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Statistical inference for diffusion type processes. (English) Zbl 0952.62077
Kendall’s Library of Statistics. 8. London: Arnold. xvi, 349 p. (1999).
The title of the book corresponds well to an important area of modern stochastics and the variety of its applications. Prof. Prakasa Rao is well known for his long standing, active and successful work in this area. In his new book, now under review, he has included almost all major results which appeared during the last about 10 years.
The book is devoted to diffusion type processes and it deals with a wide range of statistical inference problems, both parametric and nonparametric. It is needless to say that this class of processes plays an important role in many areas, such as natural sciences and population theory. Stochastic finance modelling is among the very recent areas of applications. Let us recall how important it is to estimate the so-called volatility of stocks.
The presentation is systematic. The author treats different types of inference problems, e.g. parametric or nonparametric, and the problems depend on the kind of available data, e.g. continuous in time or discrete. Another subdivision is that different methods can be used for solving the same problem and different optimality criteria can be used. In any such a case a comparison is carried out, so we can conclude which is the preferable method in one or another situation. A great attention is paid to the most important schemes when the diffusion process (its time parameter is continuous !) is observed only discretely, at fixed or at random times, so the solution of any statistical inference problem has to be based on the available discrete data. This is the most frequently met situation in practice.
The material is almost evenly distributed into the following 7 chapters whose names tell us enough about the contents: 1. Diffusion type processes (DTP). 2. Parametric inference for DTP from continuous paths. 3. Parametric inference for DTP from sampled data. 4. Nonparametric inference for DTP from continuous sample paths. 5. Nonparametric inference for DTP from sampled data. 6. Applications to stochastic modelling. 7. Numerical approximation methods for stochastic differential equations. Then there are 4 Appendixes: A. Uniform ergodic theorem. B. Stochastic integration and limit theorems for stochastic integrals. C. Wavelets. D. Gronwell-Bellman type lemma. The book ends with a comprehensive References list as well as an Author Index and Subject Index.
The material is carefully chosen and given in a smooth style. In all cases the models and the problems are explicitly described. The statements, usually formulated as theorems, are either given with their proofs, or there is a reference to original works. It is so nice to see a large number of illustrative examples. The author has included not only his own results but also results due to many other contributors in the area.
Obviously, it is expected that the reader obeys some knowledge in advanced probability, stochastic processes as well as in principles of statistical inference. Besides some brief definitions of notions and properties, indications are given in the book helping the reader to find the most appropriate sources. This timely produced book, with its encyclopedic character, will become soon a basic source of information in the area and will stimulate both, theoretical studies and applications of stochastic analysis.

62M05 Markov processes: estimation; hidden Markov models
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62M02 Markov processes: hypothesis testing
62Mxx Inference from stochastic processes
65C60 Computational problems in statistics (MSC2010)