The 1998 Laureates / Basic Sciences Category / Mathematical Science

My Sixty Years in Studies of Probability Theory 

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."