Item description for An Introduction to Markov Processes (Graduate Texts in Mathematics) by Daniel W. Stroock...
This book provides a rigorous but elementary introduction to the theory of Markov Processes on a countable state space. It should be accessible to students with a solid undergraduate background in mathematics, including students from engineering, economics, physics, and biology. Topics covered are: Doeblin's theory, general ergodic properties, and continuous time processes. A whole chapter is devoted to reversible processes and the use of their associated Dirichlet forms to estimate the rate of convergence to equilibrium.
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Est. Packaging Dimensions: Length: 9.4" Width: 6.2" Height: 0.6" Weight: 0.8 lbs.
Release Date Mar 14, 2005
ISBN 3540234993 ISBN13 9783540234999
Availability 0 units.
More About Daniel W. Stroock
Dr Daniel W. Stroock is the Simons Professor of Mathematics Emeritus at the Massachusetts Institute of Technology. He has published numerous articles and is the author of six books, most recently Probability Theory: An Analytic View, 2nd Edition (2010).
Daniel W. Stroock has an academic affiliation as follows - Massachusetts Institute of Technology.
Reviews - What do customers think about An Introduction to Markov Processes (Graduate Texts in Mathematics)?
Nice treatment of time-inhomogeneous processes Feb 17, 2007
I bought this book not because I needed to learn about Markov processes (I deal with them rather often) but because I wanted a text that discussed time-inhomogeneous Markov processes, however briefly. (I have been told that Doob does this, but I can't ever seem to bring myself to put in the work necessary to really appreciate his stochastic processes book.) Stroock gives a nice treatment of this topic in the context of simulated annealing, which is probably where most people would first encounter it these days.
I have not read most of the rest of the book, but it is clearly a more rigorous treatment than either Norris or Bremaud (both of which are nice for the beginner or non-mathematician), and it requires a bit more mathematical maturity. Nevertheless my sampling indicates that it is well-written. It is probably better suited for the mathematicians and probabilists (or those who will have to deal with more advanced topics later) than for the average user of Markov processes. I would recommend it, along with Williams' wonderful Probability with Martingales, to anyone who wants a really solid yet concise grounding in mathematical probability at the undergraduate level.
The Place to Begin Understanding Stochastic Analysis Sep 6, 2005
The book provides a solid introduction into the study of stochastic processes and fills a significant gap in the literature: a text that provides a sophisticated study of stochastic processes in general (and Markov processes in particular) without a lot of heavy prerequisites.
Stroock keeps the prerequisites very light. The reader need only have an understanding of some elementary topics. For undergraduate calculus, I recommend Apostle's two volume set Calculus, Vol. 1and Calculus,Vol. 2. Make sure you have a grasp of some basic matrix algebra, at least through eigenvectors and eigenvalues of a square matrix. You'll also need a good understanding of basic statistic/probability theory at the level of Hogg & Craig's Introduction to Mathematical Statistics, 6th Edition.
Stroock begins his book with a study of 1-dimensional random walks in Chapter 1. First passage times, first return times and the reflection principle are each introduced. (You'd study precisely the same topics from the Brownian motion point of view in an advanced measure-theoretic text). Higher dimensional random walks are introduced and transience/recurrence is studied.
A study of the Markov chain begins in earnest in Chapter 2. Stroock starts this chapter by establishing the existence of discrete time Markov chains using a construction technique based on a sequence of independent, identically distributed uniform random variables. This is a simple, yet powerful technique and different versions of this are used again in the construction of the Poisson process, the time homogenous Markov process, as well as the non-homogenous Markov process. Doeblin's Stationary Distribution Theorem is established and results from Ergodic Theory for these Doeblin chains are studied.
General (non-Doeblin) Markov chains are considered in Chapter 3. Stroock studies state communication and brings this together with some tools such as Doob's Stopping Time Theorem. Ergodic results are established in this context, starting with some simple results which are then refined.
Chapter 4 is a real highlight of the book. Continuous time Markov processes with values in a countable state space are studied in depth. This chapter focuses on the time-homogeneous case and starts with the construction of Poisson processes and compound Poisson processes. The Markov property is called out at each stage. These elementary stochastic processes are then used as the building blocks for the general time-homogenous Markov process (first with bounded, then unbounded transition rates). Ergodic Theory for these processes is then studied.
Chapter 5 builds towards a principle application, which is a Markov process study of simulated annealing. To get at this application, Stroock considers reversible Markov chains, reversible Markov processes, the Dirichlet form and Poincare's Inequality for this form. Gibbs states are studied next and the discussion of equilibriums leads quite naturally to both the study of simulated annealing and an extremely nice construction of a non-homogenous Markov process suited towards this study.
Chapter 6 is a primer for Lebesgue measure theory. Although this section is certainly not comprehensive, it is quite accessible, provides an interesting 'glimpse ahead' and suggests topics for further study.
Each chapter concludes with a number of really nice exercises of varying difficulty. The author is kind enough to provide some helpful hints for the more challenging problems.