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Theory of martingales. Transl. from the Russian by K. Dzjaparidze. (English) Zbl 0728.60048
[The Russian original appeared in 1986 (Zbl 0654.60035).]
The purpose of this voluminous monograph is to present the theory of continuous parameter real semimartingales in full/generality and then to study the weak convergence of a sequence of these semimartingales to a limiting semimartingale.
More specifically, in the first part of the book (consisting of 360 pages, divided into 4 chapters) the authors deal with real-valued stochastic processes $$X=(X_ t)_{t\geq 0}$$ defined on a filtered probability space which are semimartingales in the sense of being decomposable into a sum of a local martingale and a process with finite variation on bounded intervals. The first chapter sets down the basic facts and definitions from the theory of martingales and the general theory of stochastic processes: local martingales, square-integrable martingales, the Doob-Meyer decomposition, classification of stopping times, section and projection theorems. Some of the easier proofs here are given briefly and for the rest appropriate references are cited. A reader not already thoroughly acquainted with the material in Chapter 1 would have difficulty with the rest of the book. In Chapter 2, the stochastic integral H.X of a predictable process H with respect to a semimartingale X is defined and one standard version of Itô’s formula (for $$f(X^ 1,...,X^ d)$$, f twice differentiable, $$(X^ i)_{1\leq i\leq d}$$ d real semimartingales) is stated (but not proven). This leads to the introduction of the stochastic exponential Y of a semimartingale X defined as the solution of the (Doléans) equation $$Y=1+Y_ -\cdot X$$ (i.e. in the differential notation: $$dY_ t=Y_{t-}dX_ t$$, $$Y_ 0=1)$$. This stochastic exponential serves as an important tool later, in the study of the weak convergence of a sequence of semimartingales $$X^ n$$ to a semimartingale X - somewhat like the use of characteristic functions in the classical elementary theory of probability. The last two chapters of the first part lead to the introduction of a “triplet of predictable characteristics” associated with a real semimartingale X; these have to do with the jump discontinuities of X and the quadratic variation of its continuous martingale part. It turns out that if X is a process with stationary independent increments, then the associated triplet is non-random and is related to the familiar Lévy measure and the Brownian motion part of the process.
The stage is now set to study $$X^ n\to X$$ in the sense of weak convergence of the distributions of the semimartingales $$X^ n$$ in the Skorokhod space D; the case where X is a process with independent or conditionally independent increments is investigated in special detail. The theorems concerning $$X^ n\to X$$ occupy the second part (consisting of 4 chapters and about 300 pages) of the book; these constitute a far- reaching generalization of the classical theory of convergence in law of sums of independent (or martingale difference) random variables to infinitely divisible laws.
The third and last part of the book (consisting of 2 chapters and about 100 pages) is devoted to a study of Donsker type invariance principles and related diffusion approximations; here also the theorems generalize well-known discrete parameter results to semimartingales. Most of parts 2 and 3 of the book are of recent vintage and much of it appears in a monograph for the first time.
The book has an extensive bibliography (367 items). Many sections are followed by problems. On the whole, the book provides a somewhat overwhelming view of some of the work in progress in the theory of semimartingales. Since the general index is not very complete and since there is a total lack of any index of symbols, any attempt at rapid reading through skipping and skimming is bound to end up in continual backing up to discover what this or that term of symbol may stand for. Also the difficulty of the subject matter is such that the book could not be recommended to any inroductory or even advanced course on martingale theory. However, for active researchers in the field, the book could be very useful since it documents systematically many of the recent advances in the area of limit theorems for semimartingales.

MSC:
 60G44 Martingales with continuous parameter 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 60F99 Limit theorems in probability theory 60B10 Convergence of probability measures 60Hxx Stochastic analysis