Skip to main content
SpringerLink
Log in
Book cover

Poisson Point Processes and Their Application to Markov Processes

  • Book
  • © 2015

Overview

Authors:
  1. Kiyosi Itô

    1. Kyoto University, Japan

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • 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, 8) with an exit boundary 0
  • Includes supplementary material: sn.pub/extras

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (2 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Poisson Point Processes

    • Kiyosi Itô
    Pages 1-18

  3. Application to Markov Processes

    • Kiyosi Itô
    Pages 19-43
Back to top

Keywords

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) 

Authors and Affiliations

  • Kyoto University, Japan

    Kiyosi Itô

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation