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Poisson Point Processes and Their Application to Markov Processes (Paperback) (Kiyosi Itu00f4)

Poisson Point Processes and Their Application to Markov Processes (Paperback) (Kiyosi Itu00f4) - image 1 of 1

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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. 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.
Number of Pages: 43
Genre: Mathematics
Sub-Genre: Mathematical Analysis, Functional Analysis, Probability + Statistics / General
Series Title: SpringerBriefs in Probability and Mathematical Statistics
Format: Paperback
Publisher: Springer Verlag
Author: Kiyosi Itu00f4
Language: English
Street Date: February 1, 2016
TCIN: 51054029
UPC: 9789811002717
Item Number (DPCI): 248-15-1311
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