least a reference to a rigorous treatment of this assertion or relate it to a rigorous treatment. The Williams reference is not enough since it does not give the differentiation theory.RHarryd (talk) 15:14, 22 September 2011 (UTC)
The arguments for not having this material have not changed and will not be repeated. Mathsci (talk) 16:14, 22 September 2011 (UTC)
What is your mathematical basis? Having a personal issue with the author, is not a mathematical argument. AaronKauf (talk) 16:18, 22 September 2011 (UTC)
The mathematical notability of the work of Alouba has been already explained above. Both RHarryd and AaronKauf are WP:SPAs whose sole contributions to wikipedia have been to add content related to Allouba theory, which has not yet been recognized as being significant within mathematics. Edit warring to include this unnotable content could result in a report at WP:ANI. I would advise both of you to ask for input from WikiProject Mathematics. Attempts to insert this material have been going on for a number of years and very little has changed regarding the reception of this material within mathematics. The importance of the material is gauged from secondary sources and from mathscinet. Comparisons to the reception of work by Fields medallists so far have been without any merit whatsoever. Mathsci (talk) 16:41, 22 September 2011 (UTC)
Please read the above comments before issuing your regular veto. This behavior of vetoing and threatening is unethical and based on unsubstantiated claims. Your account along with the other British ones are WP:SPAs. AaronKauf (talk) 17:22, 22 September 2011 (UTC)
Why are you even bothering to say that? We all know how to press the "contributions" tab, and we're not stupid. You are the SPA, that is painfully obvious William M. Connolley (talk) 21:57, 22 September 2011 (UTC)
Why are you (along with Mathsci) enticing an edit war by constantly reverting and deleting any contribution we do to this section? You are not even willing to engage in a constructive mathematical discussion. It is obvious that you and your group are having personal issues with the author Allouba. And as I stated before, the Wiki is not a vehicle to settle scores. The constant harassment and threats issued by you and your friends are unethical, and against Wiki policies; and are being reported. AaronKauf (talk) 02:19, 23 September 2011 (UTC)
I know nothing of Allouba; you are completely wrong to think I am trying to settle some non-existent "score". There has been no harassment, and no threats William M. Connolley (talk) 20:57, 29 September 2011 (UTC)
Well, it's true that Hairer is the one who brought up this author and shifted a mathematical discussion into a personal one, but you followed suit. Why resort to the personal attacks instead of solving a mathematical issue? AaronKauf (talk) 17:06, 1 October 2011 (UTC)
I'm not sure this discussion is going to work very well. You started with Your account along with the other British ones are WP:SPAs which is quite wrong, indeed a PA, a a glance at our contributions will show; conversely you are an SPA; you continued by argument from spurious authority. Meanwhile, on your talk page you've been informed of a proposal to ban you indefinitely, for the reasons given here [2]. It really would be a good idea for you to respond there; a completely uninvolved admin has said of you Reasoning with this editor does no good. He won't respond at ANI, but continues to make warlike statements on article talk pages. He takes the interesting view that he doesn't need to give his real name (we should trust him as an authority) but he criticizes other editors for not giving their real names. Since he really won't listen, an indef block is appropriate William M. Connolley (talk) 18:03, 1 October 2011 (UTC)
Again, let's be clear on this. I have nothing personal against Allouba and I have no idea where you got that idea from. Also, I know neither William M. Connolley nor Mathsci and, again, I have no idea where you got the idea that I am somehow hoarding a bunch of friends to run after you and / or Allouba. Hairer (talk) 18:20, 1 October 2011 (UTC)

## RfC: Stochastic derivative

There is an ongoing dispute as to whether the section on stochastic differentiation is appropriate for this article. Should the section on the Allouba derivative be removed? Comments from knowledgeable users would be very welcome. SimonL (talk) 17:22, 19 August 2011 (UTC)

