OK, so we will now start the Ghost Prize lecture. So this lecture will, in fact, be in two parts. In the second part, Professor Hans Vollmer will describe various aspects of Professor Ito's work.

But first, Professor Junko Ito will tell us a few words about and from her father. Thank you very much for the introduction.

Since my father cannot be present on this occasion, I have prepared, together with my father, a short thank you note on this occasion. I am greatly honored that the International Mathematical Union, in cooperation with the Deutsche Mathematica Verheinigung, has awarded me the first Karl Friedrich Gauss

Prize in recognition of my work on stochastic analysis. It is difficult to express the unparalleled happiness I feel to be the recipient of this award bearing the name of the great mathematician whose work continues to inspire us all.

Because my own research on stochastic analysis is in pure mathematics, the fact that my work has been chosen for the Gauss Prize for applications of mathematics is truly unexpected and deeply gratifying. I hope, therefore, to share this great honor with my family, teachers, colleagues, and students

in mathematics, as well as with all those who took my work on stochastic analysis and extended it to areas far beyond my imagination. Professor Filmer, I am very happy that you will be giving the inaugural Gauss Prize lecture.

I have many fond memories of working with you in Zurich in the 80s when I was a visiting professor at the AJH. In fact, this picture of me and my late wife taken at that time is one of my favorites and always displayed on my desk. I'm sorry that my health situation prevents me

from attending your Gauss Prize lecture, but I look forward to reading and studying it so that I can learn about the new developments in stochastic analysis. Professor Hans Filmer, Professor Martin and the Gauss Prize Committee, D.M. Pfau, President Günter Ziegler, Professor Manuel de Leon,

and the 2006 Madrid ICM Organizing Committee, IMU President Sir John Ball, and all the participants of the 2006 ICM. Please allow me to express once again my heartfelt appreciation for the great honor that you have bestowed on me. Thank you very much.

So it's a great honor and a great pleasure for me to introduce now Professor Hans Filmer from University Humboldt in Berlin. Professor Filmer has made many fundamental contributions

in probability theory in domains such as stochastic calculus, random fields, probabilistic potential theory, or large deviations. In the last 10 or 15 years, he has become the leading expert in the world in the area of applications of stochastic analysis to mathematical finance.

So he will lecture today on the work of Ito and its impact.

So I try again. It's clearly a great honor and also a great personal pleasure for me to be able to comment, to address you on the occasion

of the first Gauss Prize which has been awarded to Kiyoshi Ito. About a week ago, by chance, I stumbled on the internet to some website where there was a discussion going on on potential candidates for the field medals.

And one statement was, unfortunately, it appears that the bias against applied mathematics will continue

and he continues, I am hoping that the Gauss Prize will correct this obvious problem and they will pick someone really wonderful like Kiyoshi Ito of Ito calculus fame. Now this has actually happened and I definitely share this feeling

that somebody really wonderful has been picked. The Gauss Prize has been awarded to Kiyoshi Ito for laying the foundations of the theory of stochastic differential equations and stochastic analysis.

Now, you may wonder why somebody who obviously cares about applications as this guy, anonymous guy, who made the statement on the internet is so enthusiastic. And in fact, why is the Gauss Prize?

Because Junko Ito, yesterday in the press conference, pointed out that her father considers himself a pure mathematician. And in the words we just heard, he expresses even surprise. Now, the statutes of the Gauss Prize

say that it is to be awarded for outstanding mathematical contributions that have found significant applications outside of mathematics or achievements that made the application

of mathematical methods to area outside of mathematics possible in an innovative way. Now, one aim of this lecture is to make the point that on both accounts,

it's a marvelous idea to pick Kiyoshi Ito as the first winner of the Gauss Prize. I mean, to those who know clearly point one, stochastic analysis and the tools and the concepts of stochastic analysis have found important implications,

significant applications in various areas outside of mathematics, but also stochastic analysis has made conceptual advances in clarifying the structure of certain situations in applications,

which then made it possible to bring in other methods like PDE, numerics of PDE. But first the conceptual insight provided by stochastic analysis had to open the door for that. So also point two is highly relevant

in this specific case. Now, I try to speak on the work of Kiyoshi Ito, its conceptual power, its beauty and its impact.

