STOCHASTIC MODELS OF CONTROL AND ECONOMIC DYNAMICS

Transcription

1 STOCHASTIC MODELS OF CONTROL AND ECONOMIC DYNAMICS V. I. ARKIN I. V. EVSTIGNEEV Central Economic Mathematics Institute Academy of Sciences of the USSR Moscow, Vavilova, USSR Translated and Edited by E. A. MEDOVA-DEMPSTER Department of Applied Mathematics Technical University of Nova Scotia Halifax, Nova Scotia, Canada M. A. H. DEMPSTER Department of Mathematics Statistics and Computing Science and School of Business Administration Dalhouse University Halifax, Nova Scotia, Canada and Balliol College Oxford, England 1987 ACADEMIC PRESS Harcourt Brace Jovanovich, Publishers. London Orlando San Diego New York Austin Boston Sydney Tokyo Toronto

2 CONTENTS Preface Preface to the English Edition Translators'Preface xi xvii xix 1 DETERMINISTIC MODELS 1. The Gale Model 1 1. Technology, Utility Functions and Programmes 1 2. Optimal Programmes over a Finite Horizon 2 3. Optimal Programmes over an Infinite Horizon 3 4. Prices 4 5. The Concept of Supporting Prices 5 6. Stationary Models 5 7. An Overview of the Theory of Stationary Models 6 8. A Parametric Form for Technology Description 7 2. Optimal Finite Horizon Programmes 8 1. The Existence of Supporting Prices 8 2. Supporting Prices as Lagrange Multipliers An Inequality Weak Turnpike Theorems The Programming Problem for the Stationary Model Stationary Prices Supporting the Turnpike Statement of the Weak Turnpike Theorem for Finite Programmes Proof of Theorem 3: The Pseudometric p Proof of Lemma Proof of Lemma The Existence of Good Infinite Programmes The Turnpike Theorem for Good Programmes 19

3 vi Contents 4. Optimal Infinite Horizon Programmes The Brock H Functional Properties of the H Functional Is an Optimal Programme Strong Turnpike Theorems and Winter Programmes Formulations and Discussion Proof of Theorem 6: Construction of Winter Programmes The Turnpike Theorem for the Pseudometric p Estimation of P Proof of Lemma Reduction of Constant Growth Models to Stationary Models and Some Examples A Reduction Scheme A Model with Utility Function Defined on the Consumption Set The One-Sector Model Optimal Control Problems and Models of Economic Dynamics Formulation of the Optimal Control Problem The Discrete Maximum Principle The Gale Model and Optimal Control A Model of the Dynamic Distribution of Resources 36 Comments on Chapter THE MAXIMUM PRINCIPLE FOR STOCHASTIC MODELS OF OPTIMAL CONTROL AND ECONOMIC DYNAMICS 1. Statement of the Optimal Control Problem and Formulation of the Stochastic Maximum Principle The Optimal Control Problem Problem Assumptions Formulation of the Maximum Principle Comments on the Problem Assumptions Smoothly Convex Optimization Problems with Operator Constraints Description of the Problem and Definition of a Local Maximum Statement of the Main Result Preliminary Comments on the Proof The Functional / Defines the Required Lagrange Multipliers The Set C Has Interior Points The Interior of C Does Not Intersect the Set M The Regular Case Proof of the Maximum Principle Reduction to a Smoothly Convex Problem The Maximum Principle with Respect to Functionals in Z,* The Integral Form of the Maximum Principle Reduction to the Pointwise Maximum Principle 62

