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This should say who Ito is. Michael Hardy 22:31, 15 Mar 2005 (UTC)
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 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)
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)
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 ( talk • contribs) 18:49, 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 ( talk • contribs) 23:02, 11 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)
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 ( talk • contribs) 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 ( talk • contribs) 02:12, 25 October 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)
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)
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)
I've been asked to give my input on this discussion. I'm a seasoned probabilist. I know the stochastic analysis literature well, books and papers. I've carefully read the two quadratic covariation differentiation papers by Allouba available on arXiv. I've also read the discussion here. I'll start with a warning to both sides, I'm busy and will not get sucked into this discussion in a time-wasting manner. The section on differentiation as it stands now mangles the facts. The section is entitled "Differentiation in Ito Calculus", and yet there is no real differentiation there. It only briefly mentions that the martingale representation theorem may be used to "formally" define the derivative as the integrand. This ostensibly covers the issue. Meanwhile, the current edits remove all mention of an existing comprehensive differentiation theory in Ito calculus and of its author. The argument for that is that citations are low. This metric is not always as definitive as one may think. As observed by one of the panelists above, two Fields medalists have two nice papers (one in a highest standard journal) that are cited just five and seven times in more than eleven years. Also, an author and his students citing his work is common to many good theories over the years, and ought not to be looked at as somehow less valuable citations. 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. I have restored the prior more factual version. It is closer to the true state of this slice of Ito's calculus. 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 ( talk) 23:14, 21 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:
Hairer ( talk) 10:12, 23 August 2011 (UTC)
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)
Should not this article be merged in Stochastic calculus? Gala.martin 22:28, 15 February 2006 (UTC)
Ito integral's properties:
70.53.188.62 00:50, 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)
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)
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)
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
where is Gaussian white noise with and Einstein summation convention is used.
If is a function of the , then the Ito chain rule has to be used
An Ito sde as above corresponds to a Stratonovich sde which reads
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)
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)
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)
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 ? 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)
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)
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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 denoting the underlying measure (probability) space, the filtration, , and the underlying probability measure. Hairer ( talk) 18:09, 21 November 2010 (UTC)
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)
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:
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)
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)
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)
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)
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)
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)
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)
How can there be a whole article on the Itô integral that does not mention the notion of progressively measurable? Isn't that the weakest class of processes that can be integrated against a given semimartingale?
It seems that the article uses adapted processes and then (sometimes) supplements them to be cadlag, which implies that they are progressively measurable, but this is not mentioned, instead the discussion goes straight to the Itô integral.
If I am mistaken on this point, then there is an error in the article Progressively measurable processes, subsection Properties, which asserts what I've said here.
I found the following chain of inclusions to be helpful in keeping these classes straight:
where all these classes are understood to be subsets of L^2(Omega x [0,T]).
I hope this chain of implications is correct; it came from stackexchange.
2A02:1210:2642:4A00:2C0E:9FB8:44BB:17B6 ( talk) 12:37, 4 November 2023 (UTC)
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This should say who Ito is. Michael Hardy 22:31, 15 Mar 2005 (UTC)
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 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)
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)
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 ( talk • contribs) 18:49, 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 ( talk • contribs) 23:02, 11 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)
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 ( talk • contribs) 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 ( talk • contribs) 02:12, 25 October 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)
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)
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)
I've been asked to give my input on this discussion. I'm a seasoned probabilist. I know the stochastic analysis literature well, books and papers. I've carefully read the two quadratic covariation differentiation papers by Allouba available on arXiv. I've also read the discussion here. I'll start with a warning to both sides, I'm busy and will not get sucked into this discussion in a time-wasting manner. The section on differentiation as it stands now mangles the facts. The section is entitled "Differentiation in Ito Calculus", and yet there is no real differentiation there. It only briefly mentions that the martingale representation theorem may be used to "formally" define the derivative as the integrand. This ostensibly covers the issue. Meanwhile, the current edits remove all mention of an existing comprehensive differentiation theory in Ito calculus and of its author. The argument for that is that citations are low. This metric is not always as definitive as one may think. As observed by one of the panelists above, two Fields medalists have two nice papers (one in a highest standard journal) that are cited just five and seven times in more than eleven years. Also, an author and his students citing his work is common to many good theories over the years, and ought not to be looked at as somehow less valuable citations. 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. I have restored the prior more factual version. It is closer to the true state of this slice of Ito's calculus. 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 ( talk) 23:14, 21 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:
Hairer ( talk) 10:12, 23 August 2011 (UTC)
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)
Should not this article be merged in Stochastic calculus? Gala.martin 22:28, 15 February 2006 (UTC)
Ito integral's properties:
70.53.188.62 00:50, 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)
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)
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)
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
where is Gaussian white noise with and Einstein summation convention is used.
If is a function of the , then the Ito chain rule has to be used
An Ito sde as above corresponds to a Stratonovich sde which reads
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)
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)
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)
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 ? 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)
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)
![]() | This article has formulas that need 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 denoting the underlying measure (probability) space, the filtration, , and the underlying probability measure. Hairer ( talk) 18:09, 21 November 2010 (UTC)
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)
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:
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)
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)
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)
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)
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)
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)
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)
How can there be a whole article on the Itô integral that does not mention the notion of progressively measurable? Isn't that the weakest class of processes that can be integrated against a given semimartingale?
It seems that the article uses adapted processes and then (sometimes) supplements them to be cadlag, which implies that they are progressively measurable, but this is not mentioned, instead the discussion goes straight to the Itô integral.
If I am mistaken on this point, then there is an error in the article Progressively measurable processes, subsection Properties, which asserts what I've said here.
I found the following chain of inclusions to be helpful in keeping these classes straight:
where all these classes are understood to be subsets of L^2(Omega x [0,T]).
I hope this chain of implications is correct; it came from stackexchange.
2A02:1210:2642:4A00:2C0E:9FB8:44BB:17B6 ( talk) 12:37, 4 November 2023 (UTC)