Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Probabilities. --- Mathematical statistics. --- Poisson processes. --- Markov processes. --- Probabilités --- Statistique mathématique --- Processus de Poisson --- Markov, Processus de --- Probabilities --- Mathematical statistics --- Poisson processes --- Markov processes --- Probabilités --- Statistique mathématique --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Processes, Poisson --- Mathematics. --- Functional analysis. --- Measure theory. --- Probability Theory and Stochastic Processes. --- Measure and Integration. --- Functional Analysis. --- Stochastic processes --- Point processes --- Distribution (Probability theory. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Distribution functions --- Frequency distribution --- Characteristic functions --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Risk

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Mathematics