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Gauss prize lecture  On Kiyosi Itôs work and its impact
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OK so renounced . 2 ghost price lecture so actually being In the 2nd spot Provost all kinds Fernmount described the various aspects of profits dose well I can't but 1st provost go tool we did as if she were about from thank you very much the introduction us is my father cannot present and this case I had prepared together with my father thank you on this occasion and greatly that the International Mathematical Union in cooperation with a dog too much much of mind has awarded 1st firstclass reduced gulls in recognition of my work on stochastic analysis it is difficult to express unparalleled happiness and feel to be the recipient of the award bearing the name of the Greek mathematician whose work continues to inspire us all because my only research of the cast analysis is in pure mathematics the fact that my work has been chosen for the guards price for applications of mathematics is truly unexpected indeed be gratifying I hope therefore to share this great
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honor with my family teachers colleagues and
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students in mathematics as well although who took my work on stochastic analysis and extended to areas far beyond my imagination professor filled will out I'm very happy to be leaving inaugural house Chrysler sir I have made a fond memories of working with viewing Series Indy 80 when I was a visiting professor at 8 to half in fact this picture of me and my late wife taken at that time is 1 of my favorites and always displayed on my desk I'm sorry that my health situation prevents me from attending your price but I look forward to reading and
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studying it so that I can learn about the new developments in stochastic analysis Professor Huntsville Professor Martin go to and the house M. farm president professor Mandarin Leone and the 2006 Madrid ICM organizing committee I area president Sir John Paul and all the participants of the
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2006 ICM please allow me to express once again my heartfelt appreciation for the great honor to have bestowed on I think so it's a great order signed a great pleasure for me to introduce now professor Huntsford from university board very proper suffered has made many fund them aboard contributions in probability theory in domains such as statistic random produced adventure series artist mediation In the last day of 15 year also has become the leading In the area applications of statistics and that is used to Matsumoto could find you elect Joto days on the work of door and it's been by fly the song that but right off nearly a great
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honor said also at great personal as for me To be able to command
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rescue deal occasion of the 1st Gulf crisis which has been voted to kill she told about a week ago by chance I stumbled on the Internet I was some Web site we are there was a discussion going on on potential candidates for the field metal and want a statement loss unfortunately it dipped years the that's body against the plight mathematics Yukon you enter contiguous I'm hoping that gulls brought the correctness of 12 invaded take someone who really wonderful Nike told of the calculus they knows this is actually have the and I definitely share the feeling that somebody really wonderful has picked the gulls Rice has been awarded to kill she told for laying foundations of the series of still has the French league they and still has big and man normal you may wonder why somebody who obviously cares the about at application as skies anonymous God who made the state and on the Internet is so and so the yesterday and infect white singles brought because Julie Cui told yesterday in a press conference pointed out that so far the considers himself a pure mathematician area world Spiegel expresses surprise no suspect you of sea gulls Price saying that it is to be awarded for outstanding mathematical contributions that telephone significant applications kind of mathematics or achievements that may vacation of mathematical methods to area outside of mathematics possibly in innovative ways not long reign of the lecture is to make the point that on both the sitcom it's the marvelous idea To take just little 1st enough got rights those community . 1 still has many uses tools and concept of the high demand is this pair phone important implications significant efficacious in various areas outside of mathematics but also still has command the hairs May concept should advances in clarifying the structure of the situation in education which then made it possible to bring in other matters like penal married so but the 1st thing conceptually solid provided by the plastic amending the metal door for it also point tool is finally red of the specific case not a try to Burma the week of the work show she told its concept should its beauty affect my goal is much more modest then both suggests the votes of kill sheet my goal is not to talk about which meant the plastic and that this is strictly about Ed under trying to explain to these numbers specialists what some of his concerns where so he just in mind you voted for it said time he had already made an important breakthrough with fundamental breaks through very fine structure of Markov processes and yet that meant completely on its all the here he has His taking the Government's statistical bureau of Japan and adjust followed by Jim he told that they simply left them he said he was not formally involved in any graduate work he could talk complicated saying and something really great came out of the so he wrote it said find a sequence of a post differentially equations determining the Markov of Brosius