• Comment - The problem is that the individual sentences in the article should be supported by WP:Reliable sources. The article is written rather loosely now, and it may run afoul of WP:Original research policy. I suggest that the article be heavily pared down (to maybe 10% of its current size) and re-built sentence-by-sentence, with a footnote in each sentence, identifying a source. Although that may seem arduous, it is a great way to guarantee an accurate article, and resolve any content disputes. This RfC is asking for comments from "knowledgeable users" but that is backwards: anyone should be able to assess the validity of the material, provided that the editor adding content provides sources. See WP:Verifiability and WP:Burden. For example, on the Allouba issue: whoever wants to add that material to the article should provide quotes from sources (here in the Talk page) that justify the material. Things should go smoothly after that. --Noleander (talk) 03:14, 25 August 2011 (UTC)
In general, sentence by sentence cites are impractical for math articles and paragraph and section level cites are used instead; see Wikipedia:Scientific citation guidelines. If a section level cite is given the added material should closely match the source, and only facts not given in there need have individual cites.--RDBury (talk) 16:07, 28 August 2011 (UTC)
Yes, I see what you are saying, but this article has no (zero) citations. The guideline you cite suggests - at a minimum - a couple of good citations in every paragraph, and suggests even more citations if the material is obscure or controversial. Since an RfC was issued on portions of this article, that would seem to indicate erring on the side of more than one citation per paragraph. --Noleander (talk) 18:38, 28 August 2011 (UTC)
You're right, the article does not come close to meeting verifiability standards. I've taken on articles like this before and it's usually the case that the majority of the material is inaccurate at best and has to be rewritten or removed, so your assessment of what will have to be done is correct for the most part. Your original point that, if I may paraphrase, it's hard to complain about adding unreferenced material when the none of the article is referenced as it is, is very true.--RDBury (talk) 08:06, 29 August 2011 (UTC)

• Comment - The Differentiation Theory is part of Ito Calculus and it should be stated in the section correctly along with the proper citations. The section is currently poorly-written and confusing to the reader. AaronKauf (talk) 01:46, 2 September 2011 (UTC)
• Citations needed - I'm hereby invoking WP:BURDEN and WP:CHALLENGE on the entire article. I've read through it, and the total lack of citations is not compliant with the WP:Verifiability policy, given that I'm challenging the material, and given the existence of an RfC questioning key parts of the article. In accordance with Wikipedia:Scientific citation guidelines, a single citation per paragraph may be acceptable, provided that the source does indeed cover the entirety of the paragraph it is supporting. --Noleander (talk) 03:05, 10 September 2011 (UTC)
• Response - As observed by AaronKauf, Noleander, and RDBurry the article as it stands is unacceptable. Also, as was observed by many (see below for reproduced quotations as per Noleander's suggestion), Allouba differentiation theory is fundamentally relevant and should be included. I've edited the section, and it is now fully referenced, verifiable, and precise. As stated in Allouba's 2006 article, his motivation for his differentiation theory is the use of the quadratic covariation to define the derivative, inspired by Ito's use of Ito to define the Ito integral. The connection to the martingale representation theorem is now more accurately explained and referenced.

Here are the quotations:

1) "The Differentiation Theory is part of Ito Calculus and it should be stated in the section correctly along with the proper citations." AaronKauf 01:46, 2 September 2011

2) "The section as it stands now can't be Wikipedia's (or any respectable encyclopedia's) version of this topic. It highlights an insignificant "formal" aspect while ignoring a more relevant one. Allouba's differentiation theory is explicit, comprehensive, rigorous, and contains the fundamental differentiation formulas. It is not merely an indirect formal definition of a derivative. This has already lead to nice Clark-Ocone and Stroock type representation results (in the second paper). There, the integrands are explicitly given by the quadratic covariation derivative of conditional expectations of the random variable being represented. These results have simpler forms under change of measure than in the classical celebrated Malliavin approach. These results also apply even in cases when the classical Malliavin derivative doesn't. This is true without additional technical arsenal like in the Hida-Malliavin calculus. For all these reasons, the theory is both notable and relevant." JoshLev247 23:14, 21 August 2011 and

