Stochastic Processes

Lectures Given at Aarhus University

By: Kiyosi Ito, Ole E Barndorff-Nielsen (Editor), Ken-iti Sato (Editor)

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Hardcover | 12 March 2004

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This is 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, today called the Levy-Ito decomposition, in a form close to Ito's original paper from 1942. 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.

Industry Reviews

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"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)

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Non-Fiction Mathematics Applied Mathematics Stochastics Probability & Statistics

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Published: 30th November 2010

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Additive processes (processes with independent increments)	p. 39
Definitions	p. 39
Decomposition of additive processes	p. 41
The Levy modification of additive processes continuous in probability	p. 45
Elementary Levy processes	p. 50
Fundamental lemma	p. 58
Structure of sample functions of Levy processes (a)	p. 61
Structure of sample functions of Levy processes (b)	p. 68
Three components of Levy processes	p. 74
Random point measures	p. 77
Homogeneous additive processes and homogeneous Levy processes	p. 83
Levy processes with increasing paths	p. 85
Stable processes	p. 88
Markov processes	p. 93
Transition probabilities and transition operators on compact metrizable spaces	p. 93
Summary of the Hille-Yosida theory of semi-groups	p. 95
Transition semi-group	p. 101
Probability law of the path	p. 103
Markov property	p. 110
The a-algebras B, B[subscript 1] and B(S)	p. 115
Strong Markov property	p. 118
An inequality of Kolmogorov type and its application	p. 124
Hitting times of closed sets	p. 129
Dynkin's formula	p. 130
Markov processes in generalized sense	p. 135
Examples	p. 143
Markov processes with a countable state space	p. 148
Fine topology	p. 154
Generator in generalized sense	p. 159
The Kac semi-group and its application to the Arcsine law	p. 164
Markov processes and potential theory	p. 170
Brownian motion and the Dirichlet problem	p. 172
Table of Contents provided by Blackwell. All Rights Reserved.

Stochastic Processes

Lectures Given at Aarhus University

At a Glance

Hardcover

Industry Reviews

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You Can Find This Book In

Other Editions and Formats

Paperback

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