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Macmillan Higher Education Palgrave Higher Education

Understanding Markov Chains

Examples and Applications

ISBN 9789814451505
Publication Date August 2013
Formats Paperback Ebook 
Publisher Springer

This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. A large focus is placed on the first step analysis technique and its applications to average hitting times and ruin probabilities. Classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes, are also covered. Two major examples (gambling processes and random walks) are treated in detail from the beginning, before the general theory itself is presented in the subsequent chapters. 

An introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times is also provided, and the book includes a chapter on spatial Poisson processes with some recent results on moment identities and deviation inequalities for Poisson stochastic integrals. The concepts presented are illustrated by examples and by 72 exercises and their complete solutions.

Nicolas Privault is an associate professor from the Nanyang Technological University (NTU) and is well-established in the field of stochastic processes and a highly respected probabilist. He has authored the book, Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales, Lecture Notes in Mathematics, Springer, 2009 and was a co-editor for the book, Stochastic Analysis with Financial Applications, Progress in Probability, Vol. 65, Springer Basel, 2011. Aside from these two Springer titles, he has authored several others. He is currently teaching the course M27004-Probability Theory and Stochastic Processes at NTU. The manuscript has been developed over the years from his courses on Stochastic Processes.

Introduction

1 Probability Background
1.1 Probability Spaces and Events
1.2 Probability Measures
1.3 Conditional Probabilities and Independence
1.4 Random Variables
1.5 Probability Distributions
1.6 Expectation of a Random Variable
1.7 Conditional Expectation
1.8 Moment and Probability Generating Functions
Exercises

2 Gambling Problems
2.1 Constrained Random Walk
2.2 Ruin Probabilities
2.3 Mean Game Duration
Exercises

3 Random Walk
3.1 Unrestricted Random Walk
3.2 Mean and Variance
3.3 Distribution
3.4 First Return to Zero
Exercises

4 Discrete-Time Markov Chains
4.1 Markov Property
4.2 Transition matrix
4.3 Examples of Markov Chains
4.4 Higher Order Transition Probabilities
4.5 The Two-State Discrete-Time Markov Chain
Exercises

5 First Step Analysis
5.1 Hitting Probabilities
5.2 Mean Hitting and Absorption Times
5.3 First Return Times
5.4 Number of Returns
Exercises

6 Classication of States
6.1 Communicating States
6.2 Recurrent States
6.3 Transient States
6.4 Positive and Null Recurrence
6.5 Periodicity and Aperiodicity
Exercises

7 Long-Run Behavior of Markov Chains
7.1 Limiting Distributions
7.2 Stationary Distributions
7.3 Markov Chain Monte Carlo
Exercises

8 Branching Processes
8.1 Defnition and Examples
8.2 Probability Generating Functions
8.3 Extinction Probabilities
Exercises

9 Continuous-Time Markov Chains
9.1 The Poisson Process
9.2 Continuous-Time Chains
9.3 Transition Semigroup9.4 Infinitesimal Generator
9.5 The Two-State Continuous-Time Markov Chain
9.6 Limiting and Stationary Distributions
9.7 The Discrete-Time Embedded Chain
9.8 Mean Absorption Time and Probabilities
Exercises

10 Discrete-Time Martingales
10.1 Filtrations and Conditional Expectations
10.2 Martingales - Definition and Properties
10.3 Ruin Probabilities
10.4 Mean Game Duration
Exercises

11 Spatial Poisson Processes
11.1 Spatial Poisson (1781-1840) Processes
11.2 Poisson Stochastic Integrals
11.3 Transformations of Poisson Measures
11.4 Moments of Poisson Stochastic Integrals
11.5 Deviation Inequalities
Exercises

12 Reliability Theory
12.1 Survival Probabilities
12.2 Poisson Process with Time-Dependent
Intensity
12.3 Mean Time to Failure
Exercises

Some Useful Identities
Solutions to the Exercises
References
Index

Reviews

“This textbook provides an elementary introduction to the classical theory of discrete and continuous time Markov chains motivated by gambling problems and covers a variety of primers on different topics … . this text may serve very well for a first undergraduate course on Markov chains for applied mathematicians, but also for students of financial engineering. It is completed by almost a hundred pages of solutions of exercises.” (Michael Högele, zbMATH 1305.60003, 2015)“The book provides an introduction to discrete and continuous-time Markov chains and their applications. … The explanation is detailed and clear. Often the reader is guided through the less trivial concepts by means of appropriate examples and additional comments, including diagrams and graphs. Also, a big plus is the presence of numerous well-chosen exercises at the end of each chapter, which are discussed in a separate ‘Solutions to the Exercises’ part at the end of the book.” (Michele Zito, Mathematical Reviews, December, 2014)
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