This treatment provides an exposition of discrete time dynamic processes evolving over an infinite horizon. Chapter 1 reviews some mathematical results from the theory of deterministic dynamical systems, with particular emphasis on applications to economics. The theory of irreducible Markov processes, especially Markov chains, is surveyed in Chapter 2. Equilibrium and long run stability of a dynamical system in which the law of motion is subject to random perturbations is the central theme of Chapters 3-5. A unified account of relatively recent results, exploiting splitting and contractions, that have found applications in many contexts is presented in detail. Chapter 6 explains how a random dynamical system may emerge from a class of dynamic programming problems. With examples and exercises, readers are guided from basic theory to the frontier of applied mathematical research.
Random Dynamical Systems: Theory and Applications
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Editorial Reviews
Review
“As I turn the pages of this new book, I come to realize how lucky graduate students of today are.”
Paul A. Samuelson, Nobel Laureate, Massachusetts Institute of Technology
“This book is a brilliant and clear synthesis of the known results on random processes in time. Any economist modeling the growth and cyclical behavior of an economy or of a smaller economic unit will need many of the results displayed and proved and will find no alternative as complete and clear. The numerous well-chosen problems are a bonus.”
Kenneth J. Arrow, Nobel Laureate, Stanford University
“This impressive and unique volume will be an excellent textbook as well as a valuable reference for economists and others interested in random dynamical systems. In addition to a rigorous treatment of the basic theory, the volume includes an introductory section on deterministic dynamical systems, an exposition of stochastic dynamic programming, and various applications to economic models, as well as many exercises and an appendix with basic mathematical background material.”
Roy Radner, Stern School, New York University
“This beautiful and elegantly written book by two world class scholars belongs on the bookshelf of any scholar who uses stochastic dynamical systems in his work. This includes ecologists, economists, mathematicians, mathematical biologists, statisticians, and many others.”
William A. Brock, University of Wisconsin, Madison
“Random Dynamical Systems is the product of the joint works of two masters, Rabi Bhattacharya and Mukul Majumdar, in mathematical statistics and mathematical economics, respectively. It presents the rigorous and yet lucid treatment of the theory of discrete time dynamical processes with applications to economics. I believe that this book will acquire the stature as the text of choice for students and researchers on this field for many years to come.”
Kazuo Nishimura, Kyoto University
"This is a beautiful book. Although mostly oriented towards economic applications, there are sufficient references to and illustrations of other application scenarios to make this of interest to a far wider readership than merely econometricians. Engineers, statisticians, and physicists would find the material of interest and value."
International Statistical Review
"The material is well chosen, well structured and distributed in six chapter each consisting of sections. The presentation is rigorous and clear.... The book is written in a smooth style... The reviewer has all the arguments to recommend the book strongly not only to institutional libraries but also to anybody who is studying, teaching or using stochastic models."
Jordan Stoyanov, Newcastle University for the Journal of Royal Statistical Society
"The book is well written, giving complete proofs of theorems and providing numerous examples that not only demonstrate the applicability of the material, but also demonstrate the bizarre behavior possible in nonlinear systems. I believe that this book will be enjoyable to anyone having a background in Markov chains or an interest in nonlinear time series models."
Peter C. Kiessler, Clemson University, Journal of the American Statistician
"This book is very well written and it is a clear synthesis of the theory of discrete time dynamical processes with many applications to economics. In particular, the book contains a large number of examples and exercises, which in my opinion, make it a useful guide for students and also for researchers."
Maria J. Garrido-Atienza, Mathematical Reviews
Paul A. Samuelson, Nobel Laureate, Massachusetts Institute of Technology
“This book is a brilliant and clear synthesis of the known results on random processes in time. Any economist modeling the growth and cyclical behavior of an economy or of a smaller economic unit will need many of the results displayed and proved and will find no alternative as complete and clear. The numerous well-chosen problems are a bonus.”
Kenneth J. Arrow, Nobel Laureate, Stanford University
“This impressive and unique volume will be an excellent textbook as well as a valuable reference for economists and others interested in random dynamical systems. In addition to a rigorous treatment of the basic theory, the volume includes an introductory section on deterministic dynamical systems, an exposition of stochastic dynamic programming, and various applications to economic models, as well as many exercises and an appendix with basic mathematical background material.”
Roy Radner, Stern School, New York University
“This beautiful and elegantly written book by two world class scholars belongs on the bookshelf of any scholar who uses stochastic dynamical systems in his work. This includes ecologists, economists, mathematicians, mathematical biologists, statisticians, and many others.”
William A. Brock, University of Wisconsin, Madison
“Random Dynamical Systems is the product of the joint works of two masters, Rabi Bhattacharya and Mukul Majumdar, in mathematical statistics and mathematical economics, respectively. It presents the rigorous and yet lucid treatment of the theory of discrete time dynamical processes with applications to economics. I believe that this book will acquire the stature as the text of choice for students and researchers on this field for many years to come.”