Watson said the menial written in Japanese it then came out much later in an extended version of the we mathematical journal and also thanks and was 1 of 1st immediately salty port of Ito's work as a memoir of the American Mathematical Society and then at the same year a separate paper on a formula concerning the plastic differential which is now known as formula which came almost in passing as 1 step In his program to know what we'll see progress he program about Markov processes so that we recall so this is not for the specialists the court those who want to learn and they have not seen you for a while both were about threats consider Markov process on our team that is usually described by transition so if you start at the position off the time she you find yourself some states based in the case with a certain probability which is specified now v Markovian Trevor does not remember what have before whatever the next only depends on his position right now and this is capture body check and Cuomo of occasions the Chol hold these different probably of Bechtel together no this is enough to construct the probability measure on test face now I'm getting more special said Ito did by focusing from now on onto continues case it was paid us are already much larger where our he passed based on the basis of continuous passed on a state based on Dean old quite easy positions at for different trajectory and see measures related to the semi who this way to probably start date given up to 20 to find a time of death article stayed in the area acres just give them by there transition probabilities depending on the present position to for this very point that these past behavior of does look intervened and these predictions for the future behavior only the present state of met until they have precise mathematical model which it also take it talk about past behavior of due process no but what Is the infinitesimal structure such picture of typically up by raising so what can we say In an identical terms we can say something starting with transition probabilities you would this transition operators we can differentiate and the continuous diffusion gave this could turn out to have form of apart from the French operator and there is called will borrow thanks which says that for a function defined may force some function as we have this
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partial differential equation in terms of the operator tho this is undertake statement In viewed as you the behavior of button probabilistic took on a level of past how can he understand the structure of these tests and Ito's inside those idea and program what's efforts to identify tension Mark process of this Peony cannot keep up and that you wait and eventual we can't struck or simply to construct circumventing CERCLA mobile construction of the process directly path warriors from attention the without attention from Member of process invite No. 1 should Leedy process the white leading processes they need processes of total speaks a straight line there incremental independent and identically distributed so that more of the in the same direction the roles Our infinite visibility and already before the paper on all describing in its he had worked on the leading processes and proved to some extent the current 20 but the small known to lead me he told decomposition on level off path of such a process known for what purposes not for the lecture only need 1 prototype of the be process since we stick with it continues case so the prototype their these roles to be ghosts and give me by Abronia mulch Brogna emotion at that time had already history 1900 budget he had written this season's cheerleaders because you theory introduced raw emotion as a model for price fluctuations on Paris Stock market 5 years later but I on very different motivation and intuition that physical Grant motion and then the rigorous mathematical construction is a measure on a test basis most is by at noted 19 23 years old but structural work you take the sequence of independent identically distributed random variables you take also homered basis in don't fool with respect for the victims of double and then some up these random affect us primitive of these function and he also homered and then you control this is uniformly column Beauchamp and he almost surely therefore limit exists that defined the continuous functions and this most are we lose trigonometric maybe fight the argument by using the Bihar functions as a basis but it's only much later that he took in a joint paper dual really gave us very elegant and then formulation approved by the general case and also going already Bellevue would say Dad a and solve a UDA of food or not 2 pickup as far as I just like construction continues but we are nowhere differential so are classic to integrate the amount of bond variation of the White House said constructed 1 example both continuous function which is low but differential but is also seen as a pathology of the quote Internet also found steel years there he says Jamaat toward evict you've walked aware the says they more dog beautiful sold leaving so I turned away a bit of a shock and disgust from this lamented of obj Lord derivative but here In this context of diffusion phenomenon boast mathematics and the real work that that politics that the typical that alert Nova differential of with the fish of the deal not the ideal the attention of forgive Markov broke his attention and makes the few ocean case will simply be such a process constant coefficients taking off a insists that this suggests to write down as it is testing the behavior of Markov process has in this form it should take off like its attention from any give them position at any given time which means it showed for reopening the that's idea and them the next step of the program struck deep pass from this stuff that differently faced mother but sold this integrated foam this occasion their position if he becomes the