{"product_id":"poisson-point-processes-and-their-application-to-markov-processes-von-kiyosi-ito","title":"Poisson Point Processes and Their Application to Markov Processes","description":"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\u0026lt; (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.\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9789811002717\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9789811002717","offer_id":39463772553309,"sku":"9789811002717","price":53.49,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/b5dbc842-4cf7-4002-b9fd-644ca3b2575c.jpg?v=1772176324","url":"https:\/\/shop.autorenwelt.de\/products\/poisson-point-processes-and-their-application-to-markov-processes-von-kiyosi-ito","provider":"Autorenwelt Shop","version":"1.0","type":"link"}