The first edition of the book was published quite long ago, in 1982. For a review of the first edition see [Zbl 0503.60062]. For three decades, the original English edition and its Russian translation became one of the most useful and most cited books in stochastic analysis and its diverse applications.
It is important to mention that Professor Elliott was and is one of the main contributors in this research area by publishing a long list of first class papers, some of them written with co-authors. This second edition is more than twice as large as the first one and it has been prepared by two authors, Professor Cohen and Professor Elliott. It is more than curious to put the two editions together and compare their contents. We easily see that there has been a huge progress in both theory and applications over the last 30 years. Interestingly, some of the developments were motivated by questions coming from the popular field of “quantitative finance”, or as this reviewer prefers to say, “stochastic financial modeling”. Most of the significant old and recent achievements are included in this edition. The presentation is both rigorous and clear. The authors pay a lot of attention to the motivational aspects by explaining what-how-why.
Completely new topics are included. Among them, there are a chapter on basic measure theory and one on an important recent topic, namely “backward stochastic differential equations”. All chapters from the first edition are entirely rewritten and essentially extended by providing new results and new, complete and transparent proofs. Many examples in the text and exercises at the end of each chapter are added. Remarkably, some of them are in fact excellent counterexamples.
Here are the names of the chapters: 1. Measure and integral. 2.Probabilities and expectation. 3. Filtrations, stopping times and stochastic processes. 4. Martingales in discrete time. 5. Martingales in continuous time. 6. The classification of stopping times. 7. The progressive, optional and predictable \(\sigma\)-algebras. 8. Processes of finite variation. 9. The Doob-Meyer decomposition. 10. The structure of square integrable martingales. 11. Quadratic variation and semimartingales. 12. The stochastic integral. 13. Random measures. 14. Itô’s differential rule. 15. The exponential formula and Girsanov’s theorem. 16. Lipschitz stochastic differential equations. 17. Markov properties of SDEs. 18. Weak solutions of SDEs. 19. Backward SDEs. 20. Control of a single jump. 21. Optimal control of drifts and jump rates. 22. Filtering.
There are also ten appendices (all are new) and a short chapter “Spaces of càdlàg adapted processes”. The book ends with a complete and representative list of references, notations and abbreviations, and an index.
This is a fundamental book in modern stochastic calculus and its applications: rich contents, well structured material, comprehensive coverage of all significant results given with complete proofs and well illustrated by examples, carefully written text.
Hence, there are more than enough reasons to strongly recommend the book to a wide audience. Among them, there are good and motivated graduate university students. Different portions of the material can be successfully combined and used by university teachers for a variety of special advanced courses in stochastics and its applications. Also, the book is an excellent reference book.
It can be predicted that this edition of the book will “live” at least as long as the first edition.