Dr. Itô made great contributions to the advancement not only of mathematical sciences, but also of physics, engineering, biology and economics, through his research in stochastic analysis, especially his invention of stochastic differential equations, which enable us to describe random motions and random phenomena in nature and society.
On stochastic processes (1) (infinitely divisible laws of probability) , Japanese Journal of Mathematics 18, 1942.
On a formula concerning stochastic differentials, Nagoya Mathematical Journal 3, 1951.
On stochastic differential equations, Memoirs of the American Mathematical Society 4, 1951.
Diffusion Processes and Their Sample Paths, Grundlehren der Math. Wiss. 125 (with H. P. McKean, Jr.) Springer-Verlag, 1965.
Selected Papers (D. W. Stroock and S. R. S. Varadhan, eds.) Springer-Verlag, 1987.
Through his study of stochastic analysis, especially his original theory regarding stochastic differential equations, Dr. Itô has made great contributions to the development not only of mathematical sciences, but also of physics, engineering, biology, and economics. His theory marked a new epoch in scientific research regarding random motion and stochastic phenomena in nature and society.
Brown, a botanist, discovered the motion of pollen particles in water. At the beginning of the twentieth century, Brownian motion was studied by Einstein, Perrin and other physicists.
In 1923, against this scientific background, Wiener defined probability measures in path spaces, and used the concept of Lebesgue integrals to lay the mathematical foundations of stochastic analysis.
In 1942, Dr. Itô began to reconstruct from scratch the concept of stochastic integrals, and its associated theory of analysis. He created the theory of stochastic differential equations,which describe motion due to random events. Since the advent of his theory, modern stochastic analysis has seen rapid progress.
The stochastic differential equation is an equation of motion that describes the continuous path of motion due to random events; Brownian motion, for example. The solutions define probability measures in path spaces that describe general diffusion motion. Since about 1950, new branches of Dr. Itô’s theory have developed and interacted with the theory of partial differential equations, the potential theory, the theory of harmonic integrals, differential geometry, harmonic analysis, and many other mathematics, as well as with theoretical physics.
Many researchers have so far contributed to establishment of the differential and integral calculus in path spaces. Recently,various attempts have been made to reconstruct differential geometry, and the theory of asymptotic expansions. Dr. Itô’s theory has survived, and continues to serve as the basic method of stochastic analysis.
Nowadays, Dr. Itô’s theory is used in various fields, in addition to mathematics, for analyzing phenomena due to random events. Calculation using the “Itô 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 in financial affairs refer to Itô calculus as “Itô’s formula.”
Dr. Itô is the father of the modern stochastic analysis that has been systematically developing during the twentieth century. This ceaseless development has been led by many, including Dr. Itô, whose work in this regard is remarkable for its mathematical depth and strong interaction with a wide range of areas. His work deserves special mention as involving one of the basic theories prominent in mathematical sciences during this century.
For these reasons, the Inamori Foundation is pleased to bestow upon Dr. Itô the 1998 Kyoto Prize in Basic Sciences.
Ever since I was a student, I have been attracted to the fact that statistical laws reside in seemingly random phenomena. Although I knew that probability theory was a means of describing such phenomena, I was not satisfied with contemporary papers or works on probability theory, since they did not clearly define the random variable, the basic element of probability theory. At that time, few mathematicians regarded probability theory as an authentic mathematical field, in the same strict sense that they regarded differential and integral calculus. With clear definition of real numbers formulated at the end of the19th century, differential and integral calculus had developed into an authentic mathematical system. When I was a student, there were few researchers in probability; among the few were Kolmogorov of Russia, and Paul Levy of France.
In1938, upon graduation from university, I joined the Cabinet Statistics Bureau, where, until I became an associate professor at Nagoya University, I worked for five years. During those five years I had much free time, thanks to the special consideration given me by then Director Kawashima (grandfather of Princess Akishino). Accordingly, I was able to continue studying probability theory, by reading Kolmogorov’s Basic Concept of Probability Theory (1933) and Paul Levy’s Theory of Sum of Independent Random Variables (1937). At that time, it was commonly believed that Levy’s works were extremely difficult, since Levy, a pioneer in the new mathematical field, explained probability theory based on his intuition. I attempted to describe Levy’s ideas, using precise logic that Kolmogorov might use. Introducing the concept of regularization, developed by Doob of the U.S., I finally devised stochastic differential equations, after painstaking solitary endeavors. My first paper was thus developed; today, it is common practice for mathematicians to use my method to describe Levy’s theory.
In precisely built mathematical structures, mathematicians find the same sort of beauty others find in enchanting pieces of music, or in magnificent architecture. There is, however, one great difference between the beauty of mathematical structures and that of great art. Music by Mozart, for instance, impresses greatly even those who do not know musical theory; the cathedral in Cologne overwhelms spectators even if they know nothing about Christianity. The beauty in mathematical structures, however, cannot be appreciated without understanding of a group of numerical formulae that express laws of logic. Only mathematicians can read “musical scores” containing many numerical formulae, and play that “music” in their hearts. Accordingly, I once believed that without numerical formulae, I could never communicate the sweet melody played in my heart.
Stochastic differential equations, called “Itô Formula,” are currently in wide use for describing phenomena of random fluctuations over time. When I first set forth stochastic differential 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 elaborate “music,” these researchers have contributed greatly to developing “Itô Formula.” In recent years, I find that my “music” is played in various fields, in addition to mathematics. Never did I expect that my “music” would be found in such various fields, its echo benefiting the practical world, as well as adding abstract beauty to the field of mathematics. On this opportunity of the Kyoto Prize lectures, I would like to express my sincerest gratitude and render homage to my senior researchers, who repeatedly encouraged me, hearing subtle sounds in my “Unfinished Symphony.”