boat From the stopping point by integrating Pierre and integrated up T classic there's no problem but is completely on deal because this depends on that and even those are made more sense as I don't months more this will be very none more so and you we don't have the classic integrator has the process is over differential so this needs a new war a new approach to integration but not known as the plastic integration now that had already been 1 step by daily which he finds the into Norway by integration by parts and this work than ever here they classic integrator so he passed here Of the integral is off on the very edge and they did it indicates that Of all the 30th of but have just seen into grants which unheeded here the construction program are by no means this tight and told Ito went about obstructing seemed to grow In this form general integral adapted in the sand said they depend only on information the fight and he has behavior point they satisfy some integral conditions but not rigorous area and the idea to right now the only months but non anticipating we months some so he is in with their related left side and that test the limits you think the basic isometry simply into like this piece but constant EL full norm here filled 2 with the expectation of the into gold which helped with into products and viruses isometry something Weizman became the that force boasts of stuff has integration of the sense of in their introduction to Ito's elected paper dance troupe and the but I'd done right everyone you could say in this room as it 3rd they say who might pick up a book that start debris lead showed that there is a subject called Suriel the plastic integration and he told Is it leads bank of this branch of integration series in brackets daily and you know there it's really much now is is not all to complete the program he still has differential equation actually solar careful for sense out of integrated former and they need the verification construction does what it but asked the Navy to provide full use of they exist verification goes by the tools formula which he of turning now now also tired of this rule and in fact outside of mathematics down holidays in many thousands of people who probably has not hurt or it sold do not care about they
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fear Yi enough but say has worked and they don't care about Beatles formula know why is that on the 1 hand it is just this turned out to be extremely use would To with old ties mathematics in many areas and also it has a certain fundamental quantity and I want to underline is a fundamental quantity of this formula be golf of course is not the 1st distinction Josie has received killed tangible prize I think in 19 8 says and not that fuel of Frost it said he has given us a fuller understanding of into became designated development of Markov tests and I've tried to to show you official summary of a whole disco this may need you would you wanted all the stuff that Corel provided a direct translation between the governing party differently creation and he underlined probabilistic mechanist its main ingredient of differential and integral calculus of functions of from most of the resulting series is the cornerstone of modem probabilities both pure and God not coming from Germany and socialized Germany and especially from the if I knew what I also think of my soul here it's together the new the resisting from an article in The Economist where describes a few would bear the book priority not bottomline probably would not extend to paraphrase of God the throws saying the nuns at that time no yeah it's something like written in Berlin and that soldiers in would is most decided blush the Infiniti to my country but more points and we did the job because was written letter but only got forward of German text not I saw you he writes heel of everybody novels integration you the product will for differentiation Anthony says out Of this formula he hollowed remaining calculus can be developed but this fall it's demonstrated as formal so mounted and he gets it Anthony said OK this week's is much smaller than devices much more smaller wanted so they will sing is much smaller than the remaining time solely for gets them and all comes to you which role product which is particularly assists and reasonable function as this stand behavior belongings trajectory it and then he writes what he'll be among nonmember nearly all libelous call this morning this barrier remarkable CRM is common for all trucks but the insightful he told lost no it's not it's not in general this formula should be written and this made in other words these extra firms cannot be forgotten in general for general curve as they turned out to to be typical into diffusion picture because here in this form something comes up which does not come up in plastic calculus that seek a directive variation of the functions Nova differential but it has a direct variation and putting me showed its equal to see Florida typical has process and do a show that for its which North is the classic differently they should to have this form and sold more generous Beatles product through it takes his form with a corresponding defined call variations and then Beatles formula for a function of class tool cases extra charm and if it failed and context of a multidimensional few follow the lead of tested differential patiently had seen it takes form and the operator Of course comes and by the way did have the choice we can either stick underlined ground and that's all we can translate but take part of all the other side To increment of few tests and then the operator changes in the sense that the drift is taken out of the operator which we had before now I want to discard some consequences which we are going to be a little the data so 1st consequences and that was the reason the reason he proved that you would assist send a check to very following qualification for the solution of at the French which