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Probability with martingales. (English) Zbl 0722.60001
Cambridge etc.: Cambridge University Press. xv, 251 p. £30.00/hbk; $59.50 /hbk; £10.95/pbk;$ 24.95 /pbk (1991).

The author demonstrates a new way of teaching basic probability theory which is at least for the German habits unusual. Normally the student will be introduced to this field first by an intuitive and historical oriented approach. In a second run finally he will be confronted with the more abstract concepts. This proceeding is timeconsuming and does not satisfy the demands of a modern teaching of probability theory. Williams offers a way out of this dilemma. He demonstrates elegantly and convincingly the “natural” relationship between “reality” and the modern means trying to conceptualize it. He stresses especially the martingale concept as a “natural one” which provides easy access to the mainpoints of probability theory. He shows what is the heart of the martingale idea and he combines this vision with a very clear intuitive perception. He visualizes the “simplicity” of ideas which normally disappears behind an impressive formalism. His philosophy seems to be: “If you are tought the simple truth first you will be able to use the machinery builded around it without being intimidated.”

Before starting to demonstrate the power and simplicity of martingales the author has to introduce the “Foundations”. This is done in Part A. There he gives a short and thorough introduction to the basics of measure theory, the concept of product measure, basic probability concepts like “event”, “random variable”, “independence” and “expectation”. Williams supplies good examples to stress the necessity and reasonableness of the use of the modern formalism even in a first encounter with probability. In Part B the author introduces right away the martingale concept. Reading the first two chapters of this part makes you wonder “Why to be afraid of conditional expectation, why to be afraid of modern stochastic integration?” And after his bold introduction of the basic concepts it does not seem to ambitious at the end of Chapter 10 to calculate hitting times for the random walk and to speak about superharmonic functions for Markov chains. After the reader’s initiation to martingale theory the author shows how nice and easy one can prove the martingale convergence theorem. By now there is collected so much power that one can attack many problems of classical probability theory.

The theory of L2 martingales is the first field to apply the new skills: Kolmogorov’s three series theorem, Kolmogorov’s strong law of large numbers. The author introduces the Doob-Meyer decomposition as well as the quadratic process of an L2 martingale. Having introduced the uniform integrability he gives a breathtaking insight in what happens when uniform integrability is combined with the martingale property. He proves Lévy’s “upward” and “downward” theorem, Kolmogorov’s 1-0 law, the strong law of large numbers, Doob’s submartingale inequality, Kakutani’s theorem on product form martingales and its relevance for LR tests. The author demonstrates exponential bounds, a special case of the law of iterated logarithm, he proves the Radon-Nikodym theorem and Doob’s optional sampling theorem for UI martingales. Chapter 15 finishes this first visit in the enchanted world of martingales by presenting applications in option pricing, control theory and filtering. Williams arranges a first rendezvous with the Black-Sholes formula and the Kalman- Bucy filter.

Part C of this book is concerned with characteristic functions and the proof of the central limit theorem. Surprisingly Part C is totally martingale free.

It follows a set of appendices where the measure-theoretic results are proved in full and in a way accessible for a newcomer in probability theory. At the end of the book one finds “Chapter E”, “the most important chapter in this book... I have left the interesting things for you to do”. Williams’ book sharpens the student’s intuition and offers an excellent teaching experience.

##### MSC:
 60-02 Research monographs (probability theory) 60Fxx Limit theorems (probability) 60Gxx Stochastic processes 60Jxx Markov processes