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The volume Stochastic Processes by K. Itö was published as No. 16 of Lecture Notes Series from Mathematics Institute, Aarhus University in August, 1969, based on Lectures given at that Institute during the academie year 1968 1969. The volume was as thick as 3.5 cm., mimeographed from typewritten manuscript and has been out of print for many years. Since its appearance, it has served, for those abIe to obtain one of the relatively few copies available, as a highly readable introduetion to basic parts of the theories of additive processes (processes with independent increments) and of Markov processes. It contains, in particular, a clear and detailed exposition of the Lévy-It ö decomposition of additive processes. Encouraged by Professor It ó we have edited the volume in the present book form, amending the text in a number of places and attaching many footnotes. We have also prepared an index. Chapter 0 is for preliminaries. Here centralized sums of independent ran dom variables are treated using the dispersion as a main tooI. Lévy's form of characteristic functions of infinitely divisible distributions and basic proper ties of martingales are given. Chapter 1 is analysis of additive processes. A fundamental structure the orem describes the decomposition of sample functions of additive processes, known today as the Lévy-Itó decomposition. This is thoroughly treated, as suming no continuity property in time, in a form close to the original 1942 paper of Itó, which gave rigorous expression to Lévy's intuitive understanding of path behavior.
List of contents
0 Preliminaries.- 1 Additive Processes (Processes with Independent Increments).- 2 Markov Processes.- Exercises.- E.0 Chapter 0.- E.1 Chapter 1.- E.2 Chapter 2.- Appendix: Solutions of Exercises.- A.0 Chapter 0.- A.1 Chapter 1.- A.2 Chapter 2.
About the author
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.
Summary
The volume Stochastic Processes by K. Itö was published as No. 16 of Lecture Notes Series from Mathematics Institute, Aarhus University in August, 1969, based on Lectures given at that Institute during the academie year 1968 1969. The volume was as thick as 3.5 cm., mimeographed from typewritten manuscript and has been out of print for many years. Since its appearance, it has served, for those abIe to obtain one of the relatively few copies available, as a highly readable introduetion to basic parts of the theories of additive processes (processes with independent increments) and of Markov processes. It contains, in particular, a clear and detailed exposition of the Lévy-It ö decomposition of additive processes. Encouraged by Professor It ó we have edited the volume in the present book form, amending the text in a number of places and attaching many footnotes. We have also prepared an index. Chapter 0 is for preliminaries. Here centralized sums of independent ran dom variables are treated using the dispersion as a main tooI. Lévy's form of characteristic functions of infinitely divisible distributions and basic proper ties of martingales are given. Chapter 1 is analysis of additive processes. A fundamental structure the orem describes the decomposition of sample functions of additive processes, known today as the Lévy-Itó decomposition. This is thoroughly treated, as suming no continuity property in time, in a form close to the original 1942 paper of Itó, which gave rigorous expression to Lévy's intuitive understanding of path behavior.
Additional text
From the reviews:
"The book can be recommended as a fine introduction to such important branches of stochastic process theory as the theories of processes with independent increments and of Markov processes. It will be a valuable acquisition for any mathematical library. The text of the book has been carefully prepared by the editors … ." (M.G. Shur, Mathematical Reviews, 2005e)
"The book under review is in fact an advanced text suitable for graduate students and based around two topics-the structure of additive processes … and the basic theory of Markov processes, which generalises Markov chains to continuous time and fairly general state spaces. … a nice introduction to Markov processes making extensive use of semigroup techniques. … The book concludes with a number of exercises accompanied by worked solutions." (David Applebaum, The Mathematical Gazette, March, 2005)
Report
From the reviews:
"The book can be recommended as a fine introduction to such important branches of stochastic process theory as the theories of processes with independent increments and of Markov processes. It will be a valuable acquisition for any mathematical library. The text of the book has been carefully prepared by the editors ... ." (M.G. Shur, Mathematical Reviews, 2005e)
"The book under review is in fact an advanced text suitable for graduate students and based around two topics-the structure of additive processes ... and the basic theory of Markov processes, which generalises Markov chains to continuous time and fairly general state spaces. ... a nice introduction to Markov processes making extensive use of semigroup techniques. ... The book concludes with a number of exercises accompanied by worked solutions." (David Applebaum, The Mathematical Gazette, March, 2005)
