This book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It first examines in detail two important examples (gambling processes and random walks) before presenting the general theory itself in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 138 exercises and 9 problems with their solutions.

Table of Contents

1 Probability Background

2 Gambling Problems

3 Random Walks

4 Discrete-Time Markov Chains

5 First Step Analysis

6 Classification of States

7 Long-Run Behavior of Markov Chains

8 Branching Processes

9 Continuous-Time Markov Chains

10 Discrete-Time Martingales

11 Spatial Poisson Processes

12 Reliability Theory

About the Author

Nicolas Privault is a 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 MH4514 Financial Mathematics at NTU. The manuscript has been developed over the years from his courses on Stochastic Processes.