{"product_id":"9789811002717","title":"SpringerBriefs in Probability and Mathematical Statistics","description":"\u003ch1\u003eSpringerBriefs in Probability and Mathematical Statistics\u003c\/h1\u003e \u003ch2\u003eItô, Kiyosi; Watanabe, Shinzo; Shigekawa, Ichiro\u003c\/h2\u003e \u003cp\u003eAn 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.\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Springer\u003c\/p\u003e \u003cp\u003ePublication Date: 2016-02-01\u003c\/p\u003e \u003cp\u003eFormat: Paperback\u003c\/p\u003e \u003cp\u003eISBN-13: 9789811002717\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-981-10-0272-4\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 43\u003c\/p\u003e ","brand":"Springer Nature Singapore","offers":[{"title":"Default Title","offer_id":46547938803852,"sku":"9789811002717","price":49.49,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9789811002717.jpg?v=1775962887","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9789811002717","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}