My goal is much more modest than was suggested in the words of Kiyoshi Ito. My goal is not to talk about new advances in stochastic analysis. This is strictly about Kiyoshi Ito. And I will try to explain to the non-specialists

what some of his concerns were. So here is Kiyoshi Ito in 1942. At that time, he had already made

an important breakthrough, a fundamental breakthrough in clarifying the structure of Markov processes. And he had done it completely on its own. Here, his photo is taken in the Government Statistical Bureau of Japan.

And we were just told by Jinko Ito that they simply left it in peace there. He was not formally involved in any graduate work. He could do completely his own thing. And something really great came out of this. So he wrote at that time,

the sequence of papers, differential equations determining a Markov process. This was in fact a mimeo written in Japanese. It then came out much later in an extended version in the Nagoya Mathematical Journal. And also thanks to Joe Dupe,

who was one of the first to immediately solve the importance of Ito's work as a memoir of the American Mathematical Society. And then at the same year, a separate paper appears on a formula concerning stochastic differentials, which is now known as Ito's formula,

which came in almost in passing as one step in his program here. Now, what was the program? The program was about Markov processes. So let me recall.

So this is not for the specialists. This is for those who want to learn and have not seen before what Ito's work is about. So let's consider a Markov process on RD. So this usually described by transition probabilities. So if you start in the position X after time T,

you find yourself in some area of the state space, in this case, RD with a certain probability, which is specified here. Now, the Markovian traveler does not remember what has happened before.

What happens next only depends on his position right now. And this is captured by the Chapman-Golmogorov equations that show how these different probabilities are patched together. Now, this is enough to construct a probability measure on path space.

Now, I'm getting more special than Ito did by focusing from now on the continuous case. Ito's papers are already much more general, where the path space would be the space of continuous paths on the state space RD. We note by XT the position at time T

for a given trajectory. And the measure is related to the semi-group in this way. The probability starting in X, given the past up to time T, to find at time T plus S, the particle, let's say, in the area A,

is just given by this transition probability, depending on the present position XT. So this clarifies the point that the past behavior does not intervene. And these predictions for the future behaviors is only the present state that matters.

And so we have a precise mathematical model which allows us to talk about the past behavior of the process. Now, but what is the infinitesimal structure here in such pictures? So these paths typically are quite crazy.

So what can we say? In analytic terms, we can say something starting with these transition probabilities viewed as transition operators. We can differentiate. And in the continuous or diffusion case, this will turn out to have this form of a partial differential operator.

And there is Kolmogorov's backward equation, which says that for a function defined in this way, for some, say, continuous function F, we have this partial differential equation here in terms of this operator. So this is an analytic statement

on infinitesimal behavior. But in probabilistic terms, on the level of paths, how can we understand the structure of these paths? And Ito's insight, Ito's idea and his program was first to identify tangents of the Markov process.

Now, this clearly cannot be done in a naive way here. And then to reconstruct or simply to construct circumventing the Kolmogorov construction, the process directly path-wise from its tangents.

Now, what are the tangents on the level of processes? Insight number one, these should be Levy processes. Now, why Levy processes? Levy processes are, so to speak, the straight lines. Their increments are independent and identically distributed, so they move, so to speak, in the same direction.

The laws are infinitely visible. And already before the papers I'm now describing, in his thesis, he had worked on Levy processes and proved, to some extent, in parallel to Levy, what is now known as the Levy-Ito decomposition

on the level of paths of such Levy processes. Now, for our purposes now for this lecture, we only need one prototype of a Levy process since we stick to the continuous case. So the prototype there, the laws will be Gaussian

and it's given by a Brownian motion. So Brownian motion at that time had already a history. 1900, Bachelier had written a thesis, Theorie de la Speculation, where he introduced Brownian motion as a model for price fluctuation on the Paris stock market.

