Stochastic Processes: Lectures Given at Aarhus University

ISBN-13: 9783642058059 / Angielski / Miękka / 2010 / 236 str.

Kiyosi Ito; Ole Eiler Barndorff-Nielsen; Ken-Iti Sato

Opis

The volume Stochastic Processes by K. Ito 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 Levy-It o decomposition of additive processes. Encouraged by Professor It o 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. Levy'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 Levy-Ito decomposition. This is thoroughly treated, as- suming no continuity property in time, in a form close to the original 1942 paper of Ito, which gave rigorous expression to Levy's intuitive understanding of path behavior.

Recenzje

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)

Spis treści

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.

O autorze

A readily accessible introduction to the theory of stochastic processes with emphasis on processes with independent increments and Markov processes. After preliminaries on infinitely divisible distributions and martingales, Chapter 1 gives a thorough treatment of the decomposition of paths of processes with independent increments. Chapter 2 contains a detailed treatment of time-homogeneous Markov processes from the viewpoint of probability measures on path space. Two separate Sections present about 70 exercises and their complete solutions. The text and exercises are carefully edited and footnoted, while retaining the style of the original lecture notes from Aarhus University.