Liptser, R. Sh.
Shiryayev, A. N.
|Title||Theory of Martingales|
|Publisher||Kluwer Academic Publishers|
|Year of publication||1989|
|Reviewed by||Petr Lachout|
The volume of Kluwer Academic Publishers is an impressive monograph summarizing present knowledge on the martingale theory. Actually, it is an enlarged and revized translation from the Russian original, Nauka Publishers, Moscow, 1986. The authors are top experts in the field, which guarantees the high level of presentation. This can be verified by a look at the table of contents.
The book is divided into three parts, which are further subdivided into chapters in order to help readers' orientation. Part I sums up the basic definitions and properties of semimartingales. Local martingales, Doob-Meyer decomposition as well as canonical representation of semimartingales and compensator of a Radon measure are introduced and discussed there. Part II is devoted to weak convergence in the space D supported by convergence of finite-dimensional distributions. Some special cases are mentioned there; e.g. convergence of semimartingales to a process with conditionally independent increments or to a semimartingale. The last section presents the invariance principle and diffusion approximation for models generated by a stationary process or by a semimartingale.
I believe that this exciting book, together with the book of Jacod and Shiryayev "Limit Theorems of Stochastic Processes", provides the standard survey of the recent martingale theory. Therefore, I do recommend the volume as a valuable contribution both to interested individuals and to scientific libraries.