Five years later, Albert Einstein was a very different motivation at intuition connected with the physical Brownian motion. And then the rigorous mathematical construction as a measure on pair space was given by Norbert Wiener in 1923.

Now here's how the construction works. You take the sequence of independent identically distributed normal random variables. You take also normal basis in L2 with respect to the Lebesgue interval on the unit interval. And then you sum up these random factors,

the primitives of these function and the also normal base. And then you can show this is uniformly convergent in T, P almost truly. Therefore the limit exists that defines a continuous function. And this was started by Wiener who choose a trigonometric basis.

Lebesgue simplified the arguments by using the hard functions as a basis, but it's only much later that Ito in a joint paper with Nijio really gave this very elegant and general formulation with the proof, the general case, and also going already to Banach

where you would pass. And so here we have the Wiener process. Now typical paths are, as I just said, by construction continuous, but they are nowhere differentiable.

So they are no classical integrators. They are not of bounded variation. Now Weierstrass had constructed one example of a continuous function which is nowhere differentiable, but this was seen as a pathology. There's this famous quote in the letter of Helmut to Stiltius

where he says, Je me ditour avec et fois et aureur de cette playe la montable difoncians sans derivées. So I turn away with shock and disgust from this lamentable plague of functions with no derivatives.

But here in this context of diffusion phenomenon, both in mathematics and in the real world, that's how it is. The typical paths are nowhere differentiable. So the question is how do we deal with this? Now the idea of a tangent of a given Markov process.

Here the tangent in X in the diffusion case will simply be such a Wiener process with constant coefficients taking off in this sense. This suggests to write down the infinitesimal behavior of the Markov process path in this form.

It should take off like it's tangent from any given position at any given time, which means it should have this form locally. So that's the idea. And then next step of the program, construct the path from this stochastic

differential equation. In other words, solve this in integrated form, this equation where the position is turned deep comes about from the starting point X by integrating up here and integrating up here. Now here it's classical, no problem, but here it's completely unclear because this depends on the path.

And even though Zingma may be smooth, the sense XS itself is non-smooth, this will be very non-smooth. And here we don't have a classical integrator. The path of a Wiener process is nowhere differentiable. So this needs a new approach to integration,

what is now known as stochastic integration. Now there had already been one step by Wiener and Paley, which defined the integral by integration by parts. And this works whenever here

you have a classical integrator. So if the path here of the integrant is of bounded variation, and they did it in the case where it's of bounded variation and deterministic. But as we have just seen, the integrants which are needed here in this construction program

by no means of this type. And so Ito went about constructing the integral in this form for general integrants, adapted in the sense that they depend only on the information up to time S, of the past behavior up to that point.

They satisfy some integrability condition, but no regularity. And the idea is to write down Riemann sums, but non anticipating Riemann sums. So here's integrant is evaluated at the left hand side, and then pass to the limit using a basic isometry

that for simple integrants like this piecewise constant, the L2 norm here will be equal to the expectation of this integral, which is a L2 norm in this product space. And why is this isometry, something reasonable came about,

and that was the birth of stochastic integration in the sense of Ito. In the introduction to Ito's selected papers, Dan Strueck and Varadan Wright, everyone, you could say in this room, has at least heard, they say,

who might pick up the book and start to read, has at least heard that there is a subject called the theory of stochastic integration, and that K Ito is the Lebesgue of this branch of integration theory, in brackets Paley and Wiener where it's Riemann.