joint looked like a fool you you wasn't tested but it's a France farreaching consequences unless all you want but should be interesting for us but it later it implies a representation Europe so let's take a functional of of diffusion process which is given by the functions more Each applied for a final position of some fine window from he wrote to teach them Eagles formula reads through this representation of a functional is a constant last and dual integral of this process if you join with death as this will you wish of falling bond problem it should is part of the French league facial and value for people to H. White because in general we have a remaining term exactly this integral all but appeals not drops all and you have the representation but shows that such simple functional of a few which can be written as the plastic integral of underlying diffusion best nor you can take products of sold picture of the results yet together In an obvious they send you get that for functional of this former that they can be represented some integral and then you had an approximation procedure full general functional and then each is the functional the nice if you wish it can be shown use the plastic integral Of that if you wish some
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integral OK so what they're I have sketched not is really it in the spirit of pure mathematics and some people have seen the field the probabilities as plight but it's a white Shield because various theoretical area as Barry by area says you you've always think of it as a further Joshi told considers himself To rather but these merry dual side of it and so in this period try to describe this program and effect but that did you official a summary of the holy book which details of you really want to know should look at the market processes for Katie those perspective written some years ago but dance troupe OK not come To impact on the reason for a voting seat belts let's 1st look at detects within mathematics and there it should be said that it took a on time at least in the West and the 5th so get talking work great weight head Daimler already in early forties then for all published and excessive weight and 0 50 but in the fifties and sixties reception was not very white knowledge exceptional door Joe immediately understood the significant make sure it published a memoir of American society and he himself wrote a check on the plastic integration in his book Purington 53 on stuff has processes in the East Devils more interest in the rest Dinkins Of those took up very fast a deal and the techniques of the press integration and
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you with them in important base for example it defining physical son of of transformation and shift both you measure transformation of the measure which corresponds with change of the drift consist of plastic differently which now he tool to print for the U.S. and 50 so he will therefore 54 256 and he describes the statement he gave a year ago and the arbiters policy In his former dead there he met Mickey is wouldn't at the time of sale and he explained to him his idea of a both pathetic differentially fish and then says Devils once location and McCain tried to Spain to face my work on behalf of French Lick Clayton and he attention it seemed to me he said that the outlook for the if significant at any rate doubles law encouraging for yet on the other hand when I explained fully it told fellow leaving local time was immediately became friends with yes made some conjecture about the price of structural fundamental fusion of of brokered talks and both sets of problems and then Ito and Michie work comes as solid problem and a company program and code diffusion processes until then would which appeared than 65 s ironically most oppressive different variations no still has integrals that for not for me as a graduate student at the time this was my 1st exposure the of Ito and I found it so formed the group of graduate students and did our own informal sending the reading section of of the book and be the horror going and I thought OK maybe the suggested truffle Bill stood but a big misunderstanding by chance and I discovered lecture notes with only by the of the plastic processes cut Institute and that arms University and take them off suddenly everything looks very clear and very easy balls with a friendly they it so I was really looking forward to meet that 1st and that that is some of 68 Ito came from oral with trying to give at University of L m this is taking 10 years later but I remember some poses poster and especially his of transmitting his own joy dealing with less methods to both because they exist both vary a very strong experience and that will 1 member of a situation where distinguished colleague from United States Will Smith Old Firm giving a lecture on probabilistic potential sealing the basis last lecture Antido will spread know this 68 so students at that time we are somewhat on Ruiz they're ready to be out for a life so we asked colleague who was just getting to finish technical proved the boat and functional Markov processes could be just forget about according not just give us the some perspective from old look for port of this took him by surprise was not prepared for the Sony has take that not eager to go in that direction and suddenly he took that up to back off instead of blood highlights difficult Our as red Stewart elite it all off the gave his views on the topic he thought they are important had potential for further research at Singapore's main persons cereal Markov process which he described excursions of the marble process he would has what point growth so was exciting stuff OK but so far most pathetic integral also in the lecture improvised lecture most of us into know 69 the situation started check and Andromache's wrote a small book Stock as integral community dedicated to kill talk and since the seventies suddenly there was ghost of the plastic and that is that in a general Martingale said yacht important paper inspired Chula now Of that could be done but not good friends with global Emilio virtually your and expanded its holders framework tremendous weighing them simply 18 infinite dimensional extensions gained a measure of value if Our 1 class of exam stalls but the amount of motivated by biological population dynamics and then very you went systematically Monday of that calculus initiated by what 1 of our which by the the way had strong schooled in Spain represented at least recently but that allow it still not that sounds and Joshi Ito himself contributed to the development of a let's just mentioned reading at ETH Zurich bearer bonds foundation of the classic differently based infinite dimensional space variant of aII view would demand that calculus as infinite
