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Stochastic Processes : Lectures given at Aarhus University - Kiyosi Ito

Stochastic Processes

Lectures given at Aarhus University

Hardcover Published: 12th March 2004
ISBN: 9783540204824
Number Of Pages: 236

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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.

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)

Additive processes (processes with independent increments)p. 39
Definitionsp. 39
Decomposition of additive processesp. 41
The Levy modification of additive processes continuous in probabilityp. 45
Elementary Levy processesp. 50
Fundamental lemmap. 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 processesp. 74
Random point measuresp. 77
Homogeneous additive processes and homogeneous Levy processesp. 83
Levy processes with increasing pathsp. 85
Stable processesp. 88
Markov processesp. 93
Transition probabilities and transition operators on compact metrizable spacesp. 93
Summary of the Hille-Yosida theory of semi-groupsp. 95
Transition semi-groupp. 101
Probability law of the pathp. 103
Markov propertyp. 110
The a-algebras B, B[subscript 1] and B(S)p. 115
Strong Markov propertyp. 118
An inequality of Kolmogorov type and its applicationp. 124
Hitting times of closed setsp. 129
Dynkin's formulap. 130
Markov processes in generalized sensep. 135
Examplesp. 143
Markov processes with a countable state spacep. 148
Fine topologyp. 154
Generator in generalized sensep. 159
The Kac semi-group and its application to the Arcsine lawp. 164
Markov processes and potential theoryp. 170
Brownian motion and the Dirichlet problemp. 172
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540204824
ISBN-10: 3540204822
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 236
Published: 12th March 2004
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.93
Weight (kg): 0.49

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