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.

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

1. The Definition of Markov Process

2. Stationary Markov Processes

3. Equivalent Markov Processes

Chapter 3 - Subprocesses

1. The Definition of Subprocess. The Connexion Between Subprocesses and Multiplicative functionals

2. Subprocesses Corresponding to Admissible Subsets. The Generation of a Part of a Process

3. Subprocesses Corresponding to Admissible Systems of Subsets

4. The Integral Type of Multiplicative Functionals and the Corresponding Subprocesses

5. Stationary Subprocesses of Stationary Markov Processes

Chapter 4 - The Construction of Markov Processes with Given Transition Functions

1. Definition of Transition Function. Examples

2. The Construction of Markov Processes with Given Transition Function

3. Stationary Transition Functions and the Corresponding Stationary Markov Processes

Chapter 5 - Strictly Markov Processes

1. Random Variables Independent of the Future and S-Past. Lemmas on Measurability

2. Definition of Strictly Markov Process

3. Stationary Strictly Markov Processes

4. Weakening the Form of the Condition for Processes continuous from the Right to be Strictly Markov

5. Strictly Markov Subprocesses

6. Criteria for a process to be Strictly Markov

Chapter 6 - Conditions for Boundedness and Continuity of a Markov Process

1. Introduction

2. Conditions for Boundedness

3. Conditions for Continuity from the Right and Absence of Discontinuities of the Second Kind

4. Jump-Type and Step Processes

5. Continuity Conditions

6. A continuity Theorem for Strictly Markov Processes

7. Examples

Addendum - A Theorem Regarding the Prolongation of Capacities, and the Properties of Measurability of the Instants of First Departure

1. A Theorem Regarding the Extension of Capacities

2. Measurability Theorems for the Instants of First Departure

Supplementary Notes

References

Alphabetical Index

Index of Lemmas and Theorems

Index of Notation