I once had a university professor explain to me that European textbooks tend to focus on theory and have few examples, applications, and solutions. The professor further stated that American textbooks tend to be heavy in examples and applications while also providing more detailed solutions. While I am sure there are many counterexamples, this provides a convenient model to apply to this textbook by Frank Beichelt of University of Witwatersrand, South Africa. Of all the formerly colonized African countries perhaps South Africa puts forward the most obvious European imprint, whereas its cultural and latitudinal distance from the Old World suggests it might be predisposed to lean toward a different, maybe even New World textbook model.
This model fits here because the textbook suggests a middle ground. Beichelt gives several examples in a typical chapter, but a reader can expect to take in just as many theorems, examples, and definitions without supporting examples. Of course, this gives less opportunity to explore applications; many types of statistical distribution, for example, are detailed without suggesting applications that have been found for them. The back of the text provides nearly complete coverage on answers to exercises and even guidance via grouping to the sections that provide the relevant theorem and definitions. However, the answers are nearly always tersely numerical making it difficult to be a self-guided tour in as many places as it could. (The rare exceptions are mercifully allotted to the very final chapters.) Take heart, the author did also produce a solution manual (ISBN: 1584886390)!
Regardless of what I am observing about the garnishes and side dishes, the meat of this entrée is present and complete. The goal of this text is a self-contained overview of probability-theoretic topics via proofs, examples, and exercises. The very listing of "Science, Engineering, and Finance" in the title emphasizes that applications to various fields are part of the content. Expect to see this work find a home in the hands of senior undergraduate and graduate students in stochastic processes as well as practitioners and researchers in mathematical finance, operations, industrial engineering, electrical engineering, and actuarial science. All these areas are touched on here, but none are delved into deeply.
The arc of this book starts in a review of basic probability theory and random variables. These preliminaries make a foundation that covers transformation of probability distributions (Laplace, etc.), inequalities in probability theory, limit theorems, and more. The groundwork laid supports forays into random point processes, focusing on Poisson and renewal processes and applications to actuarial analysis. Markov chains are looked at in discrete and continuous time. Martingales are the subject of the shortest chapter (18 pages) and the book closes with a seventh chapter on Brownian motion.
Tom Schulte is working on his own random process at Oakland University, picking a course here and a course there, winding his way toward a Masters Degree in mathematics.
Probability Theory. Basics of Stochastic Processes. Random Point Processes. Markov Chains in Discrete Time. Markov Chains in Continuous Time. Martingales. Brownian Motion.