The central and distinguishing feature shared by all the contributions made by K. Ito is the extraordinary insight which they convey. Reading his papers, one should try to picture the intellectual setting in which he was working. At the time when he was a student in Tokyo during the late 1930s, probability theory had only recently entered the age of continuous-time stochastic processes: N. Wiener had accomplished his amazing construction little more than a decade earlier (Wiener, N. , "Differential space," J. Math. Phys. 2, (1923)), Levy had hardly begun the mysterious web he was to eventually weave out of Wiener's P~!hs, the generalizations started by Kolmogorov (Kol mogorov, A. N. , "Uber die analytische Methoden in der Wahrscheinlichkeitsrechnung," Math Ann. 104 (1931)) and continued by Feller (Feller, W. , "Zur Theorie der stochastischen Prozesse," Math Ann. 113, (1936)) appeared to have little if anything to do with probability theory, and the technical measure-theoretic tours de force of J. L. Doob (Doob, J. L. , "Stochastic processes depending on a continuous parameter, " TAMS 42 (1937)) still appeared impregnable to all but the most erudite. Thus, even at the established mathematical centers in Russia, Western Europe, and America, the theory of stochastic processes was still in its infancy and the student who was asked to learn the subject had better be one who was ready to test his mettle.

Autorentext
Kiyosi Itô was born on September 1915, in Kuwana, Japan. After his undergraduate and doctoral studies at Tokyo University, he was associate professor at Nagoya University before joining the faculty of Kyoto University in 1952. He has remained there ever since and is now Professor Emeritus, but has also spent several years at each of Stanford, Aarhus and Cornell Universities and the University of Minnesota. Itô's fundamental contributions to probability theory, especially the creation of stochastic differential and integral calculus and of excursion theory, form a cornerstone of this field. They have led to a profound understanding of the infinitesimal development of Markovian sample paths, and also of applied problems and phenomena associated with the planning, control and optimization of engineering and other random systems.

Klappentext

Kiyosi Itô (1915 - 2008) was one of the pioneers of probability theory, and the originator of Ito Calculus. First published in 1942 in Japanese, this epoch-making theory of stochastic differential equations describes nondeterministic and random evolutions. The Ito formula has found applications in other branches of mathematics as well as in various other fields including e.g., conformal field theory in physics, stochastic control theory in engineering, population genetics in biology and mathematical finance in economics. Besides being renowned for his brilliant mathematical achievements during a long and productive career spanning over sixty years, Itô has been a truly inspirational teacher to many mathematicians. He has received numerous awards and honors, including the 1987 Wolf Foundation Prize in Mathematics, the 1998 Kyoto Prize in Basic Sciences and the 2003 Japanese Cultural Merit Prize. In 2006, Itô was awarded the first Carl Friedrich Gauss Prize for Applications of Mathematics. This book contains a selection of 38 papers published between 1941 and 1983.

Inhalt
[1] On Stochastic Processes (Infinitely divisible laws of probability).- [2] Differential Equations Determining a Markoff Process.- [3] On the Ergodicity of a Certain Stationary Process.- [4] A Kinematic Theory of Turbulence.- [5] On the Normal Stationary Process with no Hysteresis.- [7] Stochastic Integral.- [9] On a Stochastic Integral Equation.- [10] Stochastic Differential Equations in a Differentiable Manifold.- [11] Brownian Motions in a Lie Group.- [12] On Stochastic Differential Equations.- [13] On a Formula Concerning Stochastic Differentials.- [14] Multiple Wiener Integral.- [15] Stochastic Differential Equations in a Differentiable Manifold.- [16] Stationary Random Distributions.- [17] Complex Multiple Wiener Integral.- [18] Isotropic Random Current.- [19] Spectral Type of the Shift Transformation of Differential Processes with Stationary Increments.- [20] Potentials and the Random Walk.- [21] Wiener Integral and Feynman Integral.- [22] Construction of Diffusions.- [23] The Brownian Motion and Tensor Fields on Riemannian Manifold.- [24] Brownian Motions on a Half Line.- [25] The Expected Number of Zeros of Continuous Stationary Gaussian Processes.- [26] On Stationary Solutions of a Stochastic Differential Equation.- [27] Transformation of Markov Processes by Multiplicative Functionals.- [28] The Canonical Modification of Stochastic Processes.- [29] On the Convergence of Sums of Independent Banach Space Valued Random Variables.- [30] Generalized Uniform Complex Measures in the Hilbertian Metric Space with their Application to the Feynman Integral.- [31] On the Oscillation Functions of Gaussian Processes.- [32] Canonical Measurable Random Functions.- [33] The Topological Support of a Gauss Measure on Hilbert Space.- [34] Poisson Point Processes Attached to Markov Processes.- [37] Stochastic Differentials.- [38] Stochastic Parallel Displacement.- [40] Extension of Stochastic Integrals.- [44] Infinite Dimensional Ornstein-Uhlenbeck Processes.- [45] Regularization of Linear Random Functionals (with M. Nawata).- [46] Distribution-Valued Processes Arising from Independent Brownian Motions.