Theory of Markov Processes

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.

Inhalt

Preface
Chapter 1 - Introduction

1. Measurable Spaces and Measurable Sets
2. Measures and Integrals
3. Conditional Probabilities and Mathematical Expectations
4. Topological Measurable Spaces
5. The Construction of Probability Measures
   Chapter 2 - Markov Processes
6. The Definition of Markov Process
7. Stationary Markov Processes
8. Equivalent Markov Processes
   Chapter 3 - Subprocesses
9. The Definition of Subprocess. The Connexion Between Subprocesses and Multiplicative functionals
10. Subprocesses Corresponding to Admissible Subsets. The Generation of a Part of a Process
11. Subprocesses Corresponding to Admissible Systems of Subsets
12. The Integral Type of Multiplicative Functionals and the Corresponding Subprocesses
13. Stationary Subprocesses of Stationary Markov Processes
    Chapter 4 - The Construction of Markov Processes with Given Transition Functions
14. Definition of Transition Function. Examples
15. The Construction of Markov Processes with Given Transition Function
16. Stationary Transition Functions and the Corresponding Stationary Markov Processes
    Chapter 5 - Strictly Markov Processes
17. Random Variables Independent of the Future and S-Past. Lemmas on Measurability
18. Definition of Strictly Markov Process
19. Stationary Strictly Markov Processes
20. Weakening the Form of the Condition for Processes continuous from the Right to be Strictly Markov
21. Strictly Markov Subprocesses
22. Criteria for a process to be Strictly Markov
    Chapter 6 - Conditions for Boundedness and Continuity of a Markov Process
23. Introduction
24. Conditions for Boundedness
25. Conditions for Continuity from the Right and Absence of Discontinuities of the Second Kind
26. Jump-Type and Step Processes
27. Continuity Conditions
28. A continuity Theorem for Strictly Markov Processes
29. Examples
    Addendum - A Theorem Regarding the Prolongation of Capacities, and the Properties of Measurability of the Instants of First Departure
30. A Theorem Regarding the Extension of Capacities
31. Measurability Theorems for the Instants of First Departure
    Supplementary Notes
    References
    Alphabetical Index
    Index of Lemmas and Theorems
    Index of Notation