"Wikipedia should state the up to date facts as precisely as possible and not selectively remove pertinent ones. This is not about elevating the stature of facts, it is simply about stating them precisely and honestly." JoshLev247 23:14, 21 August 2011

3) "All's (Allouba's) QCD theory is fundamentally relevant to this section, we can't pretend it doesn't exist." .RHarryd 16:12, 19 August 2011

"So, keeping the differentiation section, pretending that the classical mart rep theorem "formal" interpretation is all there is in the context of differentiation in Ito's setting, while at the same time purposely deleting any contribution by Allouba (which is both complete with fundamental elements of differentiation and rigorous and not merely "formal") or even reference to Allouba's paper is a deliberate and unacceptable distortion to the clear history and scientific facts." RHarryd 16:12, 19 August 2011

4) "The bottom line is that Allouba made connections that none on the impressive list of authors of books about Ito's calculus made for a very long time before him (over 60 years,) thereby creating an elegantly simple stochastic differentiation theory for Ito's calculus " and "Allouba's work answers that elegantly & gives that theory, without requiring any extra settings from outside of Ito's calculus." JRMATH 00:58, 20 October 2010 (UTC)

5) "I have explained in great details that the representation theorem is not (by a long shot) Allouba's differentiation theory. He and he alone made the connection (through his definition) between the derivative of the quadratic covariation and the derivative of semimartingale wrt Brownian motion, and developing that concept into a complete differentiation theory for Ito's calculus. " AmericanProbabilist 02:12, 25 October 2010

and "Others have recognized the novelty and significance of his (Allouba's) approach, that's why they have added his contribution to the Wiki site. " AmericanProbabilist 18:49, 11 October

6) " 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." Mattrach 06:17, 10 March 2008 (UTC)

The section now is fully referenced, verifiable, and precise. The outrageous omission of Allouba's theory in a section entitled Differentiation in Ito calculus is now fixed. RHarryd (talk) 03:16, 10 September 2011 (UTC)

Comment This article on a well-trodden subject does list a good set of sources. It suffers from the same problem of many mathematics articles in lacking in-line citations for paragraphs of text. Stochastic differentiation is not dealt with in standard academic textbooks and the material there [3] is either undue (first part) or essay-like original research (second, even if it is a correct observation). The first part refers to a non-notable article which a series of single-purpose users have readded over the years; see the recent report at WP:FTN. Given the lack of coverage in academic textbooks, there seems to be no justification for a separate section on this topic (even given prior discussions on this page). Mathsci (talk) 03:29, 10 September 2011 (UTC)

The section you reverted to, mere minutes after I posted it, violates every argument you're manufacturing against my version and yet it's fine with you. This leads to only one conclusion, you want to remove Allouba's contribution at all cost, even at the expense of the section's accuracy. Allouba's article is well regarded by many well respected experts, and your irrelevant argument about textbooks has been completely answered before by others. This is disgusting bullying, pure and simple, it's unprofessional and it's unacceptable.RHarryd (talk) 04:02, 10 September 2011 (UTC)
I have Noleander's talk page on my watchlist, so saw your comment there. I edit-conflicted with you when you added the list of references here and changed the article. You have attempted to add WP:UNDUE content like a series of other single-purpose accounts. On mathscinet, there is no indication that the short paper by Allouba has had any impact whatsoever and it has not been referred to in academic textbooks. Yet over the years a series of single purpose accounts, of which you are the latest, has attempted to reinsert this material in a prominent and WP:UNDUE way in the article. Although I have commented at WP:FTN on 19 August (see above), I have never in fact edited this article or its talk page before, so the tone of the comments above is not appropriate. Please see WP:BRD. Thanks, Mathsci (talk) 04:18, 10 September 2011 (UTC)

