Kiyosi Itô, a professor of mathematics at Kyoto University, was regarded as the greatest living expert on probability theory. His chief contribution was to find a way to extend the calculus to include the dynamics of random objects. This field is now standard working equipment of, among others, mathematicians in the financial services industry, handling financial derivatives and the like.

Kiyosi Itô was born in 1915 in Hokusei-cho, Mie Prefecture, Japan. He studied mathematics at the Imperial University in Tokyo, and as a student was drawn to probability theory -- where one sees order out of chaos -- mathematics used not to predict individual random outcomes but to make overall or statistical statements, which can be very precise and informative. He devoted his life to the field, and lived to see his name attached to the everyday tools of those who model the uncertainty in the world about us.

When Itô graduated, in 1938, probability theory was not a well-developed mathematical discipline. The decisive step in harnessing the relevant modern mathematics to describe randomness and uncertainty had only recently been taken, by the Russian mathematician Kolmogorov in 1933, and outside the Russian school few mathematicians of world rank were active in the field, including Lévy in France and Doob in the US.

Itô was fortunate in spending five formative years working in the Cabinet Statistical Bureau. Here, under its enlightened director Kawashima, he had ample study time, which he used to master the works of Kolmogorov, Lévy and Doob, and then to make some of his own most outstanding contributions.

The most powerful single technique in mathematics, and indeed in the whole of science, is calculus -- the differential and integral calculus developed in the 17th century by Newton and Leibniz. This gives one the language needed to describe dynamics -- such things as the trajectory of a projectile, or the evolution of a physical system. Itô extended the calculus to include stochastic processes. The familiar Newton-Leibniz formalism was extended to include novel terms -- the Itô or correction terms, which turn out to be crucial. Itô's extension of the chain rule (or the product rule) of ordinary calculus has become known as Itô's formula, or Itô's lemma, and is at the heart of modern stochastic calculus.

In 1943 Itô became an assistant professor at Nagoya Imperial University. He had started publishing remarkable mathematical research in 1940, despite such wartime difficulties as inadequate libraries and lack of scientific contact with the West. His paper on the stochastic integral (now known as the Itô integral) appeared in 1944. He was awarded his doctorate in 1945.

For Itô the 1940s were a period of intense research activity and growing fame. In 1952 he became a professor of mathematics at Kyoto University, where he remained until his retirement in 1979, apart from visits abroad -- to Princeton in 1954-56, to Bombay, Aarhus and Cornell. He remained mathematically active after his retirement, and continued as a professor at Gakushuin University (it is common for distinguished Japanese academics to continue in the private sector after retiring from the public one). His last years were dogged by ill-health.

Itô wrote several influential books on probability theory and stochastic processes, some translated from the Japanese, and one particular classic, Diffusion Processes and Their Sample Paths, with the US mathematician Henry McKean in 1965 (diffusions are random, or stochastic, processes which evolve in time continuously and without memory).

Perhaps his other most important single contribution was his development of excursion theory, according to which the evolution in time of a stochastic process may be decomposed into excursions away from some fixed point; the resulting formalism, though powerful, is difficult, as there are in general infinitely many such excursions in finite time.

Itô's mathematical style was very direct, and appealed to probabilistic intuition rather than relying on analytic methods or mathematical formalism.

The profound impact of his work on mathematical finance is not accidental: Itô was motivated in his development of his integral in the 1940s by considering how the evolution of the return on a stock should be modelled by decomposing it into a deterministic term (the mean return, modelled as with money in the bank), and a stochastic risky term, modelling the uncertainty in the financial and economic climate. His work built on and corrected the first attempts to use such methods for financial modelling, due to the French mathematician Bachelier in 1900.

Itô will be remembered as the father of Japanese probability, as one of the very greatest probabilists and, with Bachelier, as one of the founding fathers of mathematical finance.

Itô received numerous prizes and honorary degrees. He was elected to the US National Academy of Science and to the Académie des Sciences of France. Only a week before his death he was awarded the Culture Medal of Japan, the highest prize awarded by the Emperor. The Japanese mathematical community was intensely proud of him. Largely through his influence and prestige, probability is still very strong in Japan. Japanese probabilists tended to refer to him among themselves as "the Emperor".

Kiyosi Itô is survived by his three daughters.

**Professor Kiyosi Itô, mathematician, was born on September 7, 1915. He died on November 10, 2008, aged 93**

The Times, November 20, 2008 © The Times, 2008