|
Showing 1 - 8 of
8 matches in All Departments
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. Ito, and H.
P. McKean, among others. In this book, Ito 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, Ito used, as a fundamental tool, the notion of Poisson point
processes formed of all excursions of the process on S \ {a}. This
theory of Ito'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 Ito 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 Ito's beautiful and impressive lectures in his
day.
Kosaku Yosida, born on February 7, 1909, was brought up in Tokyo.
Having majored in Mathematics at University of Tokyo, he was
appointed to Assistant at Osaka University in 1933 and promoted to
Associate Professor in 1934. He re ceived the title of Doctor of
Science from Osaka University in 1939. In 1942 he was appointed to
Professor at Nagoya University, where he worked very hard with his
colleagues to promote and expand the newly established Department
of Mathe matics. He was appointed to Professor at Osaka University
in 1953 and then to Professor at University of Tokyo in 1955. After
retiring from University of Tokyo in 1969, he was appointed to
Professor at Kyoto University, where he also acted as Director of
the Research Institute for Mathematical Sciences. He retired from
Kyoto University in 1972 and worked as Professor at Gakushuin
University until 1979. Yosida acted as President of the
Mathematical Society of Japan, as Member of the Science Council of
Japan, and as Member of the Executive Committee of the
International Mathematical Union. In 1967 he received the Japan
Academy Prize and the Imperial Prize for his famous work on the
theory of semigroups and its applications. In 1971 he was elected
Member of the Japan Academy. Yosida went abroad many times to give
series of lectures at mathematical in stitutions and to deliver
invited lectures at international mathematical symposia.
This accessible introduction to the theory of stochastic
processes emphasizes Levy processes and Markov processes. It gives
a thorough treatment of the decomposition of paths of processes
with independent increments (the Levy-Ito decomposition). It also
contains a detailed treatment of time-homogeneous Markov processes
from the viewpoint of probability measures on path space. In
addition, 70 exercises and their complete solutions are
included."
This accessible introduction to the theory of stochastic
processes emphasizes Levy processes and Markov processes. It gives
a thorough treatment of the decomposition of paths of processes
with independent increments (the Levy-Ito decomposition). It also
contains a detailed treatment of time-homogeneous Markov processes
from the viewpoint of probability measures on path space. In
addition, 70 exercises and their complete solutions are
included."
Since its first publication in 1965 in the series "Grundlehren der
mathematischen Wissenschaften" this book has had a profound and
enduring influence on research into the stochastic processes
associated with diffusion phenomena. Generations of mathematicians
have appreciated the clarity of the descriptions given of one- or
more- dimensional diffusion processes and the mathematical insight
provided into Brownian motion. Now, with its republication in the
"Classics in Mathematics" it is hoped that a new generation will be
able to enjoy the classic text of Ito and McKean."""
The central and distinguishing feature shared by all the
contributions made by K. Ito is the extraordinary insight which
they convey. Reading his papers, one should try to picture the
intellectual setting in which he was working. At the time when he
was a student in Tokyo during the late 1930s, probability theory
had only recently entered the age of continuous-time stochastic
processes: N. Wiener had accomplished his amazing construction
little more than a decade earlier (Wiener, N. , "Differential
space," J. Math. Phys. 2, (1923)), Levy had hardly begun the
mysterious web he was to eventually weave out of Wiener's P~!hs,
the generalizations started by Kolmogorov (Kol mogorov, A. N. ,
"Uber die analytische Methoden in der Wahrscheinlichkeitsrechnung,"
Math Ann. 104 (1931)) and continued by Feller (Feller, W. , "Zur
Theorie der stochastischen Prozesse," Math Ann. 113, (1936))
appeared to have little if anything to do with probability theory,
and the technical measure-theoretic tours de force of J. L. Doob
(Doob, J. L. , "Stochastic processes depending on a continuous
parameter, " TAMS 42 (1937)) still appeared impregnable to all but
the most erudite. Thus, even at the established mathematical
centers in Russia, Western Europe, and America, the theory of
stochastic processes was still in its infancy and the student who
was asked to learn the subject had better be one who was ready to
test his mettle.
A systematic, self-contained treatment of the theory of stochastic
differential equations in infinite dimensional spaces. Included is
a discussion of Schwartz spaces of distributions in relation to
probability theory and infinite dimensional stochastic analysis, as
well as the random variables and stochastic processes that take
values in infinite dimensional spaces.
|
You may like...
Barbie
Margot Robbie, Ryan Gosling
Blu-ray disc
R266
Discovery Miles 2 660
|