Dynkin / Köváry Theory of Markov Processes

1. Auflage 2014
ISBN: 978-1-4832-2610-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 220 Seiten, Web PDF

Theory of Markov Processes provides information pertinent to the logical foundations of the theory of Markov random processes. This book discusses the properties of the trajectories of Markov processes and their infinitesimal operators. Organized into six chapters, this book begins with an overview of the necessary concepts and theorems from measure theory. This text then provides 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. Other chapters consider the more complicated operation of generating a subprocess. This book discusses as well the construction of Markov processes with given transition functions. The final chapter deals with the conditions to be imposed on the transition function so that among the Markov processes corresponding to this function, there should be at least one. This book is a valuable resource for mathematicians, students, and research workers.

Autoren/Hrsg.

Dynkin, E. B.

Köváry, T.

Inhaltsverzeichnis

1;Front Cover;1
2;Theory of Markov Processes;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;8
6;Chapter 1. Introduction;12
6.1;1. Measurable spaces and measurable sets;12
6.2;2. Measures and integrals;18
6.3;3. Conditional probabilities and mathematical expectations;21
6.4;4. Topological measurable spaces;27
6.5;5. The construction of probability measures;33
7;Chapter 2. Markov Processes;36
7.1;1. The definition of Markov process;36
7.2;2. Stationary Markov processes;46
7.3;3. Equivalent Markov processes;53
8;Chapter 3. Subprocesses;64
8.1;1. The definition of subprocess. The connexion between subprocesses and multiplicative functionals;64
8.2;2. Subprocesses corresponding to admissible subsets. The generation of a part of a process;79
8.3;3. Subprocesses corresponding to admissible systems of subsets;84
8.4;4. The integral type of multiplicative functionals and the corresponding subprocesses;91
8.5;5. Stationary subprocesses of stationary Markov processes;94
9;Chapter 4. The Construction of Markov Processes with Given Transition Functions;107
9.1;1. Definition of transition function. Examples;107
9.2;2. The construction of Markov processes with given transition function;110
9.3;3. Stationary transition functions and the corresponding stationary Markov processes;112
10;Chapter 5. Strictly Markov Processes;114
10.1;1. Random variables independent of the future and s-past. Lemmas on measurability;114
10.2;2. Definition of strictly Markov process;119
10.3;3. Stationary strictly Markov processes;129
10.4;4. Weakening the form of the condition for processes continuous from the right to be strictly Markov;135
10.5;5. Strictly Markov subprocesses;139
10.6;6. Criteria for a process to be strictly Markov;145
11;Chapter 6. Conditions for Boundedness and Continuity of a Markov Process;153
11.1;1. Introduction;153
11.2;2. Conditions for boundedness;156
11.3;3. Conditions for continuity from the right and absence of discontinuities of the second kind;160
11.4;4. Jump-type and step processes;170
11.5;5. Continuity conditions;172
11.6;6. A continuity theorem for strictly Markov processes;178
11.7;7. Examples;181
12;Addendum - A Theorem Regarding the Prolongation of Capacities, and the Properties of Measurability of the Instants of First Departure;185
12.1;1. A theorem regarding the extension of capacities;185
12.2;2. Measurability theorems for the instants of first departure;194
13;Supplementary Notes;207
14;References;213
15;Alphabetical Index;215
16;Index of Lemmas and Theorems;218
17;Index of Notation;220