Essentials
of Stochastic Processes
Kiyosi Itô
Translated by: Yuji Ito
To be published by American Mathematical Society in 2006
Contents
Author’s
Preface
Translator’s
Foreword
Chapter
1. Basic Concepts
1.1. Measure Theoretic Probability (1) Intuitive
Background 1
1.2. Probability Distribution 3
1.3. Measure Theoretic Probability (2)
Mathematical Structure 6
1.4. Distribution Function,
Characteristic Function, Mean, Variance 8
1.5. Stochastic Process 13
Chapter
2. Additive
Processes
2.1. Definition of Additive Process 15
2.2. Examples of Additive Processes 16
2.3. Inequalities Concerning Sums of
Independent Random Variables 17
2.4. 0-1 Law 19
2.5. Convergence of Additive Sequences
21
2.6. Dispersion 24
2.7. Simple Properties of Additive
Processes 28
2.8. Separability
of Stochastic Processes 31
2.9. Separable Poisson Processes 33
2.10. Separable Wiener Processes 36
2.11. Additive Processes Continuous in
Probability and Infinitely Divisible Distributions 38
2.12. Structure of Separable Additive
Processes Continuous in Probability 42
2.13. Canonical Form of Infinitely
Divisible Distributions 44
2.14. Various Methods for Construction
of Poisson Processes 46
2.15. Compound Poisson Processes 48
2.16. Stable Distributions and Stable
Processes 50
Chapter
3. Stationary
Processes
3.1. Definition of Stationary Process 55
3.2. Preliminary Material Related to
Investigations of Stationary Processes 56
3.3. Spectral Decomposition of Weakly
Stationary Processes 58
3.4. Spectral Decomposition of Sample
Processes of Weakly Stationary Processes 60
3.5. Ergodic
Theorem Concerning Strongly Stationary Processes 62
3.6. Complex
3.7. Normal Stationary Processes 69
3.8. Wiener Integrals and Multiple Wiener
Integrals 71
3.9. Ergodicity
of
3.10. Generalizations of Stationary
Processes 75
Chapter
4. Markov Processes
4.1. Conditional Probability 83
4.2. Conditional Expectation 84
4.3. Martingales 86
4.4. Transition Probabilities 86
4.5. Semi-groups and Dual Semi-groups
Associated with Transition Probabilities 88
4.6. Hille-Yosida
Theory (i) 90
4.7. Hille-Yosida
Theory (ii). Construction of Semi-group 93
4.8. Generators of Transition
Probabilities (i). General Theory 96
4.9. Generators of Transition
Probabilities (ii). Examples 99
4.10. Markov Processes (i). Markov Property 102
4.11. Markov Processes (ii). Properties
of Sample Processes 104
4.12. Markov Processes (iii). Strong
Markov Property 106
4.13. Markov Times 110
4.14. Dynkin’s
Theorem on Generators 113
4.15. Examples of Markov Processes 115
4.16. Temporally Homogeneous Additive
Processes 118
4.17. Birth and Death Processes 119
Chapter
5. Diffusion
5.1. Diffusive Points 125
5.2. Ray’s Theorem 125
5.3. Local Generators 128
5.4. Classification of One-dimensional
Diffusive Points 129
5.5. Feller’s Canonical Scale 132
5.6. Feller’s Canonical Measure 136
5.7. Feller’s Canonical Form 137
5.8. Local Generators at Generalized
Shunts 141
5.9. Distribution of the First Passage
Time 143
5.10. Classical Diffusion Processes 146
5.11. Classification of Boundary Points
with respect to Feller’s Operator 149
5.12. Particular Solutions of the
Homogeneous Equation
5.13. General Solutions of the
Homogeneous Equation
5.14. Solutions of the Non-Homogeneous
Equation 156
5.15. Distributions of Various
Quantities Associated with x(a)(t) in a Regular Interval 159
5.16. Behavior of a Process at the
Boundaries of a Regular Interval 162
Postscript
Author’s
Preface
The present volume Essentials of
Stochastic Processes is an English translation of my book written in Japanese
and issued by Iwanami Shoten in 1957 in two parts:
Stochastic Processes I (from Chapter 1 to 3) and II (from Chapter 4 to 5). In
this work, I provide a unified and comprehensive account of additive processes
(or Lévy processes), stationary processes, and Markov
processes, which remain to this day the three most important classes of
stochastic processes. I had sent the Japanese original at the time of its
publication to Eugene B. Dynkin, and I was very
pleased to see A. D. Wentzell’s Russian translation
published in 1960 (Part I) and 1963 (Part II). I am also grateful to Dynkin for editing the translation and adding some
important clarification footnotes. In 1959 Shizuo Kakutani at Yale University, noting the significance of my
description of the one dimensional diffusions, advised Yuji Ito, then one of
his graduate students, to produce a translation of part II into English, which
was distributed among a limited circle of mathematicians around Yale University
as a typewritten mimeograph. On the occasion of my receiving the Kyoto Prize in
1998, Shinzo Watanbabe and
Masatoshi Fukushima encouraged me to have the entire 1957 book translated into
English and published by the American Mathematical Society. Yuji Ito graciously
agreed to take on this arduous task and revisited his earlier partial
translation, not only adding Part I, but also fully revising his original
translation of Part II.