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dimensional I don't know what based on an infinite dimensional launched wouldn't be process and In general frame at any rate the plastic connelly's now Of course should probability that no doubt but if he got rights week loyalists implications beyond mathematics sold In an adult good form by describe all our experience that 1st Interstate is that suspend test then I will assist as a young Ph.D. in instructor at MIT courses involving Eagle still plastic different facial must have might cost director check a mathematics 0 0 elected engineering for aeronautics and astronautics tool and I actually attended 1 of these tool which also aboard space flowing the effect of random disturbances by changing fluctuating densities and yet most dealers and missiles model by ghost of plastic French variations and analyzed special topics the BDT of such dynamical systems but noise but as yet no functional incredulity entangled yet you calculus problems of food drink and controller going Beyond column on was energy 1st textbook for example in Germany on the classic French but we are 73 at the Technical University of novels written primarily for engineer with something most of the satellite but I just described the randomly fluctuating that he'll get most of us want prime example not my next experience lost in 1977 when I moved to to EGH directed by next bubbles con but most of the but was double just returned from Pablo the 3rd the father Jaffe end In spite of ideals of few months it on pass into gold and the feud series and Barry Simon and this year most in Switzerland and gave a holy job Saudi Press techniques which appeared as a book and 79 and are the faulty 16 Ito's integral operator the magnetic field introduction was still has calculus very gave approval of formula so clearly importance will see according to mathematical physics and amend the idea came up in 87 to offer an honorary degree to Josie he told it was very easy because of strong support not only by the probably a joint venture of this mathematical physicist working in the field series no finally I want to describe 1 case study there'll be a little more precise name deprecation Oh my own I came into contact with this something on would David Fred Leighton class but Clark method for the best economist decals and the economy of the field level there after 84 and then deporting visited EGH and discussed Martingale aspects of the new development and finance related pricing intention of derivatives Soviet the price fluctuation on the financial market by the late this is not a recent that in some sense of closing the circle of going back to 1 of salsa remember Bushehr he wrote his 1st novel introduced most that until we have such fluctuations of the probably past they continuous case defeated financial assets this would be the best place but very quickly becomes fancy if you think of yield curve of fluctuation of yield curve the state's bases would be an infinite dimensional space and you would make a choice while still has the French based not what is she now got a big problem and his as you know and statistic econometric but also see a replica aspects related to the notion of market efficiency strong form of market efficiency said that information and expectations are in priced in so after discounting you get this property that the present crisis is equal be best get of future price and that mounting give property on the P and instead case he would be called the Martin measure not on the financial markets and trade so you can vary the number of stocks you hold it any time antic humor into net now idealized continuous trading strategy would be exactly why wouldn't be doing to grow because this is based on Monday anticipating months some and that's exactly the financial meaning you must make your investment beginning results foresight and foresee actually price increment novels such amounting to measure the system durum says that don't know opening strategies so that some of the strong and we go form both market efficiency would be absence of free dances there may be ending strategies but instead cases you have a downside risks and sisters and reverted to toward defect that although it does not have to be measure but I must admit and reverted marking such a pig now we come to a core problem and this is pricing head shingle financial derivatives defined as some functional steep price from some contract which you get paid at the end so much if seemed aerial more eager and now can see hold told comes in and a strong a crucial wait in a concrete waters this measure is unique and that OK case from my you should such socalled short then I showed you what a representation properties which was proved true Ito's formula and by the way added I 1st proved it of vapor period elected engineering because in the early seventies because electric that interested in the property but the affected authority entity financial interpretation did not mean that you have the perfect reputation of your financial derivatives by some anemic strategy and that means that this constant he'll miss the cost of the