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17 of 17 people found the following review helpful
4.0 out of 5 stars Accessible Treatment of a Demanding Topic, June 3, 2005
Paul Thurston (New York, NY USA) - See all my reviews
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The authors have assembled a very accessible treatment of Markov process theory. The text covers three principal convergence techniques in detail: the operator semigroup characterization, the solution of the martingale problem of Stroock and Varadhan and the stochastic calculus of random time changes.

The book is reasonably self-contained, although the reader is well-advised to prepare herself with an understanding of analysis and measure theory. I recommend Rudin's Real and Complex Analysis. The authors provide an appendix which catalogues the measure-theory results they require.

The introduction to the book might be a little off-putting, particularly to readers who aren't familiar with the topic. The first-time reader should skip the introduction and head straight to Chapter 1. Once you've completed the book, the Introduction makes a wonderful synopsis.

Chapter 1 focuses on operator semigroups and builds the foundations required for the remainder of the text. The Hille-Yosida Theorem characterizing those linear operators which can be generators of 'good' semigroups is studied in depth. Generalizations of Hille-Yosida are discussed, and special attention is given to operators on function spaces.

Chapter 2 develops the material needed for the analysis of stochastic processes. After covering the basic definitions, the chapter provides a self-contained treatment of the Optional Sampling Theorem. Quadratic variation of local martingales is built-up from the variation of the sample paths. Right-continuous modifications are developed as a prelude to the Doob-Meyer Theorem. The authors provide an extremely elegant and intuitive proof of the Doob-Meyer Theorem and use the results to analyze square-integrable martingales.

Chapter 3 provides the technical results needed from the theory of convergence of probability measures. This chapter hits all the highlights that you'll find in Billingsley's Convergence of Probability Measures, including the Skorohod topology, the Prohorov metric, relative compactness and the concept of a tight family.

Chapter 4 is the core of the book. The first section introduces the notion of a Markov process. Time-homogenous transition functions are defined in terms of the Chapman-Kolmogorov property. A version of Kolmogorov's extension theorem is established to provide the existence of a Markov process corresponding to a given transition function. (Analogous results are also derived for Markov chains.) The next section gives an explicit construction of a Markov process corresponding to a particular transition function via the use of Poisson processes. The so called "jump Markov process" is used in the study of Feller semigroups. The key result is that each Feller semigroup can be realized as the transition semigroup of a strong Markov process. The remainder of chapter focuses on the development of solutions to the Martingale Problem of Stroock and Varadhan which give a Markov process corresponding to a given infinitesimal generator.

Chapter 5 develops the needed material from stochastic (Ito) calculus. The authors present the definition of Brownian motion and develop its existence via the use of Feller operator semigroups. They then move on to develop the Ito Integral as well as to state and prove Ito's formula for continuous semi-martingales. The chapter finishes with an introduction to stochastic differential equations and their solutions.

Chapter 6 is a short chapter introducing the concept of a random time change and builds from the stochastic calculus material in the previous chapter.

Chapter 7 focuses on the Martingale Central Limit Theorem along with applications in the study of convergence problems. Donsker's invariance principal is discussed and sequences which limit to stochastic diffusions are studied in-depth.

Chapter 8 represents a break from the authors' self-contained format. This chapter is really a survey from the study of generators and references material covering degenerate diffusions, non-degenerate diffusions and jump processes. A diligent reader should be able to work out the results from the hints given here.

The last part of the text covers a selection of special topics. Chapter 9 is a very short chapter and covers some of the classical and modern results from branching Markov processes. Chapters 10 and 11 review the theory from the applied setting and study genetic models and population processes. The final chapter introduces the notion of a random evolution and studies properties of these families of operators from the Markov point-of-view.

The reader will find a good selection of problems and exercises at the end of each chapter. Although not required for subsequent sections, the problems solidify the material, as well as present useful techniques.
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0 of 1 people found the following review helpful
5.0 out of 5 stars Reading about Markov processes and semigroups, September 21, 2012
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This review is from: Markov Processes: Characterization and Convergence (Paperback)
It's a very good book to read about this subject. It is clear and well organized. I fully recommend it.
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Markov Processes: Characterization and Convergence
Markov Processes: Characterization and Convergence by Stewart N. Ethier (Paperback - September 14, 2005)
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