Now, this is not all. To complete the program, the stochastic differential equation is actually resolved. We have so far made sense out of the integrated form, and we need a verification that this construction does what it's asked to be,

namely to provide a solution of Kolmogorov's equation, and this verification goes via Ito's formula, to which we are turning now. Now, outside of this room, and in fact, outside of mathematics, there are nowadays many, many thousands of people

who probably have not heard, or if so, do not care about Lebesgue, Paley, Wiener, but they have heard, and they do care about Ito's formula. Now, why is that? On the one hand,

it just has turned out to be an extremely useful tool outside of mathematics in many areas, and also it has a certain fundamental quality, and I want to underline the fundamental quality of this formula. The Gauss Prize, of course,

is not the first distinction Kiyoshi Ito has received. He obtained the Wolf Prize, I think, in 1987, and in the Laudatio of the Wolf Prize, it said, he has given us a full understanding

of the infinitesimal development of Markov sample pairs, and I've tried to show you, as a short summary, how this goes. This may be viewed as Newton's Law in the stochastic realm, providing a direct translation between the governing partial differential equation

and the underlying probabilistic mechanism. Its main ingredient is a differential and integral calculus of functions of Brownian motion. The resulting theory is a cornerstone of modern probability, both pure and applied. Now, coming from Germany,

having been socialized in Germany, and especially from Berlin, if I see Newton, I also think of Leibniz. So, here is Leibniz. Together with Newton, this is taken from an article in The Economist where they describe the feud they had about priorities.

Now, the bottom line probably is that, to a large extent, to paraphrase Pascal, the truth was the same in London and Berlin at that time. Now, here is something by Leibniz, written in Berlin. Ein Merkvold jag zin Bolismus,

des algebrasch, no, des Infinitie, die Malkelts, but no point in reading it in German because it was written in Latin, but I only got hold of this German text. Now, I show you. He writes here what everybody knows, integration, he has a product rule for differentiation. And then he says, out of this formula,

the whole remaining calculus can be developed. But this formula is demonstrated as follows. So, he multiplies out and he gets this term. And then he said, okay, this dx is much smaller than x

and dy is much smaller than y, so the whole thing is much smaller than the remaining terms. So, we forget them and out comes the usual product rule,

which in particular implies this and for a reasonable function f, this standard behavior along the trajectory x. And then he writes, Quotiorima zani memorabilia omnibus quo vis communiest. So, this very remarkable theorem is common to all curves.

Now, the insight of Ito was, no, it's not. It's not. In general, this formula should be written in this way. In other words, these extra terms cannot be forgotten in general. For general curves, as they turn out to be typical

in the diffusion picture. Because here in this form, something comes up which does not come up in classical calculus, that's the quadratic variation of the function. The function is nowhere differentiable, but it has a quadratic variation. And Pauli V showed it's equal to t for a typical path of the Wiener process.

And Ito showed that for a solution of a stochastic differential equation, it will have this form. And so more generally, Ito's product rule takes this form with a correspondingly defined co-variation.

And then Ito's formula for a function f in class C2 has this extra term. And if it's spelled out in the context of a multi-dimensional diffusion solving the stochastic differential equation we have seen, it takes this form.

And here this operator L, of course comes in. And by the way, here we have a choice, we can either stick here to the underlying Brownian paths or we can translate by taking part of this over to the other side to Dx, the increment of the diffusion paths. 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 describe some consequences which we are going to need a little bit later. So first consequence is, and that was the reason, the reason he proved this. He used this then to check, to verify

Kolmogorov's equation for the solutions of his stochastic differential equation. So it looked like a tool you use in passing. But it has far reaching consequences, and I'll show you one, which will be interesting for us a little later.

It implies a representation theorem. So let's take a functional of the diffusion process, which is given by the function small h applied to the final position over some time window from zero to big T. Then Ito's formula reduces to this representation

of the functional as a constant plus an Ito integral of this process. If you choose f as a solution of the following boundary value problem, it should solve this partial differential equation, and the terminal value should be equal to h.

Why? Because in general, we have a remaining term where exactly this integral over these terms appears. Now, if this vanishes, this drops out, and you have this representation. So it shows that such simple functionals of the diffusion can be written as stochastic integrals

of the underlying diffusion path. Now you can take products of those and patch the results here together in an obvious way. Then you get it for functionals of this form, that they can be represented with some integrants, and then you add an approximation procedure for general functionals,

and then each reasonable functional of a nice diffusion X can be shown to be a stochastic integral of that diffusion with some integrant. Okay, so what I have sketched until now is really in the spirit of pure mathematics.

