Markov Processes: Characterization and Convergence 2nd Edition

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ISBN-13: 978-0471769866
ISBN-10: 047176986X
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Editorial Reviews

From the Publisher

A graduate text/reference on Markov Processes and their relationship to operator semigroups. Presents several different approaches to proving weak approximation theorems for Markov processes, emphasizing the interplay of methods of characterization and approximation. Martingale problems for general Markov processes are systematically developed for the first time in book form. --This text refers to an out of print or unavailable edition of this title.

From the Inside Flap

The recognition that each method for verifying weak convergence is closely tied to a method for characterizing the limiting process Sparked this broad study of characterization and convergence problems for Markov processes. A number of topics are presented for the first time in book form, such as Martingale problems for general Markov processes, powerful criteria for convergence in distribution in DE[O,???), multiple random time transformations, duality as a method of characterizing Markov processes, and characterizations of stationary distributions. The authors illustrate several different approaches to proving weak approximation theorems—operator semigroup convergence theorems, Martingale characterization of Markov processes, and representation of the processes as solutions of stochastic equations. The heart of the book reveals the main characterization and convergence results, with an emphasis on diffusion processes. Applications to branching and population processes, genetic models, and random evolutions, are given. Useful to the professional as a reference, suitable for the graduate student as a text, this volume features a table of the interdependencies among the theorems, an extensive bibliography, and end-of-chapter problems. --This text refers to an out of print or unavailable edition of this title.
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Product Details

  • Paperback: 552 pages
  • Publisher: Wiley-Interscience; 2nd edition (September 14, 2005)
  • Language: English
  • ISBN-10: 047176986X
  • ISBN-13: 978-0471769866
  • Product Dimensions: 6 x 1 x 10.3 inches
  • Shipping Weight: 1.5 pounds (View shipping rates and policies)
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #719,182 in Books (See Top 100 in Books)

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17 of 17 people found the following review helpful By Paul Thurston on June 3, 2005
Format: Hardcover Verified Purchase
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.
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0 of 1 people found the following review helpful By ale on September 21, 2012
Format: Paperback Verified Purchase
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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