perfect replication must be seen natural there this cannot be computed as expectation of the real measure because under the mounting measure this be a mountain goat term and the expectations broke out and in particular the special situation of chose the framework process especially derivatives this the Buchholz formula Prince really Carora arrays of those and those formulas conceptually crafty representation theorem the representation theorem gives you write concept framework and at the stroke settled the issue for any exotic derivative of both of the computation of the strategies becomes very involved with the show of Egypt calculus PD for exotic derivatives surveys of reducing it the simple or we would really advanced techniques like money bag circulars souls is nowadays financial engineering which is a way of trying has net income pretty moderates are much more complicated but all the more interesting and also here has entered provides a crucial concept and with my time is running out of money center which I've wanted to illustrate how important is you had a real genius information on financial market some no more than either of the question is what financial role of that for the financial value obtaining additional information days is small going toward the 17 of the plastic differential discusses the change of filtration whatever if you increase your information and that provides music keen to very hot topic technology news information and some of the recent work do we the the technique initiated by told for example cut about us like the summer Joshi has more than this the way in which we all the bot still has 2 processors this is a close to again take from the introduction of troop environ armed with collective this was rich already instead we always instead they probably had in mind rather that no group of specialists in the area of the plastic but we all their tax increase dramatically over the last 30 years and a number boundaries of mathematics in areas such as engineering started early find and started with a vengeance but with the developed credible momentum and amazing thing to meet home much concept not so much computational would have friends to discourse in Department of Finance and Economic nor would they a look at it says completely take over lots of toast and so I do agree that's what may need all these years the Green River that the should vote from the Internet that the Committee for the adults Reuters that's really pick someone really wonderful thank you
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Mathematik
Angewandte Mathematik
Mathematiker
Bestimmtheitsmaß
Kombinatorische Gruppentheorie
Auswahlverfahren
Stochastische Analysis
Analysis
01:38
Flächeninhalt
Mathematik
tTest
Reihe
Familie <Mathematik>
Stochastische Analysis
02:25
Statistik
Wahrscheinlichkeitsrechnung
Flächeninhalt
Zeitbereich
Reihe
Kartesische Koordinaten
Ordnung <Mathematik>
Grundraum
Stochastische Analysis
04:50
Abstimmung <Frequenz>
Folge <Mathematik>
Subtraktion
Einfügungsdämpfung
Punkt
Kalkül
Prozess <Physik>
Ortsoperator
Gruppenoperation
Zahlenbereich
Kartesische Koordinaten
Trajektorie <Mathematik>
Term
Computeranimation
Ausdruck <Logik>
Algebraische Struktur
Differential
Prognoseverfahren
Exakter Test
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DeanZahl
Wahrscheinlichkeitsmaß
Optimierung
Einflussgröße
Nichtlinearer Operator
Lineares Funktional
Fundamentalsatz der Algebra
Statistik
MarkovKette
Mathematik
Reihe
Flächeninhalt
Rechter Winkel
Mathematiker
Körper <Physik>
Aggregatzustand
Numerisches Modell
14:11
Resultante
TVDVerfahren
Kalkül
Hyperbolischer Differentialoperator
Prozess <Physik>
Punkt
Richtung
Übergang
Gruppendarstellung
PoissonKlammer
Exakter Test
Translation <Mathematik>
Bestimmtheitsmaß
Isometrie <Mathematik>
Auswahlaxiom
Einflussgröße
Gerade
Feuchteleitung
Lineares Funktional
Parametersystem
Nichtlinearer Operator
Approximation
Reihe
Klassische Physik
Ideal <Mathematik>
PfaffDifferentialform
Biprodukt
HelmholtzZerlegung
Konstante
Arithmetisches Mittel
Zufallsvariable
Identifizierbarkeit
Subtraktion
Ortsoperator
Klasse <Mathematik>
Physikalismus
Derivation <Algebra>
Term
Physikalische Theorie
Ausdruck <Logik>
Differential
Algebraische Struktur
Erwartungswert
Reelle Zahl
Inverser Limes
Delisches Problem
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Fluktuation <Physik>
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30:50
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Physikalismus
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tTest
Transformation <Mathematik>
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Physikalische Theorie
Richtung
Dynamisches System
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Optimierung
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Menge
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SmithDiagramm
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Körper <Physik>
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39:16
Resultante
Impuls
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Prozess <Physik>
Kalkül
Integraloperator
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Sondierung
RaumZeit
Übergang
Dynamisches System
Gruppendarstellung
Exakter Test
Theorem
Gruppe <Mathematik>
Randomisierung
Stützpunkt <Mathematik>
Mathematische Physik
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Formale Metadaten
Titel  Gauss prize lecture  On Kiyosi Itôs work and its impact 
Serientitel  International Congress of Mathematicians, Madrid 2006 
Anzahl der Teile  33 
Autor 
Föllmer, Hans

Lizenz 
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DOI  10.5446/15952 
Herausgeber  Instituto de Ciencias Matemáticas (ICMAT) 
Erscheinungsjahr  2006 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 
Abstract  Laudatio on the occasion of the Gauss Prize award for Applications of Mathematics to Kiyosi Itô for laying the foundations of the theory of stochastic differential equations and stochastic analysis. 