TheWork of Kyosi Itô

Kyosi Itô, 1954, when he was a Fellow at the Institute for Advanced Study, Princeton. On August 22, 2006, the International Mathematical Union awarded the Carl Friedrich Gauss Prize at the opening ceremonies of the International Congress of Mathematicians in Madrid, Spain. The prizewinner is Kyosi Itô. The Gauss prize was created to honor mathematicians whose research has had a profound impact not just on mathematics itself but also on other disciplines. To understand the achievements of Itô, it is helpful to understand the context in which they were developed. Bachelier in 1900, and Einstein in 1905, proposed mathematical models for the phenomenon known as Brownian motion. These models represent the random motion of a very small particle in a liquid suspension. Norbert Wiener and collaborators showed in the 1920s that Einstein’s model exists as a stochastic process, using the then-new ideas of Lebesgue measure theory. Many properties of the process were established in the 1930s, the most germane for this article being that its sample paths are of infinite variation on any compact time interval, no matter how small. This made the Riemann-Stieltjes integration theory inapplicable. Wienerwanted to use such integrals to study filtering theory and signal detection, important during the second world war. Despite these problems he developed a theory of integrals, known today as