was a

Japanese Japanese may refer to: * Something from or related to Japan, an island country in East Asia * Japanese language, spoken mainly in Japan * Japanese people, the ethnic group that identifies with Japan through ancestry or culture ** Japanese diaspor ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who made fundamental contributions to

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, in particular, the theory of

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...

es. He invented the concept of stochastic integral and

stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...

, and is known as the founder of so-called

Itô calculus Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central ...

Overview

Itô pioneered the theory of

stochastic integration Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created a ...

and

s, now known as

. Its basic concept is the Itô integral, and among the most important results is a change of variable formula known as

Itô's lemma In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves ...

. Itô calculus is a method used in the mathematical study of random events and is applied in various fields, and is perhaps best known for its use in

mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...

. Itô also made contributions to the study of

diffusion process In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...

es on

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s, known as stochastic differential geometry. Although the standard Hepburn romanization of his name is ''Kiyoshi Itō'', he used the spelling Kiyosi Itô (

Kunrei-shiki romanization is the Cabinet-ordered romanization system for transcribing the Japanese language into the Latin alphabet. Its name is rendered ''Kunreisiki rômazi'' in the system itself. Kunrei-shiki is sometimes known as the Monbushō system in English bec ...

). The alternative spellings Itoh and Ito are also sometimes seen in the

West West or Occident is one of the four cardinal directions or points of the compass. It is the opposite direction from east and is the direction in which the Sun sets on the Earth. Etymology The word "west" is a Germanic word passed into some ...

Biography

Seizo Ito and Kiyoshi Ito cropped Seizo Ito Kiyoshi Ito Tsuyo Ito Seitaro Ito Seihachi Ito and Mishi Ito 1937

Itô was born in Hokusei-cho in

Mie Prefecture is a prefecture of Japan located in the Kansai region of Honshu. Mie Prefecture has a population of 1,781,948 () and has a geographic area of . Mie Prefecture is bordered by Gifu Prefecture to the north, Shiga Prefecture and Kyoto Prefectur ...

on the main island of

Honshū , historically called , is the largest and most populous island of Japan. It is located south of Hokkaidō across the Tsugaru Strait, north of Shikoku across the Inland Sea, and northeast of Kyūshū across the Kanmon Straits. The island sepa ...

. He graduated with a B.S. (1938) and a Ph.D (1945) in Mathematics from the

University of Tokyo , abbreviated as or UTokyo, is a public research university located in Bunkyō, Tokyo, Japan. Established in 1877, the university was the first Imperial University and is currently a Top Type university of the Top Global University Project b ...

. Between 1938 and 1945, Itô worked for the Japanese National Statistical Bureau, where he published two of his seminal works on

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

and

es, including a series of articles in which he defined the stochastic integral and laid the foundations of the Itō calculus. After that he continued to develop his ideas on stochastic analysis with many important papers on the topic. In 1952, he became a Professor at the

University of Kyoto , mottoeng = Freedom of academic culture , established = , type = Public (National) , endowment = ¥ 316 billion (2.4 billion USD) , faculty = 3,480 (Teaching Staff) , administrative_staff = 3,978 (Total Staff) , students = 2 ...

to which he remained affiliated until his retirement in 1979. Starting in the 1950s, Itô spent long periods of time outside Japan, at

Cornell Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...

Stanford Stanford University, officially Leland Stanford Junior University, is a Private university, private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. S ...

, the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...

in Princeton, New Jersey, and

Aarhus University Aarhus University ( da, Aarhus Universitet, abbreviated AU) is a public research university with its main campus located in Aarhus, Denmark. It is the second largest and second oldest university in Denmark. The university is part of the Coimbra Gr ...

in Denmark. Itô was awarded the inaugural

Gauss Prize The Carl Friedrich Gauss Prize for Applications of Mathematics is a mathematics award, granted jointly by the International Mathematical Union and the German Mathematical Society for "outstanding mathematical contributions that have found significan ...

in 2006 by the

International Mathematical Union The International Mathematical Union (IMU) is an international non-governmental organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports ...

for his lifetime achievements. As he was unable to travel to

Madrid Madrid ( , ) is the capital and most populous city of Spain. The city has almost 3.4 million inhabitants and a metropolitan area population of approximately 6.7 million. It is the second-largest city in the European Union (EU), and ...

, his youngest daughter, Junko Itô received the Gauss Prize from the

King of Spain , coatofarms = File:Coat_of_Arms_of_Spanish_Monarch.svg , coatofarms_article = Coat of arms of the King of Spain , image = Felipe_VI_in_2020_(cropped).jpg , incumbent = Felipe VI , incumbentsince = 19 Ju ...

on his behalf. Later, International Mathematics Union (IMU) President Sir John Ball personally presented the medal to Itô at a special ceremony held in Kyoto. In October 2008, Itô was honored with Japan's

Order of Culture The is a Japanese order, established on February 11, 1937. The order has one class only, and may be awarded to men and women for contributions to Japan's art, literature, science, technology, or anything related to culture in general; recipient ...

, and an awards ceremony for the Order of Culture was held at the Imperial Palace. "Donald Keene, 7 others win Order of Culture,"
''Yomiuri Shimbun.'' October 29, 2008 (in Japanese) Itô wrote in

, Chinese,

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...

French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...

and

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...

. He died on November 10, 2008 in

Kyoto, Japan Kyoto (; Japanese: , ''Kyōto'' ), officially , is the capital city of Kyoto Prefecture in Japan. Located in the Kansai region on the island of Honshu, Kyoto forms a part of the Keihanshin metropolitan area along with Osaka and Kobe. , the cit ...

at age 93.

Scientific works of Kiyosi Itô

* * * * * * * *

Notes

References

Obituary
at ''

The New York Times ''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid ...

'' * * * *

External links

Kiyosi Itô(1915-2008) / Eightieth Birthday Lecture RIMS, Kyoto University, September 1995 / Research Institute for Mathematical Sciences, Kyoto University Kyoto

at

Research Institute for Mathematical Sciences The is a research institute attached to Kyoto University, hosting researchers in the mathematical sciences from all over Japan. RIMS was founded in April 1963. List of directors * Masuo Fukuhara (1963.5.1 – 1969.3.31) * Kōsaku Yosida (196 ...