## Theory Of Markov Processes – Dynkin

In this post, we will see the book Theory Of Markov Processes by E. B. Dynkin. The present book 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 o£ 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 o£ 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 of£ 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,

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 present work is closely allied to a monograph now in the press entitled “Infinitesimal operators of Markov processes” (Infinitezimalnye 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.

The book was translated from Russian by D. E. Brown and edited by T. Kovary. The book was published in 1961.

You can get the book here.

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# Contents

Preface vii

## Chapter 1 – Introduction 1

1. Measurable spaces and measurable sets 1
2. Measures and integrals 7
3. Conditional probabilities and mathematical expectations 10
4. Topological measurable spaces 16
5. The construction of probability measures 2

## Chapter 2 – Markov Processes 25

1. The definition of Markov process 25
2. Stationary Markov processes 35
3. Equivalent Markov processes 42

## Chapter 3 – Subprocesses 53

1. The definition of subprocess. The connexion between subprocesses and multiplicative functionals 53
2. Subprocesses corresponding to admissible subsets, The generation of a part of a process 68
3. Subprocesses corresponding to admissible systems of subsets 73
4. The integral type of multiplicative functionals and the corresponding subprocesses 80
5. Stationary subprocesses of stationary Markov processes 83

## Chapter 4 – The Construction of Markov Processes with Given Transition Functions 96

1. Definition of transition function, Examples 96
2. The construction of Markov processes with given transition function 99
3. Stationary transition functions and the corresponding stationary Markov processes 101

## Chapter 5 – Strictly Markov Processes 103

1. Random variables independent of the future and s-past, Lemmas on measurability 103
2. Definition of strictly Markov process 108
3. Stationary strictly Markov processes 118
4. Weakening the form of the condition for processes continuous from the right to be strictly Markov 124
5. Strictly Markov subprocesses 128
6. Criteria for a process to be strictly Markov 134

## Chapter 6 – Conditions for Boundedness and Continuity of a Markov ProceSS 142

1. Introduction 142
2. Conditions for boundedness 145
3. Conditions for continuity from the right and absence of discontinuities of the second kind 149
4. dJump-type and step processes 159
5. Continuity conditions 161
6. A continuity theorem for strictly Markov processes 167
7. Examples 170

Addendum – A Theorem Regarding the Prolongation of Capacities, and the Properties of Measurability of the Instants of First Departure 174
1. A theorem regarding the extension of capacities 174
2. Measurability theorems for the instants of first departure 183

Supplementary Notes 196
References 202
Alphabetical Index 204
Index of Lemmas and Theorems 207
Index of Notation 209 