I mean, some people perceive the field of probability as applied per se, no? But it's a wide field. It has very theoretical areas. It has very applied areas. It has the whole spectrum. As we have heard, Kiyoshi Ito considers himself

to be rather on the very pure side of it. And so in this spirit, I have tried to describe this program. And in fact, what I've given you is a short summary of the whole book, which does have the details. So if you really want to know, you should look at this book, Markov Processes from Kiyoshi Ito's Perspective,

written some years ago by Dan Stroup. Okay, now we come to impact, the reason for avoiding the Gauss Prize. Let's first look at impact within mathematics.

And there it should be said that it took a long time, at least in the West. In the fifth, so we are talking about work, which had been done already in the early forties, then worked out, published in an accessible way

in the early fifties. But in the fifties and sixties, the reception was not very widespread. There's a notable exception of Dupe, Joe Dupe immediately understood the significance, made sure it was published in the memoirs of the American math society.

And he himself wrote a chapter on stochastic integration in his book, appearing in 53 on stochastic processes. In the East, there was more interest in the Russian school. Dinkins, Korokhot and others took up very fast the idea and the techniques of stochastic integration

and use them in important ways. For example, in defining the Gersonov transformation and shift of the measure, transformation of the measure, which corresponds to a change of the drift in the stochastic differential equation. Now, Ito came to Princeton, to the U.S. in the fifties.

So he was there from 54 to 56. And he described in the statement he gave a year ago at the Arbel symposium in his honor, that there he met McKean, a student at the time of Feller

and explained to him his ideas about stochastic differential equation. And then he says, there was once an occasion when McKean tried to explain to Feller my work on the stochastic differential equations and the idea of tangents. It seemed to me that Feller did not fully understand

its significance. At any rate, there was no encouraging reaction. On the other hand, when I explained to Feller Levi's local time, he immediately became enthusiastic, made some conjectures, how to describe the structure of one dimensional diffusions in terms of local time

and pose that as a problem. And then, Ito and McKean worked on this, solved the problem in a way completed Feller's program and wrote a whole book, Diffusion Processes and Their Sample Paths, which appeared in 65. And ironically, no stochastic differential equations,

no stochastic integrals in that whole book. Now for me, as a graduate student at the time, this was my first exposure to the work of Ito. And I found it, so we formed a group of graduate students and did our own informal seminar,

the reading sections in this book. And we found it hard going. And I thought, okay, maybe this is the Japanese style to be tough on your students, but no, no, complete misunderstanding. By chance, I discovered lecture notes written only by Ito on stochastic processes at the Tata Institute

and at Aarhus University, and say they are marvelous. Suddenly everything looked very clear and very easy. It was written in a friendly way. And so I was really looking forward to meet that person. And that happened in the summer of 68,

when Ito came from Aarhus where he was at the time to give lectures at the University of Erlangen. This is taken 10 years later. But I remember somehow this posture, and especially his way of transmitting his own joy

in dealing with mathematics, he was explaining. And this was a very, very strong experience. And there was one memorable situation where a distinguished colleague from the United States who was visiting for a whole term, and was giving lectures on probabilistic potential theory,

gave his last lecture, and Ito was present. Now this was 68. So students at the time were somewhat unruly, and they're ready to be unpolite. So we asked this colleague who was just planning to finish

a long technical proof about additive functionals of Markov processes. Couldn't we just forget about the technicalities? And could he not just give us some perspectives, some outlook, what the important directions are? Now this took him by surprise. He was not prepared for that. So he hesitated and was not eager to go in that direction.

Then suddenly Ito went up, went to the blackboard. And that was one highlight of our time as graduate students. How Ito in one hour gave his views on topics he thought were important and had potential for further research.

