SpringerBriefs in Probability and Mathematical Statistics

Poisson Point Processes and Their Application to Markov Processes

Authors: Itô, Kiyosi

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  • Gives a beautiful elementary treatment of general Poisson point processes in Chapter 1, especially recommended for beginners
  • Shows how the notion of Poisson point processes with values in a function space of paths called excursions plays a key role in an extension problem of Markov processes in Chapter 2
  • Demonstrates how the general theory in Chapter 2 can answer completely the extension problem for the minimal diffusion on [0, ∞) with an exit boundary 0
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About this book

An extension problem (often called a boundary problem) of Markov processes has been studied, particularly in the case of one-dimensional diffusion processes, by W. Feller, K. Itô, and H. P. McKean, among others. In this book, Itô discussed a case of a general Markov process with state space S and a specified point a ∈ S called a boundary. The problem is to obtain all possible recurrent extensions of a given minimal process (i.e., the process on S \ {a} which is absorbed on reaching the boundary a). The study in this lecture is restricted to a simpler case of the boundary a being a discontinuous entrance point, leaving a more general case of a continuous entrance point to future works. He established a one-to-one correspondence between a recurrent extension and a pair of a positive measure k(db) on S \ {a} (called the jumping-in measure and a non-negative number m< (called the stagnancy rate). The necessary and sufficient conditions for a pair k, m was obtained so that the correspondence is precisely described. For this, Itô used,  as a fundamental tool, the notion of Poisson point processes formed of all excursions of  the process on S \ {a}. This theory of Itô's of Poisson point processes of excursions is indeed a breakthrough. It has been expanded and applied to more general extension problems by many succeeding researchers. Thus we may say that this lecture note by Itô is really a memorial work in the extension problems of Markov processes. Especially in Chapter 1 of this note, a general theory of Poisson point processes is given that reminds us of Itô's beautiful and impressive lectures in his day.

Reviews

“The main idea of this volume has had a profound influence on the boundary theory of Markov processes. This volume is beautifully written and it is a pleasure to read.” (Ren Ming Song, Mathematical Reviews, December, 2016) 


Table of contents (2 chapters)

Table of contents (2 chapters)

Buy this book

eBook 41,64 €
price for Spain (gross)
  • ISBN 978-981-10-0272-4
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover 51,99 €
price for Spain (gross)
  • ISBN 978-981-10-0271-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Poisson Point Processes and Their Application to Markov Processes
Authors
Series Title
SpringerBriefs in Probability and Mathematical Statistics
Copyright
2015
Publisher
Springer Singapore
Copyright Holder
The Author(s)
eBook ISBN
978-981-10-0272-4
DOI
10.1007/978-981-10-0272-4
Softcover ISBN
978-981-10-0271-7
Series ISSN
2365-4333
Edition Number
1
Number of Pages
XI, 43
Number of Illustrations
3 b/w illustrations
Topics