Random Dynamical Systems: Theory and Applications

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.

Table of Contents:
1. Dynamical systems; 2. Markov processes; 3. Random dynamical systems; 4. Random dynamical systems: special structures; 5. Invariant distributions: estimations and computation; 6. Discounted dynamic programming under uncertainty; 7. Appendix.

About the Author :
Rabi Bhattacharya is Professor of Mathematics at the University of Arizona. He has also taught at the University of California at Berkeley and Indiana University. Professor Bhattacharya has held visiting research professorships at the University of Goetttingen, the University of Bielefeld, and the Indian Statistical Institute. He is a recipient of a Guggenheim Fellowship and an Alexander Von Humboldt Forschungspreis. He is a Fellow of the Institute of Mathematical Statistics and has served on the editorial boards of a number of international journals, including the Annals of Probability, Annals of Applied Probability, Journal of Multivariate Analysis, and Statistica Sinica. He has co-authored Normal Approximations and Asymptotic Expansions (with R. Ranga Rao), Stochastic Processes with Applications (with E. C. Waymire), and Asymptotic Statistics (with M. Denker). Mukul Majumdar is H. T. and R. I. Warshow Professor of Economics at Cornell University. He has also taught at Stanford University and the London School of Economics. Professor Majumdar is a Fellow of the Econometric Society and has been a Guggenheim Fellow, a Ford Rotating Research at the University of California, Berkeley, an Erskine Fellow at the University of Canterbury, an Oskar Morgenstern Visiting Professor at New York University, a Lecturer at the College de France and an Overseas Fellow at Churchill College, Cambridge University. Professor Majumdar has served on the editorial boards of many leading journals, including The Review of Economic Studies, Journal of Economic Theory, Journal of Mathematical Economics, and Economic Theory, and edited the collection Organizations with Incomplete Information (Cambridge University Press, 1998).

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