Poisson Point Processes and Their Application to Markov Processes (eBook)

Kiyosi Itô (Autor)

eBook Download: PDF

2015 | 1st ed. 2015
XI, 43 Seiten
Springer Singapore (Verlag)
978-981-10-0272-4 (ISBN)

Infos

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.

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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. Ito, and H. P. McKean, among others. In this book, Ito 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, Ito used, as a fundamental tool, the notion of Poisson point processes formed of all excursions of the process on S / {a}. This theory of Ito'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 Ito 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 Ito's beautiful and impressive lectures in his day.

Inhalt

1. Poisson point processes.- 2. Application to Markov Process.

Produkt-Details

• Erscheint lt. Verlag: 24.12.2015
• Reihe/Serie: SpringerBriefs in Probability and Mathematical Statistics
• Reihe/Serie: SpringerBriefs in Probability and Mathematical Statistics
• Vorwort: Shinzo Watanabe, Ichiro Shigekawa
• Zusatzinfo: XI, 43 p. 3 illus.
• Verlagsort: Singapore
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: characteristic measure • discontinuous and continuous entrance points • jumping-in measure and stagnancy rate • Poisson Point Process • Poisson point process of excursions
• ISBN-10: 981-10-0272-X / 981100272X
• ISBN-13: 978-981-10-0272-4 / 9789811002724