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Probability and stochastics. (English) Zbl 1226.60001

Graduate Texts in Mathematics 261. New York, NY: Springer (ISBN 978-0-387-87858-4/hbk; 978-0-387-87859-1/ebook). xiii, 557 p. (2011).
If one publishes a textbook on basic probability theory in 2011, one needs a compelling reason why it should stand out from the bulk of textbooks that are already available. For the book at hand three such arguments could be given, namely, its completeness, its precise style, and the fact that it puts a strong accent on stochastic processes.
In nine chapters, the author goes from basic measure theory to fine properties of stochastic processes. It is remarkable that unlike in many other introductory probability books, the chapter on measure theory is very detailed and most results are rigorously proven. Measure theory is not given in the appendix, but as the first chapter of the book. The author does not treat this subject as background information, but dedicates the same attention to detail to this chapter as to the other eight chapters. He takes the time to give intuition and exercises. This makes this textbook a rewarding and capturing read from the start.
The next three chapters cover a one-semester course on basic probability for students already familiar with measure theory. (Otherwise, the first four chapters would be an appropriate choice.)
While the first chapter introduced all the measure theoretic concepts, it does not give any probabilistic intuition. This is done in Chapter II, where all the terms from the first chapter are explained heuristically. Of course the exposition is not limited to this, also many new concepts (such as independence, almost sureness, filtrations, distributions, uniform integrability) are introduced.
The third chapter treats stochastic limit theorems. To do so, first the different notions of convergence are carefully introduced, explained, and related to each other. The strong and the weak limit theorem are proven, as well as Kolmogorov’s three series theorem. The section on central limit theorems is very detailed. Apart from the standard central limit theorem, also Liapunov’s theorem and Lindeberg’s theorem are proven. The Feller-Lévy theorem is stated but not proven (this is one of very few statements in the monograph that are not proven). There are also results on convergence to Poisson and infinitely divisible distributions.
The fourth chapter is on conditional expectation. The author first treats an example where the filtration that he conditions on is countable. After having calculated the “best guess” in this setting, he examines its properties and uses that to define conditional expectations. Regular conditional probabilities are introduced and their existence on standard measurable spaces is proven; he proceeds similarly for disintegration. The end of the chapter gives the classical extension theorems for stochastic processes: the Ionescu-Tulcea theorem and Kolmogorov’s extension theorem.
Chapter five is on the classical theory of martingales in discrete time and in continuous time. The author introduces a new notion, that of a “Doob martingale”: \(X\) is a Doob martingale if it is a martingale and if the martingale property extends to general (i.e., unbounded) stopping times.
Chapter six is a very extensive (70 pages) introduction to Poisson random measures.
The chapters seven, eight and nine are on Lévy processes, Brownian motion, and Markov processes, respectively. They are carefully characterized and their path properties are examined. The prototype of a Markov process in the last chapter is a (jump-) diffusion. Nonetheless, this is not a book on stochastic calculus: The Itô integral is only treated in the appendix, and there the proofs are omitted. But this does not harm the treatment of diffusions in the least. The focus is on the different types of behavior that a Markov process can have, and on describing the process in a concise way (transition function, generator, resolvent, Kolmogorov’s equations). There is also a small section on potential theory.
The chapters five to eight could serve as the basis for a one-semester course in stochastic processes.
Every chapter is divided in 5 to 8 sections, and at the end of most sections there are exercises.
One pleasant aspect about the monograph is that the author often repeats the necessary information from previous chapters when he picks up a subject again. As stated in the introduction, “I do not assume that the reader will go through the book line by line from the beginning to the end. Some things are re-called or re-introduced when they are needed.”
Overall, the monograph is very complete in the sense that the chosen subjects are treated in great detail, and nearly no proofs are omitted. Also, the amount of material covered in 540 pages is impressive.
The style is very precise and mathematically rigorous, which makes the book a pleasure to read.
So, in conclusion, this is a valuable addition to the family of probability textbooks. Because of its completeness, it is also a good reference for researchers.

MSC:

60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60Axx Foundations of probability theory
60Gxx Stochastic processes
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