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Economic.Dynamics.Theory.and.Computation
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Contents Preface xiii Common Symbols xvii 1 Introduction 1 I Introduction to Dynamics 9 2 Introduction to Programming 11 2.1 Basic Techniques 11 2.1.1 Algorithms 11 2.1.2 Coding: First Steps 14 2.1.3 Modules and Scripts 19 2.1.4 Flow Control 21 2.2 Program Design 25 2.2.1 User-Defined Functions 25 2.2.2 More Data Types 27 2.2.3 Object-Oriented Programming 29 2.3 Commentary 33 3 Analysis in Metric Space 35 3.1 A First Look at Metric Space 35 3.1.1 Distances and Norms 36 3.1.2 Sequences 38 3.1.3 Open Sets, Closed Sets 41 3.2 Further Properties 44 3.2.1 Completeness 44 3.2.2 Compactness 46 vii viii Contents 3.2.3 Optimization, Equivalence 48 3.2.4 Fixed Points 51 3.3 Commentary 54 4 Introduction to Dynamics 55 4.1 Deterministic Dynamical Systems 55 4.1.1 The Basic Model 55 4.1.2 Global Stability 59 4.1.3 Chaotic Dynamic Systems 62 4.1.4 Equivalent Dynamics and Linearization 66 4.2 Finite State Markov Chains 68 4.2.1 Definition 68 4.2.2 Marginal Distributions 72 4.2.3 Other Identities 76 4.2.4 Constructing Joint Distributions 80 4.3 Stability of Finite State MCs 83 4.3.1 Stationary Distributions 83 4.3.2 The Dobrushin Coefficient 88 4.3.3 Stability 90 4.3.4 The Law of Large Numbers 93 4.4 Commentary 96 5 Further Topics for Finite MCs 99 5.1 Optimization 99 5.1.1 Outline of the Problem 99 5.1.2 Value Iteration 102 5.1.3 Policy Iteration 105 5.2 MCs and SRSs 107 5.2.1 From MCs to SRSs 107 5.2.2 Application: Equilibrium Selection 110 5.2.3 The Coupling Method 112 5.3 Commentary 116 6 Infinite State Space 117 6.1 First Steps 117 6.1.1 Basic Models and Simulation 117 6.1.2 Distribution Dynamics 122 6.1.3 Density Dynamics 125 6.1.4 Stationary Densities: First Pass 129 6.2 Optimal Growth, Infinite State 133 Contents ix 6.2.1 Optimization 133 6.2.2 Fitted Value Iteration 135 6.2.3 Policy Iteration 142 6.3 Stochastic Speculative Price 145 6.3.1 The Model 145 6.3.2 Numerical Solution 150 6.3.3 Equilibria and Optima 154 6.4 Commentary 156 II Advanced Techniques 157 7 Integration 159 7.1 Measure Theory 159 7.1.1 Lebesgue Measure 159 7.1.2 Measurable Spaces 163 7.1.3 General Measures and Probabilities 166 7.1.4 Existence of Measures 168 7.2 Definition of the Integral 171 7.2.1 Integrating Simple Functions 171 7.2.2 Measurable Functions 173 7.2.3 Integrating Measurable Functions 177 7.3 Properties of the Integral 178 7.3.1 Basic Properties 179 7.3.2 Finishing Touches 180 7.3.3 The Space L 1 183 7.4 Commentary 186 8 Density Markov Chains 187 8.1 Outline 187 8.1.1 Stochastic Density Kernels 187 8.1.2 Connection with SRSs 189 8.1.3 The Markov Operator 195 8.2 Stability 197 8.2.1 The Big Picture 197 8.2.2 Dobrushin Revisited 201 8.2.3 Drift Conditions 204 8.2.4 Applications 207 8.3 Commentary 210 x Contents 9 Measure-Theoretic Probability 211 9.1 Random Variables 211 9.1.1 Basic Definitions 211 9.1.2 Independence 215 9.1.3 Back to Densities 216 9.2 General State Markov Chains 218 9.2.1 Stochastic Kernels 218 9.2.2 The Fundamental Recursion, Again 223 9.2.3 Expectations 225 9.3 Commentary 227 10 Stochastic Dynamic Programming 229 10.1 Theory 229 10.1.1 Statement of the Problem 229 10.1.2 Optimality 231 10.1.3 Proofs 235 10.2 Numerical Methods 238 10.2.1 Value Iteration 238 10.2.2 Policy Iteration 241 10.2.3 Fitted Value Iteration 244 10.3 Commentary 246 11 Stochastic Dynamics 247 11.1 Notions of Convergence 247 11.1.1 Convergence of Sample Paths 247 11.1.2 Strong Convergence of Measures 252 11.1.3 Weak Convergence of Measures 254 11.2 Stability: Analytical Methods 257 11.2.1 Stationary Distributions 257 11.2.2 Testing for Existence 260 11.2.3 The Dobrushin Coefficient, Measure Case 263 11.2.4 Application: Credit-Constrained Growth 266 11.3 Stability: Probabilistic Methods 271 11.3.1 Coupling with Regeneration 272 11.3.2 Coupling and the Dobrushin Coefficient 276 11.3.3 Stability via Monotonicity 279 11.3.4 More on Monotonicity 283 11.3.5 Further Stability Theory 288 11.4 Commentary 293 Contents xi 12 More Stochastic Dynamic Programming 295 12.1 Monotonicity and Concavity 295 12.1.1 Monotonicity 295 12.1.2 Concavity and Differentiability 299 12.1.3 Optimal Growth Dynamics 302 12.2 Unbounded Rewards 306 12.2.1 Weighted Supremum Norms 306 12.2.2 Results and Applications 308 12.2.3 Proofs 311 12.3 Commentary 312 III Appendixes 315 A Real Analysis 317 A.1 The Nuts and Bolts 317 A.1.1 Sets and Logic 317 A.1.2 Functions 320 A.1.3 Basic Probability 324 A.2 The Real Numbers 327 A.2.1 Real Sequences 327 A.2.2 Max, Min, Sup, and Inf 331 A.2.3 Functions of a Real Variable 334 B Chapter Appendixes 339 B.1 Appendix to Chapter 3 339 B.2 Appendix to Chapter 4 342 B.3 Appendix to Chapter 6 344 B.4 Appendix to Chapter 8 345 B.5 Appendix to Chapter 10 347 B.6 Appendix to Chapter 11 349 B.7 Appendix to Chapter 12 350 Bibliography 357 Index 367 |
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感谢分享！简介如下： The aim of this book is to teach topics in economic dynamics such as simulation, stability theory, and dynamic programming. The focus is primarily on stochastic systems in discrete time. Most of the models we meet will be nonlinear, and the emphasis is on getting to grips with nonlinear systems in their original form, rather than using crude approximation techniques such as linearization. As we travel down this path, we will delve into a variety of related fields, including fixed point theory, laws of large numbers, function approximation, and coupling. |
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