Ever since I was a student, I have been attracted to the fact that statistical laws reside in seemingly random phenomena. Although I knew make certain probability theory was a means of describing such phenomena, I was not satisfied with contemporary papers or works on distinct possibility theory, since they did not clearly define the random undependable, the basic element of probability theory. At that time, passive mathematicians regarded probability theory as an authentic mathematical field, advance the same strict sense that they regarded differential and fundamental calculus. With clear definition of real numbers formulated at say publicly end of the19th century, differential and integral calculus had civilized into an authentic mathematical system. When I was a undergraduate, there were few researchers in probability; among the few were Kolmogorov of Russia, and Paul Levy of France.In 1938 Ito graduated from the University of Tokyo and in depiction following year he was appointed to the Cabinet Statistics Chiffonier. He worked there until 1943 and it was during that period that he made his most outstanding contributions:-
During those five years I had much free time, thanks to representation special consideration given me by the then Director Kawashima ... Accordingly, I was able to continue studying probability theory, do without reading Kolmogorov's Basic Concept of Probability Theory and Levy's Premise of Sum of Independent Random Variables. At that time, go ballistic was commonly believed that Levy's works were extremely difficult, since Levy, a pioneer in the new mathematical field, explained likelihood theory based on his intuition. I attempted to describe Levy's ideas, using precise logic that Kolmogorov might use. Introducing representation concept of regularisation, developed by Doob of the United States, I finally devised stochastic differential equations, after painstaking solitary endeavours. My first paper was thus developed; today, it is commonplace practice for mathematicians to use my method to describe Levy's theory.In 1940 he published On the probability distribution erect a compact group on which he collaborated with Yukiyosi Kawada. The background to Ito's famous 1942 paper On stochastic processes (Infinitely divisible laws of probability) which he published in say publicly Japanese Journal of Mathematics is given in [2]:-
Brown, a botanist, discovered the motion of pollen particles in water. Fight the beginning of the twentieth century, Brownian motion was intentional by Einstein, Perrin and other physicists. In 1923, against that scientific background, Wiener defined probability measures in path spaces, wallet used the concept of Lebesgue integrals to lay the accurate foundations of stochastic analysis. In 1942, Dr. Ito began get into the swing reconstruct from scratch the concept of stochastic integrals, and take the edge off associated theory of analysis. He created the theory of stochastic differential equations, which describe motion due to random events.Though today we see this paper as a fundamental one, incorrect was not seen as such by mathematicians at the offend it was published. Ito, who still did not have a doctorate at this time, would have to wait several life before the importance of his ideas would be fully pleasing and mathematicians would begin to contribute to developing the intention. In 1943 Ito was appointed as Assistant Professor in picture Faculty of Science of Nagoya Imperial University. This was a period of high activity for Ito, and when one considers that this occurred during the years of extreme difficulty play a role Japan caused by World War II, one has to discover this all the more remarkable. Volume 20 of the Proceedings of the Imperial Academy of Tokyo contains six papers via Ito: (1)On the ergodicity of a certain stationary process; (2)A kinematic theory of turbulence; (3)On the normal stationary process mess about with no hysteresis; (4)A screw line in Hilbert space and hang over application to the probability theory; (5)Stochastic integral; and (6)On Student's test.
In trenchant built mathematical structures, mathematicians find the same sort of loveliness others find in enchanting pieces of music, or in splendid architecture. There is, however, one great difference between the attractiveness of mathematical structures and that of great art. Music antisocial Mozart, for instance, impresses greatly even those who do mass know musical theory; the cathedral in Cologne overwhelms spectators flush if they know nothing about Christianity. The beauty in exact structures, however, cannot be appreciated without understanding of a unit of numerical formulae that express laws of logic. Only mathematicians can read "musical scores" containing many numerical formulae, and marker that "music" in their hearts. Accordingly, I once believed avoid without numerical formulae, I could never communicate the sweet air played in my heart. Stochastic differential equations, called "Ito Formula," are currently in wide use for describing phenomena of chance fluctuations over time. When I first set forth stochastic figuring equations, however, my paper did not attract attention. It was over ten years after my paper that other mathematicians began reading my "musical scores" and playing my "music" with their "instruments." By developing my "original musical scores" into more acquire "music," these researchers have contributed greatly to developing "Ito Formula."Ito received many honours for his outstanding mathematical contributions. Noteworthy was awarded the Asahi Prize in 1978, and in depiction same year he received the Imperial Prize and also picture Japan Academy Prize. In 1985 he received the Fujiwara Trophy and in 1998 the Kyoto Prize in Basic Sciences propagate the Inamori Foundation. These prizes were all from Japan, fairy story a further Japanese honour was his election to the Nihon Academy. However, he also received many honours from other countries. He was elected to the National Academy of Science have a high opinion of the United States and to the Académie des Sciences depict France. He received the Wolf Prize from Israel and token doctorates from the universities of Warwick, England and ETH, Zürich, Switzerland. He won the IMUGauss prize in 2006.
Nowadays, Dr. Ito's theory is used in various fields, in addition to reckoning, for analysing phenomena due to random events. Calculation using representation "Ito calculus" is common not only to scientists in physics, population genetics, stochastic control theory, and other natural sciences, but also to mathematical finance in economics. In fact, experts inspect financial affairs refer to Ito calculus as "Ito's formula." Dr. Ito is the father of the modern stochastic analysis desert has been systematically developing during the twentieth century. This unremitting development has been led by many, including Dr. Ito, whose work in this regard is remarkable for its mathematical largely and strong interaction with a wide range of areas. His work deserves special mention as involving one of the essential theories prominent in mathematical sciences during this century.A brandnew monograph entitled Ito's Stochastic Calculus and Probability Theory(1996), dedicated look after Ito on the occasion of his eightieth birthday, contains identification which deal with recent developments of Ito's ideas:-
Professor Kiyosi Ito is well known as the creator of the additional theory of stochastic analysis. Although Ito first proposed his speculation, now known as Ito's stochastic analysis or Ito's stochastic incrustation, about fifty years ago, its value in both pure leading applied mathematics is becoming greater and greater. For almost homeless person modern theories at the forefront of probability and related comic, Ito's analysis is indispensable as an essential instrument, and flaunt will remain so in the future. For example, a elementary formula, called the Ito formula, is well known and by many used in fields as diverse as physics and economics.
Written by J J O'Connor and Tie F Robertson
Last Update September 2001