Poisson Point Processes and Their Application to Markov Processes (SpringerBriefs in Probability and Mathematical Statistics) - Softcover

Itô, Kiyosi

Softcover

ISBN 10: 9811002711 ISBN 13: 9789811002717

Verlag: Springer, 2016

Inhaltsangabe

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.

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Biography of Kiyosi Itô Kiyosi Itô was born on September 7, 1915, in Kuwana, Japan. After his undergraduate and doctoral studies at Tokyo University, he was associate professor at Nagoya University before joining the faculty of Kyoto University in 1952. He has remained there ever since and is now Professor Emeritus, but has also spent several years at each of Stanford, Aarhus and Cornell Universities and the University of Minnesota. Itô's fundamental contributions to probability theory, especially the creation of stochastic differential and integral calculus and of excursion theory, form a cornerstone of this field. They have led to a profound understanding of the infinitesimal development of Markovian sample paths, and also of applied problems and phenomena associated with the planning, control and optimization of engineering and other random systems. Professor Itô has been the inspirer and teacher of an entire generation of Japanese probabilists. Biography of Henry McKean Henry McKean was born on December 14, 1930, in Wenham, Massachusetts. He studied mathematics at Dartmouth College, Cambridge University, and Princeton University; he received his degree from the last in 1955. He has held professional positions at Kyoto University, MIT, Rockefeller University, Weizmann Institute, Balliol College, Oxford, and the Courant Institute of Mathematical Sciences (1969 to present). His main interests are probability, Hamiltonian mechanics, complex function theory, and nonlinear partial differential equations.

Details zur Ausgabe

• Verlag: Springer
• Erscheinungsdatum: 2016
• Sprache: Englisch
• ISBN 10: 9811002711
• ISBN 13: 9789811002717
• Einband: Taschenbuch
• Auflage: 1
• Anzahl der Seiten: 56
• Kontakt zum Hersteller: ProductSafety@springernature.com; ProductSafety@springernature.com; ; Poststr. 9; Darmstadt; 64293; Deutschland