• Comment Mathsci: Something is not kosher here. You hastily removed the well-written, mathematically-sound revised version posted by RHarryd; and resorted to the old poorly-written and misleading one that we were complaining about in the first place. You've done this without giving other editors and users a chance to read it or discuss it. This is an un-democratic and domineering act that is against Wiki policies. You might have personal issues with the author Allouba and/or his work. However, we are here to state mathematical facts and not settle a score. Thank you. AaronKauf (talk) 17:41, 10 September 2011 (UTC)
I think you'll find that if you want to add something to the article....you have to discuss it here first, on the talk page. Before you have added it. And Tag-teaming is about as Kosher as Pork around here. --Τασουλα (Almira) (talk) 22:27, 22 September 2011 (UTC)
• No, wikipedia is not supposed to require discussion prior to expanding articles. (And I never did understand how crying "tag-teaming" is supposed to differ from conceding that it may just be your own view which nobody else shares..) Present arguments better instead of wikilawyering. And if there's a conflict, rather than just erasing a large section of apparently well-written material, I suggest spinning it off into a separate sub-article (link it in See Also for starters), to test if it survives (threat of AfD may be an incentive for the proponents to better scan the recent textbooks): that way, it will prevent your debate from being incarnated as an edit war which is unconstructive to wikipedia's article space content. To me, it seems this has been distracting from the fact that the article overall could be far more useful and understandable than it is, and I think more examples would help explain the topic better. Cesiumfrog (talk) 00:01, 23 September 2011 (UTC)

• We are talking about the quadratic covariation derivative which has been published in peer-reviewed articles. We are contributing to this article like every other editor. However, each time the aforementioned theory is written, accounts like Mathsci and William M. Connolley, quickly either revert it or delete it. This has been their "single mission" these days. I always discuss my contribution on the talk page, and those accounts refuse to engage in a constructive mathematical discussion. They instead resort to harassment and threats. This behavior is not democratic and doesn't adhere to Wiki policy and should be reported. AaronKauf (talk) 02:42, 23 September 2011 (UTC)
Rather than ad hominems and posturing, why not concentrate harder on finding independent authorative sources to support your contention that that content deserves such a large fraction of the attention in a general article on ito calculus? Or else avoid this particular debate by moving your content into its own specific article? Cesiumfrog (talk) 03:06, 23 September 2011 (UTC)
We are experts in this field and I can tell you that this theory belongs to this section. We are contributing mathematical facts in a collegial environment, this is not a pageant. However every time these users from England see that any contribution doesn't satisfy their "personal agenda", they immediately collaborate with each other to revert, delete, and fudge the section without engaging in constructive discussions. Yeah talk about tag-teaming indeed. And your point of putting the section in an article by itself: Of course we can do that, but believe me the same group will be wikihouding us WP:HOUND to satisfy their mission. This aggressive behavior WP:CIV constitutes harassment WP:HA, and is against Wiki policy. In fact, it is this type of behavior (which is not confined to this section by the way), if left unchecked, will lead to diminishing the credibility of the Wiki. It is a matter of principle. AaronKauf (talk) 18:37, 26 September 2011 (UTC)
Please do not attribute me (or anyone else for that matter) a "personal agenda" which I do not have. I have nothing personal against Allouba since I don't know him. AFAIK, our closest "personal" link is that his former PhD advisor (R. Durrett) is a colleague of a close friend and collaborator of mine (J. Mattingly).
The reason why I removed that section was that I felt that it distorted the article and that it would give non-experts a misleading view of the subject. I stand by this position, which was confirmed for example by SimonL who seems to be a student who was confused by a previous version of the article. You are of course welcome to challenge this position and to provide objective arguments for inclusion of that section. If you do so however, please provide some hard evidence, otherwise we get again locked in a sterile discussion. Simply self-proclaiming yourself an "expert in this field" does not carry much weight if you are not willing to actually share your identity. (Presumably Aaron Kauf is not your real name since I found no mathematical article signed by that name.) As for lumping together several independent editors into some sort of "English conspiracy", let's just say that this comment wasn't very helpful. Hairer (talk) 08:28, 29 September 2011 (UTC)

