Format:
Online-Ressource (XI, 43 p. 3 illus. in color, online resource)
Edition:
1st ed. 2015
ISBN:
9789811002724
Series Statement:
SpringerBriefs in Probability and Mathematical Statistics
Content:
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
Note:
Description based upon print version of record
,
Foreword; Preface; References; Contents; 1 Poisson Point Processes; 1.1 Point Functions; 1.2 Point Processes; 1.3 Poisson Point Processes; 1.4 The Structure of Poisson Point Processes (1) the Discrete Case; 1.5 The Structure of Poisson Point Processes (2) the General Case; 1.6 Transformation of Poisson Point Processes; 1.7 Summable Point Processes; 1.8 The Strong Renewal Property of Poisson Point Processes; References; 2 Application to Markov Processes; 2.1 Problem; 2.2 The Poisson Point Process Attached to a Markov Process at a State a; 2.3 The Jumping-In Measure and the Stagnancy Rate
,
2.4 The Existence and Uniqueness Theorem2.5 The Resolvent Operator and the Generator ; 2.6 Examples; Reference
Additional Edition:
ISBN 9789811002717
Additional Edition:
Erscheint auch als Druck-Ausgabe Itō, Kiyosi Poisson point processes and their application to Markov processes Singapore : Springer Singapore, 2015 ISBN 9789811002717
Language:
English
DOI:
10.1007/978-981-10-0272-4
URL:
Volltext
(lizenzpflichtig)
Bookmarklink
