Markov Decision Processes: Discrete Stochastic Dynamic Programming (Wiley Series in Probability and Statistics)

Book Description

The Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists.
"This text is unique in bringing together so many results hitherto found only in part in other texts and papers. . . . The text is fairly self-contained, inclusive of some basic mathematical results needed, and provides a rich diet of examples, applications, and exercises. The bibliographical material at the end of each chapter is excellent, not only from a historical perspective, but because it is valuable for researchers in acquiring a good perspective of the MDP research potential."
-Zentralblatt fur Mathematik
". . . it is of great value to advanced-level students, researchers, and professional practitioners of this field to have now a complete volume (with more than 600 pages) devoted to this topic. . . . Markov Decision Processes: Discrete Stochastic Dynamic Programming represents an up-to-date, unified, and rigorous treatment of theoretical and computational aspects of discrete-time Markov decision processes."
-Journal of the American Statistical Association

Customer Reviews:

Excellent and detailed, although focusing on exact algorithms only.......2007-06-04

Anyone working with Markov Decision Processes should have this book. It has detailed explanations of several algorithms for MDPs: linear programming, value iteration and policy iteration for finite and infinite horizon; total-reward and average reward criteria, and there's one last chapter on continuous-time MDPs (SMDPs).

However, it does not cover some new ideas like partitioning and some faster approximated algorithms. But still, it is a great book!

Make sure to also get Bertsekas' "Dynamic Programming and Optimal Control".

Numerical Solution of Stochastic Differential Equations (Stochastic Modelling and Applied Probability)

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Excellent

Numerical Solution of Stochastic Differential Equations (Stochastic Modelling and Applied Probability)
Peter E. Kloeden , and Eckhard Platen
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540540628

Book Description

The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations, due to the peculiarities of stochastic calculus. The book proposes to the reader whose background knowledge is limited to undergraduate level methods for engineering and physics, and easily accessible introductions to SDE and then applications as well as the numerical methods for dealing with them. To help the reader develop an intuitive understanding and hand-on numerical skills, numerous exercises including PC-exercises are included.

Customer Reviews:

Excellent.......2002-04-10

This book is one of the finest written on the subject and is suitable for readers in a wide variety of fields, including mathematical finance, random dynamical systems, constructive quantum field theory, and mathematical biology. It is certainly well-suited for classroom use, and it includes computer exercises what are definitely helpful for those who need to develop actual computer code to solve the relevant equations of interest. Since it emphasizes the numerical solution of stochastic differential equations, the authors do not give the details behind the theory, but references are given for the interested reader.

As preparation for the study of SDEs, the authors detail some preliminary background on probability, statistics, and stochastic processes in Part 1 of the book. Particularly well-written is the discussion on random number generators and efficient methods for generating random numbers, such as the Box-Muller and Polar Marsaglia methods. Both discrete and continuous Markov processes are discussed, and the authors review the connection between Weiner processes (Brownian motion for the physicist reader) and white noise. The measure-theory foundations of the subject are outlined briefly for the interested reader.

Part 2 begins naturally with an overview of stochastic calculus, with the Ito calculus chosen to show how to generalize ordinary calculus to the stochastic realm. The authors motivate the subject as one in which the functional form of stochastic processes was emphasized, with Ito attempting to find out just when local properties such as the drift and diffusion coefficients can characterize the stochastic process. The Ito formula is shown to be a generalization of the chain rule of ordinary calculus to the case where stochasticity is present. The authors are also careful to distinguish between "random" differential equations and "stochastic" differential equations. The former can be solved by integrating over differentiable sample paths, but in the latter one has to face the nondifferentiability of the sample paths, and hence solutions are more difficult to obtain. The authors give many examples of SDEs that can be solved explicitly, and prove existence and uniqueness theorems for strong solutions of the SDEs. And since ordinary differential equations are usually tackled by Taylor series expansions, it is perhaps not surprising that this technique would be generalized to SDEs, which the authors do in detail in this part. They also outline the differences between the Ito and Stratonovich interpretations of stochastic integrals and SDEs.

Part 3 is definitely of great interest to those who must develop mathematical models using SDEs. The authors carefully outline the reasons where Ito versus the Stratonovich formulations are used, this being largely dependent on the degree of autocorrelation in the processes at hand. The Stratonovich SDE is recommended for cases when the white noise is used as an idealization of a (smooth) real noise process. The authors also show how to approximate Markov chain problems with diffusion processes, which are the solutions of Ito SDEs. Several very interesting examples are given of the applications of stochastic differential equations; the particular ones of direct interest to me were the ones on population dynamics, protein kinetics, and genetics; option pricing, and blood clotting dynamics/cellular energetics.

After a review of discrete time approzimations in ordinary deterministic differential equations, in part 4 the authors show to solve SDEs using this approximation. The familiar Euler approximation is considered, with a simple example having an explicit solution compared with its Euler approximate solution. They also show how to use simulations when an explicit solution is lacking. The importance notions of strong and weak convergence of the approximate solutions are discussed in detail. Strong convergence is basically a convergence in norm (absolute value), while weak convergence is taken with respect to a collection of test functions. Both of these types of convergence reduce to the ordinary deterministic sense of convergence when the random elements are removed.

The discussion of convergence in part 4 leads to a very extensive discussion of strongly convergent approximations in part 5, and weakly convergent approximations in part 6. Stochastic Taylor expansions done with respect to the strong convergence criterion are discussed, beginning with the Euler approximation. More complicated strongly convergent stochastic approximation schemes are also considered, such as the Milstein scheme, which reduces to the Euler scheme when the diffusion coefficients only depend on time. The strong Taylor schemes of all orders are treated in detail. Since Taylor approximations make evaluations of the derivatives necessary, which is computational intensive, the authors discuss strong approximation schemes that do not require this, much like the Runge-Kutta methods in the deterministic case , but the authors are careful to point out that the Runge-Kutta analogy is problematic in the stochastic case. Several of these "derivative-free" schemes are considered by the authors. The authors also consider implicit strong approximation schemes for stiff SDEs, wherein numerical instabilities are problematic. Interesting applications are given for strong approximations for SDEs, such as the Duffing-Van der Pol oscillator, which is very important system in engineering mechanics and phyics, and has been subjected to an incredible amount of research.

Book Description

For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. This book introduces the basic concepts in the design and analysis of randomized algorithms. The first part of the text presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications. Algorithmic examples are also given to illustrate the use of each tool in a concrete setting. In the second part of the book, each chapter focuses on an important area to which randomized algorithms can be applied, providing a comprehensive and representative selection of the algorithms that might be used in each of these areas. Although written primarily as a text for advanced undergraduates and graduate students, this book should also prove invaluable as a reference for professionals and researchers.

Customer Reviews:

Book that didn't meet my expectations.......2006-09-18

Algorithms are my specialty, and I'm really interest in everything that is connected with them. This is one of the few books from the field of algorithms that I was a problem to read. I found this book hard to read because of several reasons.

Firstly, i have a problem with the composition of material from the book. The material is in the many places presented in the unnatural way. Book is method oriented, so often same problem is treated in several places in the book. On the other hand the book is not fully method oriented, so there are chapters of the book that don't present any method of building randomized algorithms. There are several chapters that are organized around some concept from the probability theory. I don't see the reason for these two orientation to be mixed.

Often I have a feeling that authors are not particulary interested in randomized algorithms, and that thet their main interest is to show probability methods in the theory of algorithms. So, there are, for example, chapters in the book named "Moments and Deviations" and "Tail Inequalities". I don't want to say that these concepts are not important for the randomized algorithm complexity claculations, but I think that such chapters belongs to book on probability theory, not randomized algorithms book. On the other side, therms of Monte Carlo and Las Vegas algorithms get together one section in the chapter in which they are described. It is true that in these chapters contain randomized algorithms as examples of usage of mathematical concepts, but the question is: should this book present general mathematical concepts, or randomized algorithms.

The second big drawback is lack of precise mathematical notion in many places in the book. For example, in the chapter on game theory the reader get impression that the whole game theory are game trees. Yet, authors fail to define what game tree is. The definition they give is more lausy desciption than definition. They don't say which kind of tree is game tree. Is it binary? Of course it is not, but authors in this section work only with binary trees. Further, in the text authors said that this tree is uniform. I have to admit that I never heard about uniform trees. The problem is that all definitions in the book is given in this way, by the paragraph of the text, which describe the term, not define it. In fact, the only concepts that are properly defined are ones form the probability theory. None of the concepts from the algorithms theory or data structures theory is not defined as it should be.

The third great problem with the book is that these concepts are never ilustrated with the concrete example. There is a section about the game trees, for example, but there is no single game tree for some game generated in this section. This is not a single case. All examples in the book are about mathmatical, or nore precisely probability theory concepts, and all of them looks like they are taken from the workbook on probability theory, and doesn't have any connection with algorithms.

Another problem is that all chapters are not builded in the same manner. There are chapters (unfortenately very little of them) that have theoretical overview of the method they deal with, but in the other chapters there are no proper theoretical description of the method of the matter.

To resume, this book shows the lack of concept and system in the writting, as well as the interest of authors more in mathmatics than in algorithm field.

My opinion is that there are much better books on the randomized algorithms tnan this one.

More work should be done in proofs.......2004-11-02

Overall, the authors explain core concepts, the examples and the possible applications well. However, the readibility of their proof is far from that of the above three. Honestly some proofs should be re-written completely.

For example, in page 116, they try to use the induction method to prove Lova(')sz Local Lemma. After reading that page many times, I still didn't understand the structure of their proof.

I was TA for under-grad level algorithm course, got A+ in advanced Calculus II and A in intro. to PDE (both in under-grad level), really knew something about induction method and a little bit about algorithm. I am not smart, but far from stupid.

In the end, I google the internet and found a 3-page proof for the same thing. That's easy to catch in few minutes, and then, I understand the 1-page proof in the book. Is it ironic?

A subtle introduction to probablistic algoritms.......2002-01-14

This book is a jewel. It demonstrates how clever and beautifully simple probabilistic ideas can lead to the design of very efficient algorithms. I like its very verbal intuitive style,
with proof strategies being always transparently explained.
For computer scientists, this is *the* reference work in randomized algorithms, by now a major paradigm of algorithms design. For classical probabilists, this
could serve as an eye-opener on unsuspected applications of their field to important areas of computer science.

An enciclopedia for randomized algorithms........2001-07-21

The book has an exoustive amount of algorithms. Not everything is proved. Sometimes the proof contains to few steps to be understood. There are many algorithms explained well. After reading this book it is easy to create your own randomized algorithms.

extremely informative but obscure.......1999-10-16

I've taken two CS classes that use this book and I always felt like this book was very informative. The algorithms and concepts that Motwani brings forth are extremely insightful and interesting. However, the presentation of the proofs has a lot of room for improvement. Notation is carried over from previous chapters and is sometimes unexplained, which makes it very difficult for someone who does not have a lot of familiarity with the material presented. The book presents very interesting topics and leaves a lot of open (unresolved) questions to the reader's curiosity and challenge.

Stochastic Finance: An Introduction In Discrete Time 2 (De Gruyter Studies in Mathematics)

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Excellent book on mathematical finance

Stochastic Finance: An Introduction In Discrete Time 2 (De Gruyter Studies in Mathematics)
Hans Follmer , and Alexander Schied
Manufacturer: Walter de Gruyter
ProductGroup: Book
Binding: Hardcover

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ASIN: 3110183463

Book Description

This book is an introduction to financial mathematics for mathematicians. It is intended both for graduate students with a certain background in probability theory as well as for professional mathematicians in industry and academia. In contrast to many textbooks on mathematical finance, only discrete-time stochastic models are considered. This setting has the advantage that the text can concentrate from the beginning on typical problems which are suggested by financial applications. Moreover, certain principles, such as the general incompleteness of realistic market models, become thus more transparent and visible. On the other hand, all models are based on general probability spaces, and so the text captures the interplay between probability theory and functional analysis which is typical for modern mathematical finance.

The first part of the book contains a study of financial investments in a static one-period market model. Here, an investor faces intrinsic risk and uncertainty, which cannot be hedged away. The tools presented to deal with this situation range from the classical theory of expected utility until the more recent development of measures of risk.

In the second part of the book, the idea of dynamic hedging and arbitrage-free pricing of contingent claims is developed in a multi-period framework. Such market models are typically incomplete, and particular focus is given to

methods combining the dynamic hedging of a risky position with the tools of assessing risk and uncertainty as presented in part.

Contents: Mathematical finance in one period: Arbitrage theory. Expected utility. Optimal investments. Measures of risk Dynamic Arbitrage Theory: Dynamic hedging of contingent claims. American contingent claims. Optional decomposition and super-hedging. Efficient hedging in incomplete markets. Minimizing the hedging error. Hedging under constraints References. Index

Customer Reviews:

Excellent book on mathematical finance.......2004-04-20

It is well-known that the mathematical study of finance has, over the last two decades, led to a number of discoveries in stochastic analysis whose import extends beyond the boundaries of finance to other areas of mathematics. There are, currently, many good text-books which treat the mathematics of financial markets (e.g. Pliska, Bingham&Kiesel, Elliott&Kopp, Musiela&Rutkowski, Karatzas&Shreve, roughly in increasing order of difficulty. Pliska's text works only in the discrete-time framework, whereas the others move quickly to continuous-time). The text by Follmer and Schied deals only with the discrete-time case, but covers a large amount of material which you won't find in any of the other books: A thorough introduction to utility theory, excellent coverage of coherent and convex risk measures, and various approaches to hedging in incomplete markets. Each chapter quickly brings the reader close to the frontiers of research. Future research in these areas also promises to overflow the boundaries, providing new applications to other branches of functional analysis.

A word of caution: Though the text restricts itself to the "simpler" discrete-time case, thus avoiding stochastic integration, it nevertheless demands a solid background in analysis, including graduate level probability theory and functional analysis. Though not technically a requirement, some background in mathematical finance is necessary in order to understand what this book is about.

Book Description

Stochastic processes are found in probabilistic systems that evolve with time. Discrete stochastic processes change by only integer time steps (for some time scale), or are characterized by discrete occurrences at arbitrary times. Discrete Stochastic Processes helps the reader develop the understanding and intuition necessary to apply stochastic process theory in engineering, science and operations research. The book approaches the subject via many simple examples which build insight into the structure of stochastic processes and the general effect of these phenomena in real systems. The book presents mathematical ideas without recourse to measure theory, using only minimal mathematical analysis. In the proofs and explanations, clarity is favored over formal rigor, and simplicity over generality. Numerous examples are given to show how results fail to hold when all the conditions are not satisfied. Audience: An excellent textbook for a graduate level course in engineering and operations research. Also an invaluable reference for all those requiring a deeper understanding of the subject.

Customer Reviews:

Terrific Text, better than going to class.......2007-03-14

This is a great textbook. It gives a great treatment of the subject, with the right mix of intuition and mathematical rigor. It is one of those rare textbooks that you can read carefully and simply forget about going to the lectures.

Provides excellent intuitive explanations.......2005-12-15

While Gallager is lacking in mathematical rigor and in typeface aesthetics, it is a superb book for understanding the intuition underlying the theorems. Sheldon Ross's book is more rigorous but does not offer as much intuition. The two books together make an excellent team.

Must have prior knowledge.......2005-11-05

This book is being used in my graduate class. It is very condensed to the point where many mathematical expressions and arguments are squeezed inside the paragraph rather than being stated on a separate line. As a student I believe this book is of little help without prior knowledge of stochastic theory.

Worth having!.......2003-04-30

Although the aesthetics (print, figures etc) of the book leave something to be desired, this is definitely a book one can get a lot out of. Gallager's treatment of the subject matter favors simplicity and his thought process is usually easy to follow. Occasionally one finds less than water tight proofs but all in all this is a great addition to the bookshelf.

Great Book.......1999-10-15

A great book to have. The book gives you real deep insight into the theorems of probability, queing systems and discrete stochastic processes. What most advanced level books on stochastic processes dont have is the explaination of the theorems because authors assume the audiance to be mathematicians. Gallegers book is more for engineers and less for mathematicicans.

Applications Of Multi-Objective Evolutionary Algorithms (Advances in Natural Computation)

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Applications Of Multi-Objective Evolutionary Algorithms (Advances in Natural Computation)

Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Hardcover

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ASIN: 9812561064

Book Description

This book presents an extensive variety of multi-objective problems across diverse disciplines, along with statistical solutions using multi-objective evolutionary algorithms (MOEAs). The topics discussed serve to promote a wider understanding as well as the use of MOEAs, the aim being to find good solutions for high-dimensional real-world design applications. The book contains a large collection of MOEA applications from many researchers, and thus provides the practitioner with detailed algorithmic direction to achieve good results in their selected problem domain.

Design and Analysis of Randomized Algorithms: Introduction to Design Paradigms (Texts in Theoretical Computer Science. An EATCS Series)

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Design and Analysis of Randomized Algorithms: Introduction to Design Paradigms (Texts in Theoretical Computer Science. An EATCS Series)
J. Hromkovic
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 3540239499

Book Description

Randomness is a powerful phenomenon that can be harnessed to solve various problems in all areas of computer science. Randomized algorithms are often more efficient, simpler and, surprisingly, also more reliable than their deterministic counterparts. Computing tasks exist that require billions of years of computer work when solved using the fastest known deterministic algorithms, but they can be solved using randomized algorithms in a few minutes with negligible error probabilities.

Introducing the fascinating world of randomness, this book systematically teaches the main algorithm design paradigms – foiling an adversary, abundance of witnesses, fingerprinting, amplification, and random sampling, etc. – while also providing a deep insight into the nature of success in randomization. Taking sufficient time to present motivations and to develop the reader's intuition, while being rigorous throughout, this text is a very effective and efficient introduction to this exciting field.

Introduction to the Numerical Solution of Markov Chains

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Introduction to the Numerical Solution of Markov Chains
William J. Stewart
Manufacturer: Princeton University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0691036993

Book Description

A cornerstone of applied probability, Markov chains can be used to help model how plants grow, chemicals react, and atoms diffuse--and applications are increasingly being found in such areas as engineering, computer science, economics, and education. To apply the techniques to real problems, however, it is necessary to understand how Markov chains can be solved numerically. In this book, the first to offer a systematic and detailed treatment of the numerical solution of Markov chains, William Stewart provides scientists on many levels with the power to put this theory to use in the actual world, where it has applications in areas as diverse as engineering, economics, and education. His efforts make for essential reading in a rapidly growing field.

Here Stewart explores all aspects of numerically computing solutions of Markov chains, especially when the state is huge. He provides extensive background to both discrete-time and continuous-time Markov chains and examines many different numerical computing methods--direct, single-and multi-vector iterative, and projection methods. More specifically, he considers recursive methods often used when the structure of the Markov chain is upper Hessenberg, iterative aggregation/disaggregation methods that are particularly appropriate when it is NCD (nearly completely decomposable), and reduced schemes for cases in which the chain is periodic. There are chapters on methods for computing transient solutions, on stochastic automata networks, and, finally, on currently available software. Throughout Stewart draws on numerous examples and comparisons among the methods he so thoroughly explains.

Numerical Solution of SDE Through Computer Experiments (Universitext)

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Numerical Solution of SDE Through Computer Experiments (Universitext)
Peter Eris Kloeden , Eckhard Platen , and Henri Schurz
Manufacturer: Springer
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Binding: Paperback

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ASIN: 3540570748

Book Description

The book provides an easily accessible computationally oriented introduction into the numerical solution of stochastic differential equations using computer experiments. It develops in the reader an ability to apply numerical methods solving stochastic differential equations in their own fields. Furthermore, it creates an intuitive understanding of the necessary theoretical background from stochastic and numeric analysis. A downloadable softward containing programs for over 100 problems is provided at each of the following homepages:

http://www.math.uni-frankfurt.de/~numerik/kloeden/
http://www.business.uts.edu.au/finance/staff/eckhard.html
http.//www.math.siu.edu/schurz/SOFTWARE/

to enable the reader to develop an intuitive understanding of the issues involved. Applications include stochastic dynamical systems, filtering, parametric estimation and finance modeling.

The book is intended for readers without specialist stochastic background who want to apply such numerical methods to stochastic differential equations that arise in their own filed.

Maximum Entropy and Bayesian Methods Garching, Germany 1998 (Fundamental Theories of Physics)

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Maximum Entropy and Bayesian Methods Garching, Germany 1998 (Fundamental Theories of Physics)

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ASIN: 0792357663

Book Description

This volume, arising from the 1998 MaxEnt conference, contains a wide range of applications of Bayesian probability theory and maximum entropy methods to problems of concern in such fields as physics, image processing, coding theory, machine learning, economics, data analysis and various other problems. It presents papers by the leading researchers in the field of Bayesian statistics and maximum entropy methods, and represents the latest developments in the field.

Audience: This book will be of interest to researchers in applied statistics, information theory, coding theory, image and signal processing.

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