Showing papers in "Japanese Journal of Mathematics in 2007"
[...]
TL;DR: In this paper, the authors present three examples of the mean-field approach to modelling in economics and finance (or other related subjects) and show that these nonlinear problems are essentially well-posed problems with unique solutions.
Abstract: We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.
2,385 citations
TL;DR: In this paper, the authors describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which such convergence occurs.
Abstract: We describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which such convergence occurs. Both continuous and discrete time are considered and a number of particular examples, notably the way in which a population of animals move together, are considered as particular instances of our setting.
543 citations
TL;DR: In this paper, positive and negative results on the Hodge conjecture are discussed for integral Hodge classes, and possible extensions of the conjecture to the general Kahler setting, and the structure of the so-called locus of Hodge class.
Abstract: I will discuss positive and negative results on the Hodge conjecture. The negative aspects come on one side from the study of the Hodge conjecture for integral Hodge classes, and on the other side from the study of possible extensions of the conjecture to the general Kahler setting. The positive aspects come from algebraic geometry. They concern the structure of the so-called locus of Hodge classes, and of the Hodge loci.
115 citations
TL;DR: In this article, it was shown that the law of the length of the excursion straddling an independent exponential time is infinitely divisible, i.e., it is polynomial in the time of the last zero before the exponential time.
Abstract: We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein’s representations that, e.g. the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss the Ornstein–Uhlenbeck processes.
60 citations
TL;DR: A report on recent results and outstanding problems concerning motives associated to graphs can be found in this article, where the authors present a survey of the state-of-the-art.
Abstract: A report on recent results and outstanding problems concerning motives associated to graphs.
51 citations
TL;DR: In this paper, the authors introduce several homotopy invariants of words and give a classification of words of length five, based on the Reidemeister moves on knot diagrams.
Abstract: We discuss a topological approach to words introduced by the author in [Tu2]–[Tu4]. Words on an arbitrary alphabet are approximated by Gauss words and then studied up to natural modifications inspired by the Reidemeister moves on knot diagrams. This leads us to a notion of homotopy for words. We introduce several homotopy invariants of words and give a homotopy classification of words of length five.
41 citations
TL;DR: In this paper, a presentation of Ito's excursion theory for general Markov processes is given, with several applications to Brownian motion and related processes, and several applications are discussed.
Abstract: A presentation of Ito’s excursion theory for general Markov processes is given, with several applications to Brownian motion and related processes.
27 citations
TL;DR: In this article, the authors describe a story of how a Russian mathematician solved an important problem in ergodic theory using Ito multiple stochastic integrals, and describe how a Moscow mathematician solved this problem using a single graph.
Abstract: In this paper we describe a story how a Moscow mathematician solved an important problem in ergodic theory using Ito multiple stochastic integrals.
24 citations
TL;DR: In this paper, the authors show that the geometry of the Heath-Jarrow-Morton interest rates market dynamics can be nonparametrically calibrated by the observation of a single trajectory of the market evolution.
Abstract: We show that the geometry of the Heath–Jarrow–Morton interest rates market dynamics can be non-parametrically calibrated by the observation of a single trajectory of the market evolution. Then the hypoellipticity of the infinitesimal generator can be exactly measured. On a data set of actual interest rates we show the prevalence of the hypoelliptic effect together with a sharp change of regime. Volatilities are computed by applying the Fourier cross-volatility estimation methodology.
19 citations
TL;DR: In this paper, the matrix Euler congruence conjecture was shown to hold for any integer matrix A, prime p, and natural number n. In fact the conjecture immediately follows from a result of C.J. Smyth.
Abstract: In [1], V.I. Arnold conjectured “the matrix Euler congruence”
$${\rm tr} A^{p^n}\equiv {\rm tr} A^{p^{n-1}}\,(\text{mod}\,{p^{n}})$$
for any integer matrix A, prime p, and natural number n. He proved it for p ≤ 5, n ≤ 4. In fact the conjecture immediately follows from a result of C.J. Smyth [5]. We give a simple proof of this result and discuss a related conjecture of Arnold concerning some congruences for multinomial coefficients.
8 citations
TL;DR: A brief historical report on Teiji Takagi which was prepared at the commencement of 'Takagi Lectures' of The Mathematical Society of Japan is given in this paper, where careful readers may find some hints for the questions how and why Takagi was able to establish his class field theory.
Abstract: This article is a brief historical report on Teiji Takagi which was prepared at the commencement of ‘Takagi Lectures’ of The Mathematical Society of Japan. The first of its two purposes is to give some informations on the circumstances of education and research of mathematics in Japan surrounding Takagi who could finally established himself as the founder of the Japanese school of modern mathematics. The other is a brief overview on Takagi’s works of mathematics some of which are still attractive to and influential on especially ambitious students of mathematics. The author hopes that careful readers may find some hints for the questions how and why Takagi was able to establish his class field theory. At the end of this article the readers will find an English translation of the preface of his book Algebraic theory of numbers (in Japanese) which is the only thing that he left for us to see his total view over class field theory after the establishment of Artin’s reciprocity law.
TL;DR: A common feature that can be consistently found in the works of Professor Kiyosi Ito is a leap from the analysis in distribution family level toward the analysis and synthesis in sample paths level, which has turned analytic descriptions into thoroughly stochastic ones.
Abstract: A common feature that can be consistently found in the works of Professor Kiyosi Ito is a leap from the analysis in distribution family level toward the analysis and synthesis in sample paths level, which has turned analytic descriptions into thoroughly stochastic ones.
TL;DR: The Takagi Lectures as discussed by the authors are the first series of lectures in mathematics to be crowned with a Japanese mathematician's name, and they have been used for many years in Japanese mathematics.
Abstract: The Takagi Lectures are the first series of lectures in mathematics to be crowned with a Japanese mathematician's name. The author provides some historical background of the Takagi Lectures.
TL;DR: In this article, the author recollects the author's encounter with Ito in Princeton 1954 and their collaboration on diffusion process and their sample path and their cooperation on sample path.
Abstract: This paper recollects the author’s encounter with Ito in Princeton 1954 and their collaboration on diffusion process and their sample path.
TL;DR: In this article, the boundary of the crown domain of a Riemannian symmetric space X is investigated and in case X is of Hermitian type, an affirmative answer is given.
Abstract: In this paper we raise a question about the boundary of the crown domain of a Riemannian symmetric space X. In case X is of Hermitian type we give an affirmative answer.
TL;DR: The author expresses his sincere gratitude for receiving the first Gauss Prize from the International Mathematical Union, at the special occasion of the ceremonial event which was held at the Research Institute for Mathematical Sciences, Kyoto University, on September 14, 2006.
Abstract: The author expresses his sincere gratitude for receiving the first Gauss Prize from the International Mathematical Union, at the special occasion of the ceremonial event which was held at the Research Institute for Mathematical Sciences, Kyoto University, on September 14, 2006.
TL;DR: The main facts of K. Ito's stochastic integration as well as excursion theory are presented in this paper, together with a number of applications, and the main features of the K.
Abstract: The main facts of K. Ito’s stochastic integration as well as excursion theory are presented, together with a number of applications.