I think it was mainly excursion theory of Markov process, which he described excursions of a Markov process viewed as a past-valued Poisson point process. So it was absolutely exciting stuff. Okay, but so far, no stochastic integrals. Also in that lecture, this improvised lecture,

no stochastic integrals. Now, 69, the situation started to change. Henry McKean wrote a small book, Stochastic Integrals, clearly dedicated to Kiyoshi Ito. And since the 70s, suddenly there was an explosion of stochastic analysis in a general martingale setting

via an important paper inspired, I'm sure, by Ito of Konita and Watanabe. And then the French school took over, Mayor de la Cherie, Jacques Cordure, and expanded the whole framework in a tremendous way.

Then since the 80s, infinite dimensional extensions came, measure value diffusions are one class of examples, Dawson, Watanabe, and others motivated by biological population dynamics. And then in a very new and systematic way,

in Maliaven calculus initiated by Paul Maliaven, which by the way has a strong school in Spain represented, at least until recently, by David Buellar and still by Marta Sanz. And Kiyoshi Ito himself contributed to this development

the lectures he mentioned in his greeting at the ETH Zurich, where on foundation of stochastic differential equations in infinite dimensional spaces, where in a way he viewed to Maliaven calculus as an infinite dimensional Ito calculus based on an infinite dimensional Ornstein-Uenbeck process

and embedded that in a general framework. At any rate, stochastic analysis is now clearly a core chapter of probability. There is no doubt about it. But the Gauss Prize requires applications beyond mathematics.

So in an anatomical form, I will describe how I experienced that. The first thing to say is that this went faster. When I was, as a young PhD, an instructor at MIT,

courses involving Ito stochastic differential equation, I still have my course directory, so I checked. In mathematics, zero. Electrical engineering, four. Aeronautics and astronautics, two. And I actually attended one of these two,

which was about space flight and the effect of random disturbances by changing fluctuating densities in the atmosphere. And this was modeled by Ito stochastic differential equations and analyzed special topics, stability of such dynamical systems perturbed by noise,

stochastic Lyapunov functions introduced and handled via Ito calculus, problems of filtering and control. Going beyond the Kalman Busey filter. And the first textbook, for example, in Germany

on stochastic differential equation by Ludwig Arnold appeared in 73 at the technical university Stuttgart was written primarily for engineers. For example, the motion of a satellite, what I just described, randomly fluctuating, density of the atmosphere was one prime example here. Now, my next experience was in 1977

when I moved to ETH Zurich, my next door neighbor was Conrad Oosterwalder. Conrad Oosterwalder had just returned from Harvard where he worked with Schroder, Jaffe and inspired by ideas of Shemansic on pass integrals in quantum field theory.

And Barry Simon in this year was in Switzerland and gave a whole lecture course on pass space techniques, which appeared as a book in 79. And here are the sections 14 to 16, Ito's integral Schrodinger operators with magnetic fields

introduction to stochastic calculus where he gave a proof of Ito's formula. So clearly the importance was seen by colleagues in mathematical physics. And then the idea came up in 87 to offer an honorary degree to Kiyoshi Ito. It was very easy because it had strong support, not only by the probabilists,

it was a joint venture with mathematical physicists working in quantum field theory. Now, finally, I want to describe one case study where I'm being a little bit more precise, namely the applications to finance. Now, my own, I came into contact with this

in something. Yeah, sorry. Through a student, David Krebs, who later got the Clark Medal for the best economist under 40. So some analog economics of the field medal after 84. And then at 84, he visited ETH

and we discussed martingale aspects of a new development and finance related to pricing and hedging of derivatives. So we have a price fluctuation on the liquid financial market, by the way, this is not a recent set. In some sense, we are closing the circle. We are going back to one of the sources.

Remember, Bashir Yee wrote his first, as though they introduced Brownian motion with that purpose. And so we have such fluctuations. We have a probability measure on pair space. Let's stick to the continuous case. If we have D financial assets, this would be the pair space, but very quickly it becomes fancy.