Although almost half a century has
passed since the initial publication in Japanese, I hope there is enough of
value in this work to merit its publication in English at this time. It should
be noted some detailed introductions to additive processes and Markov processes
are given in two of my lecture notes published later on:
Lectures on Stochastic Processes, Tata
Institute of Fundamental Research, Bombay, 1960.
Stochastic Processes, edited by Ole E. Barndorff-Nielsen
and Ken-iti Sato, Springer, 2004 (originally
published as Lecture Notes from
However, the present volume is the only
one among my English textbooks that includes an introduction to stationary
processes.
Chapter 5 is devoted to the one
dimensional diffusion theory which is important as a basic prototype of the
study of Markov processes. This chapter starts with a presentation of the local
generator of a one dimensional diffusion process as a generalized second order
differential operator discovered by William Feller several years before I wrote
this book. It then proceeds to a detailed description of the boundary behaviors
of the solutions of the associated homogeneous and inhomogeneous equations in
an analytical way, followed by their probabilistic implications on the path
properties of the diffusion near the boundaries.
My lecture notes from the Tata Institute mentioned above contain another detailed
explanation of the Feller local generator. Section 4.6 of my joint book with H.
P. McKean,
Diffusion Processes and Their Sample Paths,
Springer, 1965; in Classics in Mathematics, Springer, 1996.
also exhibits the boundary behaviors
with some probabilistic proof, while sections 5.12, 5.13 and 5.14 of the
present volume are readily understood even by readers unfamiliar with
probability theory. When I wrote the original Japanese version of this book,
the real study of stochastic processes had just begun, and not much related
literature was available as noted in the Postscript. In the five decades since
then, there have been significant developments in the theory of stochastic
processes with many important subsequent publications, some of which are listed
in the Preface to the Original and the Foreword by the Editors in the above
mentioned book published in 2004 based on my Aarhus
Lecture Notes.
I am very much indebted to those who
have helped me bring this translation project to a successful completion. My
gratitude, first and foremost, goes to Yuji Ito for the precise yet elegant
translation which far exceeded my expectations, and I sincerely wish to thank
him once again for his time and efforts. My thanks are due to M. Fukushima, K. Ichihara, and S. Watanabe for the meticulous care they took
in proof-reading and editing the translated manuscript. This English version is
in many ways superior to the original in that it eliminates minor
inconsistencies and updates some of the discussion. In particular, the original
version in Japanese, written when I had just started my work on paths in Markov
processes, contains discussions of the general theory in Chapters 4 and 5 that
are in hindsight somewhat unclear and misleading. I am grateful to M. Fukushima
and S. Watanabe for suggesting the appropriate amendments in these chapters.
In view of the fact that Professor Shizuo Kakutani had first suggested,
shortly after its Japanese publication in 1957, that my book be translated into
English, I had hoped to be able to finally present him with this English
version published by the American Mathematical Society. It was with great
sadness that I learned of his passing away in the summer of 2004 in
Translator’s
Foreword
It is my great pleasure to present an
English translation of “Essentials of Stochastic Processes” written by
Professor Kiyosi Itô. It
was almost half a century ago when the original Japanese version of this book
was published by Iwanami Shoten. As it is mentioned
by Professor It.o in the Author’s Preface, I took up
the translaton of Part II (Chapters 4 and 5 of the
book) into English only a couple of years after the publication of the original
with the urging of the late Professor Shizuo Kakutani of Yale University. I was a graduate student in
mathematics at Yale at the time, trying to write a Ph. D. thesis under
Professor Kakutani’s supervision, and he probably
thought that I should look into the possibility of working in the field of
continuous parameter Markov processes, which was undergoing a rapid development
at the time. No doubt, he felt that the best place to follow this development
is to read the account by Professor It.o, who was one
of the central figures spearheading this development. Professor Kakutani himself was very much interested in the materials
contained in this book, and he thought there may be people around Yale and
elsewhere in the
A couple of years ago, Masatoshi
Fukushima approached me and asked whether I would be interested in having my
old translation (possibly adding a new translation of Part I) published in a
more formal manner, as there are materials in it which had never been published
in English elsewhere and continue to draw interests of the specialists in the
field. I was delighted to hear this proposal with the additional information
that it is the wishes of Professor It.o also to have
a formal publication of an English translation of this book, and would like me
to take up the task of actual translation of the entire book. As I was not sure
whether Professor Kakutani had asked for permission
from Professor It.o to translate the portion of the
book before he told me to take up the task and decided to circulate copies of
the product through the Mathematics Department of Yale, I was very pleased and
honored to learn Professor Itô’s wishes, and decided
to embark on the new translation project with his blessings.
I had thought that I would be able to
finish the project within a year or so, but it took much longer than I had
expected, partly because I decided, in addition to translating Part I, to
retranslate Part II to make the entire manuscript consistent and easier to
read. Lack of my previous experience in writing articles in AMS-Latex format
also forced me to spend considerable amount of extra time. I am truly grateful
to