Random dynamical systems

Cover......Page 1
Half-title......Page 3
Dedication......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Acknowledgment......Page 15
Notation......Page 17
1.1 Introduction......Page 19
1.2 Basic Definitions: Fixed and Periodic Points......Page 21
1.3.1 Li–Yorke Chaos and Sarkovskii Theorem......Page 29
1.3.2 A Remark on Robustness of Li–Yorke Complexity......Page 32
1.3.3 Complexity: Alternative Approaches......Page 34
1.4 Linear Difference Equations......Page 35
1.5 Increasing Laws of Motion......Page 38
1.6 Thresholds and Critical Stocks......Page 44
1.7 The Quadratic Family......Page 50
1.7.1 Stable Periodic Orbits......Page 51
1.8 Comparative Statics and Dynamics......Page 56
1.8.1 Bifurcation Theory......Page 57
1.9.1 The Harrod–Domar Model......Page 64
1.9.2.1 Homogeneous Functions: A Digression......Page 65
1.9.2.2 The Model......Page 66
1.9.3 Balanced Growth and Multiplicative Processes......Page 71
1.9.4 Models of Intertemporal Optimization with a Single Decision Maker......Page 77
1.9.4.1 Optimal Growth: The Aggregative Model......Page 78
1.9.4.2 On the Optimality of Competitive Programs......Page 84
1.9.4.3 Stationary Optimal Programs......Page 89
1.9.4.4 Turnpike Properties: Long-Run Stability......Page 90
1.9.5.1 Periodic Optimal Programs......Page 95
1.9.5.2 Chaotic Optimal Programs......Page 97
1.9.5.3 Robustness of Topological Chaos......Page 98
1.9.6 Dynamic Programming......Page 101
1.9.6.1 The Value Function......Page 103
1.9.6.2 The Optimal Transition Function......Page 106
1.9.6.3 Monotonicity With Respect to the Initial State......Page 108
1.9.7 Dynamic Games......Page 113
1.9.8 Intertemporal Equilibrium......Page 116
1.9.9 Chaos in Cobb–Douglas Economies......Page 119
1.10 Complements and Details......Page 122
1.11 Supplementary Exercises......Page 131
2.1 Introduction......Page 137
2.2 Construction of Stochastic Processes......Page 140
2.3 Markov Processes with a Countable Number of States......Page 144
2.4 Essential, Inessential, and Periodic States of a Markov Chain......Page 149
2.5 Convergence to Steady States for Markov Processes on Finite State Spaces......Page 151
2.6 Stopping Times and the Strong Markov Property of Markov Chains......Page 161
2.7 Transient and Recurrent Chains......Page 168
2.8 Positive Recurrence and Steady State Distributions of Markov Chains......Page 177
2.9 Markov Processes on Measurable State Spaces: Existence of and Convergence to Unique Steady States......Page 194
2.10 Strong Law of Large Numbers and Central Limit Theorem......Page 203
2.11 Markov Processes on Metric Spaces: Existence of Steady States......Page 209
2.12 Asymptotic Stationarity......Page 214
2.13 Complements and Details......Page 219
2.14 Supplementary Exercises......Page 257
3.1 Introduction......Page 263
3.2 Random Dynamical Systems......Page 264
3.3 Evolution......Page 265
3.4 The Role of Uncertainty: Two Examples......Page 266
3.5.1 Splitting and Monotone Maps......Page 268
3.5.2 Splitting: A Generalization......Page 273
3.5.3 The Doeblin Minorization Theorem Once Again......Page 278
3.6.1 First-Order Nonlinear Autoregressive Processes (NLAR(1))......Page 280
3.6.2 Stability of Invariant Distributions in Models of Economic Growth......Page 281
3.6.3 Interaction of Growth and Cycles......Page 285
3.6.4 Comparative Dynamics......Page 291
3.7.1 Iteration of Random Lipschitz Maps......Page 293
3.7.2 A Variant Due to Dubins and Freedman......Page 299
3.8 Complements and Details......Page 302
3.9 Supplementary Exercises......Page 312
4.1 Introduction......Page 314
4.2 Iterates of Real-Valued Affine Maps (AR(1) Models)......Page 315
4.3 Linear Autoregressive (LAR(k)) and Other Linear Time Series Models......Page 322
4.4 Iterates of Quadratic Maps......Page 328
4.5 NLAR(k) and NLARCH(k) Models......Page 335
4.6 Random Continued Fractions......Page 341
4.6.1 Continued Fractions: Euclid’s Algorithm and the Dynamical System of Gauss......Page 342
4.6.2 General Continued Fractions and Random Continued Fractions......Page 343
4.6.3 Bernoulli Innovation......Page 348
4.7 Nonnegativity Constraints......Page 354
4.8 A Model with Multiplicative Shocks, and the Survival Probability of an Economic Agent......Page 356
4.9 Complements and Details......Page 360
5.1 Introduction......Page 367
5.2 Estimating the Invariant Distribution......Page 368
5.3 A Sufficient Condition for…Consistency......Page 369
5.3.1 …Consistency......Page 370
5.4 Central Limit Theorems......Page 378
5.5 The Nature of the Invariant Distribution......Page 383
5.5.1 Random Iterations of Two Quadratic Maps......Page 385
5.6 Complements and Details......Page 387
5.7 Supplementary Exercises......Page 393
6.1 Introduction......Page 397
6.2 The Model......Page 398
6.2.1 Optimality and the Functional Equation of Dynamic Programming......Page 399
6.3.1 Continuous Correspondences......Page 403
6.3.2 The Maximum Theorem and the Existence of a Measurable Selection......Page 404
6.4 Dynamic Programming with a Compact Action Space......Page 406
6.5.1 The Aggregative Model of Optimal Growth Under Uncertainty: The Discounted Case......Page 408
6.5.2 Interior Optimal Processes......Page 415
6.5.3 The Random Dynamical System of Optimal Inputs......Page 420
6.5.4 Accumulation of Risky Capital......Page 425
6.6.1 Upper Semicontinuous Model......Page 427
6.6.2 The Controlled Semi-Markov Model......Page 428
6.6.3 State-Dependent Actions......Page 433
A1. METRIC SPACES: SEPARABILITY, COMPLETENESS, AND COMPACTNESS......Page 437
A1.2. Completeness......Page 438
A1.3. Compactness......Page 440
A2. INFINITE PRODUCTS OF METRIC SPACES AND THE DIAGONALIZATION ARGUMENT......Page 441
A3. MEASURABILITY......Page 443
A3.2. Product Spaces: Separability Once Again......Page 444
A3.4. Change of Variable......Page 446
A4. BOREL-CANTELLI LEMMA......Page 448
A5. CONVERGENCE......Page 449
Bibliography......Page 453
Author Index......Page 471
Subject Index......Page 475