If you think of yield curves, the fluctuation of yield curves, the state space would be an infinite dimensional space and you could make a choice here via stochastic differential equations. Now, what is P? Yeah, that's a big problem. There's business and this has statistical econometric,

but also theoretical aspects related to the notion of market efficiency. It's a strong form of market efficiency says that information and expectations are immediately priced in. So after discounting, you get this property that the present price is equal to the best guess of the future price.

And that's the martingale property under P. And in that case, P would be called the martingale measure. Now on the financial market, you can trade. So you can vary with the number of stocks you hold at any time. And the cumulative net gain of an idealized continuous trading strategy

would be exactly an Ito integral. Why would it be an Ito integral? Because this is based on non anticipating Riemann sums, and that's exactly the financial meaning. You must make your investment at the beginning without foresight and before the actually price increment happens.

Now for such a martingale measure, dupe system theorem says that there are no winning strategies. So that's somehow too strong. And a weaker form of market efficiency would be the absence of free lunches. There may be winning strategies,

but in that case, you have a downside risk. And this is an equivalent to the fact that the model does not have to be a martingale measure, but it must admit an equivalent martingale measure. Okay, now we come to a core problem in this business pricing catching of financial derivatives, defined as some functionals of the price process.

So some contract which says you get paid at the end so much if the scenario small omega happens. And now we can see how Ito comes in, in a strong and crucial way. Okay, in a complete model, this measure is unique

and that's okay for nice diffusion such as a so-called Black-Scholes model. Then I showed you a representation property which was proved through Ito's formula. And by the way, where did I first see this proof? In a paper appearing in electrical engineering

because in the early 70s, because the electrical engineers were interested in this property. But the fact itself was already known. And in the financial interpretation, it now means that you have a perfect replication of your financial derivative by some dynamic strategy.

And that means that this initial constant here, the initial costs of this perfect replication must be the natural price. And this can all be computed as the expectation of the equivalent martingale measure because under the martingale measure, this will be a martingale term

and the expectations will drop out. And in particular, it's a special situation of the Black-Scholes framework, special process, special derivatives. This is the Black-Scholes formula. But it's really a corollary of Ito's formula and Ito's formula is conceptually

plus the representation theorem. The representation theorem gives the right conceptual framework and at the stroke settles the issue for any exotic derivatives. Of course, now the computation of the strategies becomes very involved. It's a mixture of ideal calculus and PDEs.

For exotic derivatives, there are ways of reducing it to the simple case or to use really advanced techniques like Malia van calculus. So all this is nowadays financial engineering, which is a way of applied stochastic analysis. Incomplete models are much more complicated,

but all the more interesting. And also here, Ito's stochastic analysis provides the crucial concepts and tools. My time is running out, so I'm skipping one example, which I wanted to illustrate how an important issue, heterogeneous information on financial markets.

Some people know more than others. The question is what is the financial role of that for the financial value of obtaining certain additional information. There is a small paper by Ito in the seventies on stochastic differentials where he discusses a change of filtration. What happens if you increase your information?

And that provides immediately the key to this very hot topic of heterogeneous information. And some of the recent work do use those techniques initiated by Ito, for example, Karatsas, Imkeller, Schweitzer. To summarize, Kiyoshi Ito

has molded the way in which we all think about stochastic processes. This is a quote, again, taken from the introduction of Strueck and Varadan to his collective papers. Now this was written already in 78. We all in 78, they probably had in mind

a rather known group of specialists in the area of stochastic analysis. But we all, that has increased dramatically over the last 30 years and beyond the boundaries of mathematics

in areas such as engineering, where it started early, finance, where it started with a vengeance and really developed an incredible momentum. And the amazing thing is to me, how much the concepts, not so much the computational tools

have framed the discourse in departments of finance and economics nowadays. It has completely taken over, so to speak. And so I do agree, and I'm sure many of all these will agree with that initial quote from the internet

that the Committee for the Gauss Prize has really picked someone really wonderful. Okay, thank you.