E-book: Random Dynamical Systems: Theory and Applications

Preface		ix
Acknowledgment		xiii
Notation		xv
1 Dynamical Systems		1
1.1 Introduction		1
1.2 Basic Definitions: Fixed and Periodic Points		3
1.3 Complexity		11
1.3.1 Li—Yorke Chaos and Sarkovskii Theorem		11
1.3.2 A Remark on Robustness of Li—Yorke Complexity		14
1.3.3 Complexity: Alternative Approaches		16
1.4 Linear Difference Equations		17
1.5 Increasing Laws of Motion		20
1.6 Thresholds and Critical Stocks		26
1.7 The Quadratic Family		32
1.7.1 Stable Periodic Orbits		33
1.8 Comparative Statics and Dynamics		38
1.8.1 Bifurcation Theory		39
1.9 Some Applications		46
1.9.1 The Harrod—Domar Model		46
1.9.2 The Solow Model		47
1.9.3 Balanced Growth and Multiplicative Processes		53
1.9.4 Models of Intertemporal Optimization with a Single Decision Maker		59
1.9.5 Optimization with Wealth Effects: Periodicity and Chaos		77
1.9.6 Dynamic Programming		83
1.9.7 Dynamic Games		95
1.9.8 Intertemporal Equilibrium		98
1.9.9 Chaos in Cobb—Douglas Economies		101
1.10 Complements and Details		104
1.11 Supplementary Exercises		113
2 Markov Processes		119
2.1 Introduction		119
2.2 Construction of Stochastic Processes		122
2.3 Markov Processes with a Countable Number of States		126
2.4 Essential, Inessential, and Periodic States of a Markov Chain		131
2.5 Convergence to Steady States for Markov Processes on Finite State Spaces		133
2.6 Stopping Times and the Strong Markov Property of Markov Chains		143
2.7 Transient and Recurrent Chains		150
2.8 Positive Recurrence and Steady State Distributions of Markov Chains		159
2.9 Markov Processes on Measurable State Spaces: Existence of and Convergence to Unique Steady States		176
2.10 Strong Law of Large Numbers and Central Limit Theorem		185
2.11 Markov Processes on Metric Spaces: Existence of Steady States		191
2.12 Asymptotic Stationarity		196
2.13 Complements and Details		201
2.13.1 Irreducibility and Harris Recurrent Markov Processes		210
2.14 Supplementary Exercises		239
3 Random Dynamical Systems		245
3.1 Introduction		245
3.2 Random Dynamical Systems		246
3.3 Evolution		247
3.4 The Role of Uncertainty: Two Examples		248
3.5 Splitting		250
3.5.1 Splitting and Monotone Maps		250
3.5.2 Splitting: A Generalization		255
3.5.3 The Doeblin Minorization Theorem Once Again		260
3.6 Applications		262
3.6.1 First-Order Nonlinear Autoregressive Processes (NLAR(1))		262
3.6.2 Stability of Invariant Distributions in Models of Economic Growth		263
3.6.3 Interaction of Growth and Cycles		267
3.6.4 Comparative Dynamics		273
3.7 Contractions		275
3.7.1 Iteration of Random Lipschitz Maps		275
3.7.2 A Variant Due to Dubins and Freedman		281
3.8 Complements and Details		284
3.9 Supplementary Exercises		294
4 Random Dynamical Systems: Special Structures		296
4.1 Introduction		296
4.2 Iterates of Real-Valued Affine Maps (AR(1) Models)		297
4.3 Linear Autoregressive (LAR(k)) and Other Linear Time Series Models		304
4.4 Iterates of Quadratic Maps		310
4.5 NLAR (k) and NLARCH (k) Models		317
4.6 Random Continued Fractions		323
4.6.1 Continued Fractions: Euclid's Algorithm and the Dynamical System of Gauss		324
4.6.2 General Continued Fractions and Random Continued Fractions		325
4.6.3 Bernoulli Innovation		330
4.7 Nonnegativity Constraints		336
4.8 A Model with Multiplicative Shocks, and the Survival Probability of an Economic Agent		338
4.9 Complements, and Details		342
5 Invariant Distributions: Estimation and Computation		349
5.1 Introduction		349
5.2 Estimating the Invariant Distribution		350
5.3 A Sufficient Condition for square root n-Consistency		351
5.3.1 square root n-Consistency		352
5.4 Central Limit Theorems		360
5.5 The Nature of the Invariant Distribution		365
5.5.1 Random Iterations of Two Quadratic Maps		367
5.6 Complements and Details		369
5.7 Supplementary Exercises		375
6 Discounted Dynamic Programming Under Uncertainty		379
6.1 Introduction		379
6.2 The Model		380
6.2.1 Optimality and the Functional Equation of Dynamic Programming		381
6.3 The Maximum Theorem: A Digression		385
6.3.1 Continuous Correspondences		385
6.3.2 The Maximum Theorem and the Existence of a Measurable Selection		386
6.4 Dynamic Programming with a Compact Action Space		388
6.5 Applications		390
6.5.1 The Aggregative Model of Optimal Growth Under Uncertainty: The Discounted Case		390
6.5.2 Interior Optimal Processes		397
6.5.3 The Random Dynamical System of Optimal Inputs		402
6.5.4 Accumulation of Risky Capital		407
6.6 Complements and Details		409
6.6.1 Upper Semicontinuous Model		409
6.6.2 The Controlled Semi-Markov Model		410
6.6.3 State-Dependent Actions		415
A Appendix		419
A1. Metric Spaces: Separability, Completeness, and Compactness		419
A1.1. Separability		420
A1.2. Completeness		420
A1.3. Compactness		422
A2. Infinite Products of Metric Spaces and the Diagonalization Argument		423
A3. Measurability		425
A3.1. Subspaces		426
A3.2. Product Spaces: Separability Once Again		426
A3.3. The Support of a Measure		428
A3.4. Change of Variable		428
A4. Borel-Cantelli Lemma		430
A5. Convergence		431
Bibliography		435
Author Index		453
Subject Index		457

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).