4 Contents vii 4. Stochastic Analogues of the Gale Model Technology, Objective Functionals and Programmes The Existence of Optimal Programmes Supporting Prices Generalized Prices Construction of Prices of Integral Type When Prices of Integral Type Do Not Exist Model Definition in Parametric Form The Relations between Parametric and Functional Conditions Reduction to the General Problem of Optimal Control Models Described in Terms of Elementary Technological Processes 77 Comments on Chapter MARKOV CONTROL: THE MAXIMUM PRINCIPLE AND DYNAMIC PROGRAMMING 1. The Sufficiency of Markov Control Markov Optimal Control The Basic Lemma The Induction Hypothesis for the Proof of the Sufficiency Theorem Preparation for the Application of the Basic Lemma Application of the Basic Lemma Completion of the Proof of Theorem The Case of a Process of Independent Random Variables The Maximum Principle for Markov Controls Statement of the Maximum Principle Auxiliary Results Proof of Theorem A Simple Problem of Optimal Control The Method of Dynamic Programming and Its Connection with the Maximum Principle The Basic Idea of Dynamic Programming The Bellman Value Function The Bellman Functional Equation Proof of Theorem The Connection between the Bellman Equation and the Maximum Principle Construction of Markov Controls The Problem of Constructing Markov Controls The Linear Convex Model A Lemma on Markov Dependency Proof of Theorem Markov Programmes in the Gale Model Two Lemmas Proof of Theorem 7 112

5 viii Contents 5. Markov Prices in Models of Economic Dynamics Markov Prices in the Gale Model The Stochastic Analogue of the Model of the Dynamic Distribution of Resources 115 Comments on Chapter OPTIMAL ECONOMIC PLANNING OVER AN INFINITE HORIZON: WEAK TURNPIKE THEOREMS 1. The Stationary Infinite Horizon Model Preliminary Comments Definition of the Stationary Model Stationary Programmes and Prices Assumptions for Stationary Models The Turnpike and Its Supporting Prices The Existence of Turnpike Programmes Stationary Generalized Prices Supporting the Turnpike Stationary Prices Supporting the Turnpike Uniform Strict Concavity and Uniform Continuity Conditions for Utility Functionals Definition of Concavity Conditions Integral Functionals Possessing Properties (F.I) and (F.2) Nonintegral Functionals Possessing Properties (F.I) and (F.2) Uniform Continuity Conditions Weak Turnpike Theorems Preliminary Comments Statement and Discussion of Results Proof of Theorems 4 and The Turnpike Theorem for Good Programmes ' The Existence of an Optimal Programme over an Infinite Horizon Statement of the Existence Theorem and Construction of the Brock H Functional Transition from the H Functional to the Sum Y.F, '41 Comments on Chapter APPROXIMATION OF PROGRAMMES-AND STRONG TURNPIKE THEOREMS 1. Approximation of Programmes The Canonical Approximation Scheme A Sufficient Condition for the Canonical Approximation to Be a Programme Some Estimates Forward a-approximation Backward a-approximation 148

6 Contents ix 2. Winter Programmes Assumptions and Statement of the Existence Result Sketch of the Proof Formal Description of the Programme C X, Is a Programme Utility Estimates for The Strong Turnpike Theorem Statement of the Result The Turnpike Theorem for the Pseudometric p, Estimation of 2 p, Proof of Lemma The Model with History Beginning at Description of the Model The Model M' with History Beginning at oo and Corresponding to the Model M Existence of Majorizing M-Programmes The Turnpike Results Concerning Prices 161 Comments on Chapter APPENDICES I II MEASURABLE SELECTION THEOREMS AND THEIR APPLICATIONS 1. Basic Definitions Lemmas Needed for the Reduction of the Measurable Selection Theorem to a Special Case The Special Case of the Measurable Selection Theorem Proof of the Measurable Selection Theorem Some Corollaries of the Measurable Selection Theorem 169 CONDITIONAL DISTRIBUTIONS 1. Existence Theorem for Conditional Distributions Reduction of the Existence Theorem to a Special Case Liftings Proof of the Special Case of the Existence Theorem Random Convex Sets and the Generalized Jensen's Inequality 176 III SOME GENERAL RESULTS FROM MEASURE THEORY AND FUNCTIONAL ANALYSIS 1. The Separation Theorem The Kuhn-Tucker Theorem Liusternik's Theorem L p Spaces Komlos's Theorem and Its Application to Optimization Problems The Yosida Hewitt Theorem Monotone Class Theorems 184 References. 187 Further References 195 Index 203