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This updated third edition of Shiryaev´s work on probability contains a systematic treatment from the ground up, starting with intuitive ideas, then developing more sophisticated subjects. Examples are discussed in detail, and there are a large number of exercises.

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Advanced maths students have been waiting for this, the third edition of a text that deals with one of the fundamentals of their field. This book contains a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks and the Kalman-Bucy filter. Examples are discussed in detail, and there are a large number of exercises. This third edition contains new problems and exercises, new proofs, expanded material on financial mathematics, financial engineering, and mathematical statistics, and a final chapter on the history of probability theory.

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Introduction.- Elementary Probability Theory.- Mathematical Foundations of Probability Theory.- Convergence of Probability Measures. Central Limit Theorem.

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"This book is axiomatic and abstract, and presents a comprehensive but pure-math approach to probability theory. ... There are good exercises throughout the book ... . I think this is a good book if you like this kind of very abstract approach." (Allen Stenger, MAA Reviews, August, 2017)

"This book provides a general introduction to probability theory, and covers several advanced topics. ... Numerous examples and problems help the reader to understand the topics. The book is recommended to master and PhD students in mathematics." (László Viharos, Acta Scientiarum Mathematicarum, Vol. 83 (1-2), 2017)

It is clear that this book contains important and interesting results obtained through a long time period, beginning with the classical Bernoulli´s law of large numbers, and ending with very recent results concerning convergence of martingales and absolute continuity of probability measures. Let us note especially that the great number of ideas, notions and statements in the book are well-motivated, explained in detail and illustrated by suitably chosen examples and a large number of exercises. Thus, the present book is a synthesis of all significant classical ideas and results, and many of the major achievements of modern probability theory. In the whole it is a welcome addition to mathematical literature and can become an indispensable textbook for courses in stochastics.

- J. Stoyanov, Zentralblatt

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