First of all, let's make it clear that the Quadratic Covariation Derivative was in this section for over 4 years until you came about three months ago and removed it. Do you remove each and every section in the Wiki that a student doesn't understand? My position is that we work on writing a comprehensible and accurate section instead of taking the sole liberty of deleting it.
Second, not everyone likes to advertise his or her name like you do. This doesn't mean that they are not experts in their fields. I personally know expert mathematicians from Cornell University who are contributing to the Wiki without using their real names. On the other hand, Mathsci, and William M. Connolley have been sanctioned by you to delete and revert any contribution that mentions the Quadratic Covariation Derivative. In your books, they are "experts". Mathsci is not using his real name, and William M. Connolley has one mathematical article on MathSciNet.
Finally, if you don't have a personal issue with the author Allouba, why do you and your friends keep plastering his name all over the Wiki articles and on the noticeboards intentionally? This defamation is liable here in the US. You don't need to know a person to discriminate against him or her. On my recent visit to Europe, I was faced with anti-Semitic acts from people who don't know anything about me other than I'm jewish. History has showed us that those issues should be taken seriously. It is a matter of principle. AaronKauf (talk) 18:57, 29 September 2011 (UTC)
You just made what can be perceived as a legal threat. If you didn't really mean it, I suggest trying to rescind it lest your account ban is initiated. (As for playing the racism card here, nonsequitor.)
I just made my own naive search, and failed to find any independent books that discuss Allouba derivative in connection with Ito calculus, although I found plenty that discuss Malliavin derivative instead. For that reason alone (but also because no valid arguments seem to have been raised in opposition), I suggest that this article must give significantly less space to discussion of Allouba derivatives than it does to Malliavin. (This currently means less than one line, although it might not preclude starting a separate page.) Cesiumfrog (talk) 00:04, 30 September 2011 (UTC)

No, I'm stating facts. The Wiki is an American organization and is bounded by American laws. Rather than engaging in a mathematical discussion, this group always resorts to slanders and is constantly harassing and threatening any contributor who doesn't conform with their ambiguous editing. The conduct of one user controlling a section, and sending other users to revert and delete edits that challenge his misleading interpretation, is against Wiki policy. All their actions are logged for everyone to see. AaronKauf (talk) 19:53, 3 October 2011 (UTC)

## Software / libraries

Is there (open-source) software available which integrates Itō stochastic differential equations? References to this software and perhaps applications would be really valuable. Andy (talk) 08:20, 25 August 2011 (UTC)

## Citations needed: see WP:BURDEN

I'm hereby invoking WP:BURDEN and WP:CHALLENGE on the entire article. I've read through it, and the total lack of citations is not compliant with the WP:Verifiability policy, given that I'm challenging the material, and given the existence of an RfC questioning key parts of the article. In accordance with Wikipedia:Scientific citation guidelines, a single citation per paragraph may be acceptable, provided that the source does indeed cover the entirety of the paragraph it is supporting. (PS: I originally made this comment above in the RfC section, but Im repeating it here to be more prominent). --Noleander (talk) 03:19, 10 September 2011 (UTC)

## Drift and Diffusion

Neither is mentioned, yet a link ftom the Ito's Lemma page says "Assume X_t is a a Itô drift-diffusion process that..." and links here, but drift is not mentioned. Could someone make this explicit? — Preceding unsigned comment added by 193.52.24.38 (talk) 08:44, 5 June 2016 (UTC)

## Supersymmetric theory of SDEs

This whole section looks more like an attempt at self-promotion than anything else to me. This perspective on SDEs is certainly not mainstream and does not really seem to bring anything new from a mathematical perspective. The main point seems to be to promote the "superiority" of the Stratonovich interpretation of SDEs over their Itô interpretation, which is simply not a mathematical question. The fact that the main (and very long) article as well as the addition on this page were entirely created by a single account operating under an obviously fake name certainly raises red flags... Hairer (talk) 21:18, 5 June 2017 (UTC)