Semimartingale Theory and Stochastic Calculus

Rating:
Summary: a readable book on martingale theory of Strasbourg school
Review: The main contents of this book can be found in other more famous monographs: Probabilities and Potential, Vol. 1 and 2, by Dellacherie/Meyer; Limit Theorems for Stochastic Processes, by Jacod/Shiryaev. Unfortunately, these monographs are very hard to read. So the main merit of He-Wang-Yan can be said: it's suitable for self-study and accessible to any persistent graduate student with certain mathematical maturity.
1. It contains the essence of Dellacherie/Meyer, but debugs/simplifies many proofs.
2. It gives a quick and clear presentation of Dellacherie's capacity theory, with application to section theorems.
3. This book is supplemented and further developed in the form of
problems. Some of the problems are useful results and some of them are difficult.

Contents: 1. Preliminaries. 2. Classical martingale theory. 3. Processes and stopping times. 4. Section theorems and their applications. 5. Projections of processes. 6. Martingales with integrable variation and square integrable martingales. 7. Local martingales. 8. Semimartingales and quasimartingales. 9. Stochastic integrals. 10. Martingale spaces H1 and BMO. 11. The characteristics of semimartingales. 12. Changes of measures. 13. Predictable representation property. 14. Absolute continuity and contiguity of measures. 15.Weak convergence for cadlag processes. 16.Weak convergence for semimartingales.

The first ten chapters (about 400 pages) are pretty easy to read. From Chapter 11 on, things get dramatically complicated as the authors try to work under the most general framework. For Girsanov's theorem, local absolute continuity is considered, and Jacod's random measures become the common language. These stuffs seem too complicated to a beginner. Other books should be consulted to see the most useful forms of these theorems. For example, I'm more happy with Protter's presentation of PRP (predictable presentation property), in the second edition of his book on SDE.

As to the last three chapters, I didn't really read them, but chapter 15 seems very nice and quite self-contained.

For the follow-up reading of He-Wang-Yan, I would recommend Revuz/Yor: Continuous Martingales and Brownian Motion, and Vol.2 of Rogers/Williams: Diffusions, Martingales and Markov Processes. They will show the applications, as well as intuitions, which are often obscured by the heavy machineries invented to takcle the most general cases.

A final remark: Frank Knight gave an interesting review of the book by Revuz/Yor, which is worth of looking. The review can be accessed at MathSciNet.