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Foundations of modern probability. (English) Zbl 0892.60001
Probability and Its Applications. New York, NY: Springer. xii, 523 p. (1997).
The monograph is a comprehensive treatment of large parts of probability theory from the basic theory to advanced subjects – mainly in stochastic analysis. There are chapters on elements of measure theory (including the construction of Lebesgue’s integral), convergence concepts for r.v.’s, processes and measures, characteristic functions, conditional expectations and disintegration, renewal theory (on \(\mathbb{R}\)) and random walks, (subadditive) ergodic theorems, point processes, Skorokhod embedding and invariance principles, Feller processes and semigroups. A substantial part of the book is devoted to stochastic analysis from martingale theory to the general theory of stochastic integration w.r.t. (possibly) discontinuous semimartingales and the Bichteler-Dellacherie theorem. Further, stochastic differential equations (strong and weak solutions) with a special chapter on one-dimensional equations, Ray-Knight theorems, additive functionals and the connection to PDEs and potential theory are treated. All results (with the exception of a few theorems in the appendix such as the projection theorem) are presented with proofs. The book is very carefully written. Occasionally new and shorter and/or more elegant proofs are provided even for classical results. Topics which are not covered (and which the author regrets not to have covered) are large deviations, Gibbs measures, stochastic differential geometry, Malliavin calculus and branching and superprocesses.
It is truly surprising how much material the author has managed to cover in the book. Of course a certain price had to be paid: proofs are often very concise and examples are not numerous. Therefore beginners in probability theory may find the book too hard to follow. On the other hand more advanced readers are likely to regard the book as an ideal reference. Indeed the monograph has the potential to become a (possibly even “the”) major reference book on large parts of probability theory for the next decade or more.

MSC:
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60Hxx Stochastic analysis
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