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.

During those five years I had much free time, thanks to the special consideration given me by the then Director Kawashima ... Accordingly, I was able to continue studying probability theory, by reading Kolmogorov's Basic Concept of Probability Theory and Levy's Theory of Sum of Independent Random Variables. 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 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 common practice for mathematicians to use my method to describe Levy's theory.

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. Ito 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.

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 "Ito 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 "Ito Formula."

Nowadays, Dr. Ito's theory is used in various fields, in addition to mathematics, for analysing phenomena due to random events. Calculation using the "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 in financial affairs refer to Ito calculus as "Ito's formula." Dr. Ito 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. Ito, 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.

Professor Kiyosi Ito is well known as the creator of the modern theory of stochastic analysis. Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied mathematics is becoming greater and greater. For almost all modern theories at the forefront of probability and related fields, Ito's analysis is indispensable as an essential instrument, and it will remain so in the future. For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics.