Bhattacharya R., Majumdar M. Random Dynamical Systems: Theory and Applications

Подождите немного. Документ загружается.

This page intentionally left blank

RANDOM DYNAMICAL SYSTEMS

This book provides an exposition of discrete time dynamic processes evolv-

ing over an inﬁnite horizon. Chapter 1 reviews some mathematical results

from the theory of deterministic dynamical systems, with particular empha-

sis on applications to economics. The theory of irreducible Markov pro-

cesses, especially Markov chains, is surveyed in Chapter 2. Equilibrium and

long-run stability of a dynamical system in which the law of motion is sub-

ject to random perturbations are 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 dy-

namic programming problems. With examples and exercises, readers are

guided from basic theory to the frontier of applied mathematical research.

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 Goettingen, 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 Ap-

plied Probability, Journal of Multivariate Analysis, Econometric Theory, and

Statistica Sinica. He has co-authored Normal Approximations and Asymp-

totic Expansions (with R. Ranga Rao), Stochastic Processes with Applica-

tions (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 Economet-

ric Society and has been a Guggenheim Fellow, a Ford Rotating Research

Professor at the University of California at 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 he has edited the collection Organi-

zations with Incomplete Information (Cambridge University Press, 1998).

To Urmi, Deepta, and Aveek

Random Dynamical Systems

Theory and Applications

RABI BHATTACHARYA

University of Arizona

MUKUL MAJUMDAR

Cornell University

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-82565-8

ISBN-13 978-0-521-53272-3

ISBN-13 978-0-511-27353-7

2007

Information on this title: www.cambrid

e.or

/9780521825658

This publication is in copyright. Subject to statutory exception and to the provision of

relevant collective licensing agreements, no reproduction of any part may take place

without the written permission of Cambridge University Press.

ISBN-10 0-511-27353-3

ISBN-10 0-521-82565-2

ISBN-10 0-521-53272-8

Cambridge University Press has no responsibility for the persistence or accuracy of urls

for external or third-party internet websites referred to in this publication, and does not

guarantee that any content on such websites is, or will remain, accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

hardback

erback

eBook (EBL)

hardback

Contents

Preface page ix

Acknowledgment xiii

Notation xv

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

vi Contents

1.10 Complements and Details 104

1.11 Supplementary Exercises 113

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

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

Contents vii

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

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

Invariant Distributions: Estimation and Computation 349

5.1 Introduction 349

5.2 Estimating the Invariant Distribution 350

5.3 A Sufﬁcient Condition for

√

n-Consistency 351

5.3.1

√

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

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

viii Contents

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

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