Poisson Point Processes and Their Application to Markov Processes (Paperback, 1st ed. 2015)

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.

General

Imprint:	Springer Verlag, Singapore
Country of origin:	Singapore
Series:	SpringerBriefs in Probability and Mathematical Statistics
Release date:	2016
First published:	2015
Authors:	Kiyosi Ito
Foreword by:	Shinzo Watanabe • Ichiro Shigekawa
Dimensions:	235 x 155 x 3mm (L x W x T)
Format:	Paperback
Pages:	43
Edition:	1st ed. 2015
ISBN-13:	978-981-10-0271-7
Categories:	Books > Science & Mathematics > Mathematics > Probability & statistics Books > Science & Mathematics > Mathematics > Applied mathematics > Stochastics
LSN:	981-10-0271-1
Barcode:	9789811002717

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