© 2013

Understanding Markov Chains

Examples and Applications

  • Easily accessible to both mathematics and non-mathematics majors who are taking an introductory course on Stochastic Processes

  • Filled with numerous exercises to test students' understanding of key concepts

  • A gentle introduction to help students ease into later chapters, also suitable for self-study


Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Nicolas Privault
    Pages 1-6
  3. Nicolas Privault
    Pages 7-36
  4. Nicolas Privault
    Pages 37-60
  5. Nicolas Privault
    Pages 61-75
  6. Nicolas Privault
    Pages 77-94
  7. Nicolas Privault
    Pages 95-116
  8. Nicolas Privault
    Pages 117-128
  9. Nicolas Privault
    Pages 129-148
  10. Nicolas Privault
    Pages 149-166
  11. Nicolas Privault
    Pages 167-209
  12. Nicolas Privault
    Pages 211-223
  13. Nicolas Privault
    Pages 225-239
  14. Nicolas Privault
    Pages 241-245
  15. Back Matter
    Pages 247-354

About this book


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.


Applications of Stochastic Processes Discrete and continuous-time Markov Chains First-step analysis in Markov Chains Gambling Processes and random walks in Markov Chains Highly accessible textbook on Stochastic Processes Introduction to Stochastic Processes Markov Chains self-study Markov Chains textbook Markov Chains textbook with examples Modern textbook on Stochastic Processes Nicolas Privault Stochastic Processes Solved problems in Markov Chains

Authors and affiliations

  1. 1.School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

About the authors

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.

Bibliographic information

Industry Sectors
IT & Software
Finance, Business & Banking
Energy, Utilities & Environment
Oil, Gas & Geosciences


“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)