An Introduction to Continuous-Time Stochastic Processes

This book serves as a rigorous introduction to the theory of continuous-time stochastic processes. Since the subject of stochastic processes is so vast, the authors have chosen a particular subset of the theory to focus on, with particular attention to applications, especially in finance, biology, and medicine.

 

The audience for this particular book ranges from graduate students or undergraduate students with an advanced background in stochastic analysis or something similar to other mathematicians and scientists working in a different area. As such a mathematician, I found the authors’ choice to skip proofs to avoid distractions (yet provide concrete references) quite reasonable. Proofs are given when they serve a purpose. Additionally, plenty of references are provided. This book might be tough, however, for undergraduate students, even with a background in probability or basic stochastic analysis. Indeed, the first chapter gives a brief rundown of the fundamentals of probability, assuming the reader is comfortable with the essentials of measure theory. That being said, this book could certainly serve as a textbook, say, for a special topics course in statistical methods. 

 

The authors break up the book into two main parts. The first part is devoted to an introduction to the theories of stochastic processes, and as the title suggests, the focus is on continuous-time processes. The second part is devoted to applications. The authors do a good job of demonstrating the applicability of the theories to a wide range of areas. Exercises are provided at the end of each chapter; the difficulty ranges from basic applications to more advanced ideas which introduce new applications or processes.

 

Overall this book is a nice way to get into the basics of stochastic processes for someone working in a different field. It is quite reasonable that this could serve as either a main textbook or secondary reference for a graduate course. Sufficient details on each topic are provided by the authors, which makes this possible.

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Eric Stachura is currently an Assistant Professor of Mathematics at Kennesaw State University. He is generally interested in analysis and partial differential equations.