El profesor Kijosí Itō fue uno de los teóricos de la probabilidad más distinguidos del mundo. Es el creador de una rama de las matemáticas que se ocupa de la estocástica y las probabilidades, ahora conocida como cálculo de Itō en su honor. Una de sus herramientas principales, la integral estocástica, también se conoce como integral de Itō. Este cálculo desempeña un papel fundamental en las matemáticas financieras modernas.
The book highlights K. Ito's remarkable insights within the context of his intellectual environment during the late 1930s in Tokyo, a period when probability theory was evolving with the advent of continuous-time stochastic processes. It discusses the foundational work of key figures like N. Wiener, A. N. Kolmogorov, and W. Feller, illustrating how their contributions were pivotal yet complex, often challenging for students. The narrative emphasizes the nascent state of stochastic process theory, requiring perseverance and intellect from those who sought to master it.
Stochastic analysis, a branch of probability theory stemming from the theory of stochastic differential equations, is becoming increasingly important in connection with partial differential equations, non-linear functional analysis, control theory and statistical mechanics.
The volume by K. Itö, published in August 1969 as part of the Lecture Notes Series from the Mathematics Institute, Aarhus University, is based on lectures from the academic year 1968-1969. This 3.5 cm thick, mimeographed text has been out of print for years, serving as a valuable introduction to additive processes and Markov processes for those fortunate enough to obtain one of the few copies. It features a clear exposition of the Lévy-Itö decomposition of additive processes. Encouraged by Professor Itö, the volume has been edited into its current form, with amendments and additional footnotes, along with an index. Chapter 0 covers preliminaries, discussing centralized sums of independent random variables and utilizing dispersion as a key tool. It also presents Lévy's characteristic functions of infinitely divisible distributions and essential properties of martingales. Chapter 1 focuses on the analysis of additive processes, detailing a fundamental theorem that describes the decomposition of sample functions, known as the Lévy-Itö decomposition. This is explored thoroughly, without assuming continuity in time, closely aligning with Itö's original 1942 paper that articulated Lévy's intuitive understanding of path behavior.
Since its first publication in 1965 in the series Grundlehren der mathematischen Wissenschaften this book has had a profound and enduring influence on research into the stochastic processes associated with diffusion phenomena. Generations of mathematicians have appreciated the clarity of the descriptions given of one- or more- dimensional diffusion processes and the mathematical insight provided into Brownian motion. Now, with its republication in the Classics in Mathematics it is hoped that a new generation will be able to enjoy the classic text of Itô and McKean .
