Poisson Point Processes and Their Application to Markov Processes PDF
by Kiyosi Ito
Part of the SpringerBriefs in Probability and Mathematical Statistics series
Description
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.
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Download - Immediately Available
- Format:PDF
- Publisher:Springer Nature Singapore
- Publication Date:24/12/2015
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- ISBN:9789811002724
Information
-
Download - Immediately Available
- Format:PDF
- Publisher:Springer Nature Singapore
- Publication Date:24/12/2015
- Category:
- ISBN:9789811002724