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Introduction to stochastic calculus. (English) Zbl 1434.60003
Indian Statistical Institute Series. Singapore: Springer (ISBN 978-981-10-8317-4/hbk; 978-981-10-8318-1/ebook). xiii, 441 p. (2018).
This book is a volume of the Indian Statistical Institute Series published by Springer. It is written by well-known scientists and contributors in stochastic calculus and its applications. It is remarkable to see that the book is dedicated to the memory of the late Professor Gopinath Kallianpur (1925–2015).
Based on measure-theoretic probability, the authors have written a comprehensive course on stochastic calculus. Let us list the chapter names (some key words and/or phrases are given in brackets):
1. Discrete parameter martingales (filtration, Doob’s maximal inequality).
2. Continuous-time processes (martingales and stopping times).
3. The Itô’s Integral (Brownian motion, quadratic variation, multidimensional Itô’s integral).
4. Stochastic Integration (quadratic variation, stochastic integrators).
5. Semimartingales (semimartingales, Dellacherie-Meyer-Mokobodzky-Bichteler theorem)
6. Pathwise formula for the stochastic integral (stochastic integral).
7. Continuous semimartingales (random time change, weak solution of SDE).
8. Predictable increasing processes (predictable stopping time, Doob-Meyer decomposition).
9. The Davis inequality (Burkholder-Davis-Gundy inequality, sigma-martingale).
10. Integral representation of martingales (multidimensional semimartingale, integral representation, connection to mathematical finance).
11. Dominating process of a semimartingale (optimization result, Metivier-Pellamail inequality, Emery topology).
12. SDE Driven by r.c.l.l. semimartingales (right-continuous, left-hand-limit, stochastic differential equation, Euler-Peano approximation).
13. Girsanov theorem (Cameron-Martin formula, Girsanov theorem, Girsanov-Meyer theorem). There is a Bibliography and Index.
The material included into this book is well chosen and well organized. Precise definitions are given to all notions followed by a series of important results (theorems, lemmas, corollaries). Complete proofs are provided for almost all statements. The style is compact and clear. The presentation is well complemented by a large number of useful remarks and exercises.
Graduate students attending university courses in modern probability theory and its applications can benefit a lot from working with this book. There are good reasons to expect that the book will be met positively by students, university teachers and young researchers.

MSC:
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G48 Generalizations of martingales
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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