Theory of Markov Processes
By E. B. Dynkin
()
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Starting with a brief survey of relevant concepts and theorems from measure theory, the text investigates operations that permit an inspection of the class of Markov processes corresponding to a given transition function. It advances to the more complicated operations of generating a subprocess, followed by examinations of the construction of Markov processes with given transition functions, the concept of a strictly "Markov process," and the conditions required for boundedness and continuity of a Markov process. Addenda, notes, references, and indexes supplement the text.
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Theory of Markov Processes - E. B. Dynkin
Leach
PREFACE
The present hook aims at investigating the logical foundations of the theory of Markov random processes.
The theory of Markov processes has developed rapidly in recent years. The properties of the trajectories of such processes and their infinitesimal operators have been studied, and intimate connexions have been discovered between the behaviour of the trajectories and the properties of the differential equations corresponding to the process. These connexions are useful for studying differential equations as well as Markov processes. The material thus accumulated has made necessary a critical survey of the fundamentals of the theory. In particular, the usual statement of the Markov principle of absence of after-effects
has been found to be inadequate and various authors have proposed different forms for a strengthened principle whereby a process is strictly Markov.
It has become obvious that the most natural subject for study is presented by Markov processes cut off at a random instant. All these and other ideas were originally introduced by different authors in different forms, according to the specific purposes of their specialized works - in which stationary Markov processes are considered almost exclusively.
A general theory is built up in the present book which also covers non-stationary processes. Stationary processes are . regarded as an important special case. Non-stationary processes are well known to be reducible to the stationary type by an artificial method requiring the passage to a more complicated phase space*). However, the stationary processes thus obtained are in a certain sense degenerate, so that this type of reduction is by no means always suitable. Then again, a concept of the Markov process that is more general in essence is closer to first principles than the concept of stationary Markov process. There is a canonical time scale for stationary Markov processes. In the general process there is no such scale and all the definitions have to be invariant with respect to any monotonic continuous transformation of time.
The theory cannot be adequately developed by extending the concept of a Markov process as a random function of a special type.
For we are usually concerned, when studying Markov processes, not with a single probability measure but with a whole collection of such measures, corresponding to all the possible initial instants and all the possible initial states; in other wos, we are concerned, not with one random function, but with a whole collection of such functions, with definite inter-relationships. This is one of the reasons why the theory of Markov processes has to possess its familiar autonomy with respect to the general theory of probability processes. The theory of Markov processes is built up in the present book without any reference whatever to the general theory of probability processes.
This book cannot be used by the student to make his first acquaintance with the theory of Markov processes. Although we have not assumed formally any previous acquaintance with the theory of probability, in fact a reading of the book can only prove of value to someone already acquainted with an elementary exposition of the theory of Markov processes, such as is contained, for instance, in Feller’s Introduction to probability theory and its applications,
Vol. 1 (Vvedenie v teoriyu veroyatnostei i ee prilozheniya), or Gnedenko’s Course of probability theory
(Kurs teorii veroyatnostei).
The first introductory chapter contains a brief survey of the necessary concepts and theorems from measure theory. Any proofs that can be found in text-books are omitted here. The second chapter gives a general definition of Markov process and investigates the operations that make possible an inspection of the class of Markov processes corresponding to a given transition function. The more complicated operation of generating a subprocess is studied in Chapter 3. The connexion is revealed between the subprocesses of a Markov process and the multiplicative functionals of its trajectory. The most important classes of multiplicative functionals and subprocesses are investigated. Chapter 4 is concerned with the construction of Markov processes with given transition functions. The concept of strictly Markov process is discussed in Chapter 5. Finally, Chapter 6 is devoted to a study of the conditions to he imposed on the transition function so that among the Markov processes corresponding to this function, there should he at "least one, all the trajectories of which possess some type of continuity or bounded-ness. The supplement describes some of Choquet’s results concerning the general theory of capacities. Measurability theorems for the instants of first departure are deduced from these results. A historical and bibliographical index will be found at the end of the book.
The present work is closely allied to a monograph now in the press entitled Infinitesimal operators of Markov processes
(inf initezimalnye operatory markovskikh protsessov), which is devoted to the task of classifying Markov processes. The two works should be regarded as the two parts of a single monograph on the theory of Markov processes.
The present material comes from a series of papers and special courses given by the author at Moscow and Pekin universities. The author is grateful to his audience for a number of observations which he made use of during the final preparation of the manuscript.
I must express my indebtedness and sincere gratitude to Mr. A.A. Yushkevich for his careful reading of the manuscript and various comments that made it possible to eliminate a number of inaccuracies and obscurities.
E.B. Dynkin
*See article 3 of Chapter 4.
THEORY OF
MARKOV PROCESSES
CHAPTER 1
INTRODUCTION
1. Measurable Spaces and Measurable Sets
satisfying the following conditions:
1.1.A1. If A*).
1.1.A2. If A
).
, the aggregate of all sets A].
a π-system if:
1.1.B1. It follows from A1, A2∈ that A1∩ A2∈ **).
a λ-system if it satisfies the following conditions:
.
1.1.C2. If A1, A2∈ and A1∩ A2 = ø.
1.1.C3. If A1, A2∈ and. A1⊇A2, then A1\A2∈ .
1.1.C4. If A1, …,An, … ∈ and An↑A.
is simultaneously a Π-system and a λ-system, it is a σ-algebra. For 1.1.A1 follows from 1.1.C1 and 1.1.C3. Moreover, it follows from the relationship A∪B = A∪(B\AB)and properties 1.1.B1, 1.1.C3 and 1.1.C2 that, if A, then A∪B = A∪(B\AB)∈ , and consequently, if A1, A2, …, An. Now let An(n by 1.1.C.4. It follows from the relationship
Condition 1.1.A2 is therefore satisfied.
).
is obviously a λ-system. We shall show that this intersection is at the same time a Π-system. The assertion of the lemma will follow from this.
1 of all the sets A such that A∩B∈ for all B∈ This means that, if A∈ , B∈ , then A∩B∈ .
for all Ais a Π-system.
1.2. is called a measurable space.
. The values of s and t can be infinite as well as finite, and we put
.
.
where Cn).
).
:
Since the intervals (t-measurable is simply that, for any t,
.
, satisfying the condition
and
.
-system if the following conditions are fulfilled:
.
.
contains the characteristic).
by lemma 1.1.
). we put
and, in accordance with .
In view to .
1.3. (i=1, 2, …, nfor the σ-algebra generated by subsets of the form A1× … ×An, where A(it may be noticed that the sets A1×…×An , we shall write Ωn .
×… for the σ-algebra in this space generated by the subsets
∞ for brevity. It may be noticed that the class of subsets of type (1.1) is a π-system.
) (i can run either through the values 1,2,…,n×…), defined by the expression
,
is measurable.
Proof. Suppose for clarity that i , clearly forms a σ-algebra. This σ-algebra contains all the sets of: type