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Autore: | Itô Kiyosi |
Titolo: | Poisson Point Processes and Their Application to Markov Processes / / by Kiyosi Itô |
Pubblicazione: | Singapore : , : Springer Singapore : , : Imprint : Springer, , 2015 |
Edizione: | 1st ed. 2015. |
Descrizione fisica: | 1 online resource (54 p.) |
Disciplina: | 519.23 |
Soggetto topico: | Probabilities |
Measure theory | |
Functional analysis | |
Probability Theory and Stochastic Processes | |
Measure and Integration | |
Functional Analysis | |
Persona (resp. second.): | WatanabeShinzo <1935-> |
ShigekawaIchirō <1953-> | |
Note generali: | Description based upon print version of record. |
Nota di bibliografia: | Includes bibliographical references at the end of each chapters. |
Nota di contenuto: | 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 |
Sommario/riassunto: | 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. |
Titolo autorizzato: | Poisson Point Processes and Their Application to Markov Processes |
ISBN: | 981-10-0272-X |
Formato: | Materiale a stampa |
Livello bibliografico | Monografia |
Lingua di pubblicazione: | Inglese |
Record Nr.: | 9910300248603321 |
Lo trovi qui: | Univ. Federico II |
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