Kazuo Nishimura, Kyoto University
"This is a beautiful book. Although mostly oriented towards economic applications, there are sufficient references to and illustrations of other application scenarios to make this of interest to a far wider readership than merely econometricians. Engineers, statisticians, and physicists would find the material of interest and value."
International Statistical Review
"The material is well chosen, well structured and distributed in six chapter each consisting of sections. The presentation is rigorous and clear.... The book is written in a smooth style... The reviewer has all the arguments to recommend the book strongly not only to institutional libraries but also to anybody who is studying, teaching or using stochastic models."
Jordan Stoyanov, Newcastle University for the Journal of Royal Statistical Society
"The book is well written, giving complete proofs of theorems and providing numerous examples that not only demonstrate the applicability of the material, but also demonstrate the bizarre behavior possible in nonlinear systems. I believe that this book will be enjoyable to anyone having a background in Markov chains or an interest in nonlinear time series models."
Peter C. Kiessler, Clemson University, Journal of the American Statistician
"This book is very well written and it is a clear synthesis of the theory of discrete time dynamical processes with many applications to economics. In particular, the book contains a large number of examples and exercises, which in my opinion, make it a useful guide for students and also for researchers."
Maria J. Garrido-Atienza, Mathematical Reviews
Book Description
Covers a range of topics related to the relatively recent literature on the theory and applications of dynamical systems subject to random shocks. It can serve as the basic text for courses on dynamic economics or stochastic processes, and is a valuable tool for further exploration.
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3.0 out of 5 stars
Serviceable,
This review is from: Random Dynamical Systems: Theory and Applications (Paperback)
This book is a serviceable treatment of dynamics in general. The first chapter covers discrete, deterministic dynamical systems and chaos. The second chapter is independent from the first and covers the general theory of Markov processes. The third chapter introduces (Markovian) random dynamical systems. The fourth chapter is on special cases of random dynamical systems. The fifth chapter covers some asymptotic stationarity results. The last chapter is on stochastic dynamic programming.
There are a few typos in the book and the way it has been typeset remind me of a draft rather than a final, polished textbook. The book is probably an OK entry point into the field if you have a solid background in undergraduate analysis and are already familiar with measure theory and probability. The exposition becomes quite dry and hard to follow as the book goes on. There are exercises in the book but they tend to be either too easy or too open ended. There are no solutions. The book has definitely been written with economics in mind, almost all examples are derived from economics. Having said that, they're not particularly interesting examples. All in all, this is a good but not stellar book. It's hard to put my finger on it, but the book just didn't "excite" me enough.
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Key Phrases - Statistically Improbable Phrases (SIPs):
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unique optimal program, optimal transition function, rth return time, uniformly strict contraction, stationary optimal stock, unique invariant probability, ergodic chaos, discounted dynamic programming, unique invariant distribution, splitting hypothesis, random dynamical systems, splitting condition, nondecreasing maps, quadratic family, extinction program, asymptotic stationarity, simple symmetric random walk, stationary optimal policy, consumption good sector, positive recurrent, investment good sector, maximum theorem, null recurrent, separable metric space, total variation distance Key Phrases - Capitalized Phrases (CAPs): (learn more)
Markov Processes, Some Applications, Special Structures, Supplementary Exercises, Proof of Step, Discounted Dynamic Programming Under Uncertainty, Iterates of Real-Valued Affine Maps, Basic Definitions, Metric Spaces, The Maximum Theorem, Increasing Laws of Motion, Iterates of Quadratic Maps, Construction of Stochastic Processes, Bibliographical Notes, Random Continued Fractions, Measurable State Spaces New!
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Front Cover | Table of Contents | First Pages | Index | Surprise Me!
unique optimal program, optimal transition function, rth return time, uniformly strict contraction, stationary optimal stock, unique invariant probability, ergodic chaos, discounted dynamic programming, unique invariant distribution, splitting hypothesis, random dynamical systems, splitting condition, nondecreasing maps, quadratic family, extinction program, asymptotic stationarity, simple symmetric random walk, stationary optimal policy, consumption good sector, positive recurrent, investment good sector, maximum theorem, null recurrent, separable metric space, total variation distance Key Phrases - Capitalized Phrases (CAPs): (learn more)
Markov Processes, Some Applications, Special Structures, Supplementary Exercises, Proof of Step, Discounted Dynamic Programming Under Uncertainty, Iterates of Real-Valued Affine Maps, Basic Definitions, Metric Spaces, The Maximum Theorem, Increasing Laws of Motion, Iterates of Quadratic Maps, Construction of Stochastic Processes, Bibliographical Notes, Random Continued Fractions, Measurable State Spaces New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
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