Stochastic models of control and economic dynamics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | Undetermined |
| Published: |
London u.a.
Academic Pr.
1987
|
| Series: | Economic theory, econometrics, and mathematical economics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | EST: Verojatnostnye modeli upravlenija i ėkonomičeskoj dinamiki <engl.> |
| Physical Description: | XXI, 208 S. |
| ISBN: | 0120620804 |
Staff View
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| adam_text | IMAGE 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
IMAGE 2
CONTENTS
PREFACE XI
PREFACE TO THE ENGLISH EDITION XVII
TRANSLATORS PREFACE 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 11
3. AN INEQUALITY 12
3. WEAK TURNPIKE THEOREMS 12
1. THE PROGRAMMING PROBLEM FOR THE STATIONARY MODEL 12
2. STATIONARY PRICES SUPPORTING THE TURNPIKE 13
3. STATEMENT OF THE WEAK TURNPIKE THEOREM FOR FINITE PROGRAMMES 14 4.
PROOF OF THEOREM 3: THE PSEUDOMETRIC P 15
5. PROOF OF LEMMA 3 16
6. PROOF OF LEMMA 4 17
7. THE EXISTENCE OF GOOD INFINITE PROGRAMMES 18
8. THE TURNPIKE THEOREM FOR GOOD PROGRAMMES 19
IMAGE 3
VI CONTENTS
4. OPTIMAL INFINITE HORIZON PROGRAMMES 20
1. THE BROCK H FUNCTIONAL 20
2. PROPERTIES OF THE H FUNCTIONAL 21
3. IS AN OPTIMAL PROGRAMME 22
5. STRONG TURNPIKE THEOREMS AND WINTER PROGRAMMES 23
1. FORMULATIONS AND DISCUSSION 23
2. PROOF OF THEOREM 6: CONSTRUCTION OF WINTER PROGRAMMES 24 3. THE
TURNPIKE THEOREM FOR THE PSEUDOMETRIC P 25
4. ESTIMATION OF P 25
5. PROOF OF LEMMA 10 26
6. REDUCTION OF CONSTANT GROWTH MODELS TO STATIONARY MODELS AND SOME
EXAMPLES 27
1. A REDUCTION SCHEME 27
2. A MODEL WITH UTILITY FUNCTION DEFINED ON THE CONSUMPTION SET 28 3.
THE ONE-SECTOR MODEL 29
7. OPTIMAL CONTROL PROBLEMS AND MODELS OF ECONOMIC DYNAMICS 30
1. FORMULATION OF THE OPTIMAL CONTROL PROBLEM 30
2. THE DISCRETE MAXIMUM PRINCIPLE 31
3. THE GALE MODEL AND OPTIMAL CONTROL 35
4. A MODEL OF THE DYNAMIC DISTRIBUTION OF RESOURCES 36
COMMENTS ON CHAPTER 1 37
2 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 39
1. THE OPTIMAL CONTROL PROBLEM 39
2. PROBLEM ASSUMPTIONS 41
3. FORMULATION OF THE MAXIMUM PRINCIPLE 42
4. COMMENTS ON THE PROBLEM ASSUMPTIONS 43
2. SMOOTHLY CONVEX OPTIMIZATION PROBLEMS WITH OPERATOR CONSTRAINTS 46
1. DESCRIPTION OF THE PROBLEM AND DEFINITION OF A LOCAL MAXIMUM 46 2.
STATEMENT OF THE MAIN RESULT 47
3. PRELIMINARY COMMENTS ON THE PROOF 48
4. THE FUNCTIONAL / DEFINES THE REQUIRED LAGRANGE MULTIPLIERS 49 5. THE
SET C HAS INTERIOR POINTS 50
6. THE INTERIOR OF C DOES NOT INTERSECT THE SET M 50
7. THE REGULAR CASE 52
3. PROOF OF THE MAXIMUM PRINCIPLE 53
1. REDUCTION TO A SMOOTHLY CONVEX PROBLEM 53
2. THE MAXIMUM PRINCIPLE WITH RESPECT TO FUNCTIONALS IN Z,* 57 3. THE
INTEGRAL FORM OF THE MAXIMUM PRINCIPLE 59
4. REDUCTION TO THE POINTWISE MAXIMUM PRINCIPLE 62
IMAGE 4
CONTENTS VII
4. STOCHASTIC ANALOGUES OF THE GALE MODEL 63
1. TECHNOLOGY, OBJECTIVE FUNCTIONALS AND PROGRAMMES 63
2. THE EXISTENCE OF OPTIMAL PROGRAMMES 65
3. SUPPORTING PRICES 66
4. GENERALIZED PRICES 67
5. CONSTRUCTION OF PRICES OF INTEGRAL TYPE 68
6. WHEN PRICES OF INTEGRAL TYPE DO NOT EXIST 70
7. MODEL DEFINITION IN PARAMETRIC FORM 72
8. THE RELATIONS BETWEEN PARAMETRIC AND FUNCTIONAL CONDITIONS 73 9.
REDUCTION TO THE GENERAL PROBLEM OF OPTIMAL CONTROL 76
10. MODELS DESCRIBED IN TERMS OF ELEMENTARY TECHNOLOGICAL PROCESSES 77
COMMENTS ON CHAPTER 2 78
3 MARKOV CONTROL: THE MAXIMUM PRINCIPLE AND DYNAMIC PROGRAMMING
1. THE SUFFICIENCY OF MARKOV CONTROL 81
1. MARKOV OPTIMAL CONTROL 81
2. THE BASIC LEMMA 83
3. THE INDUCTION HYPOTHESIS FOR THE PROOF OF THE SUFFICIENCY THEOREM 84
4. PREPARATION FOR THE APPLICATION OF THE BASIC LEMMA 85
5. APPLICATION OF THE BASIC LEMMA 86
6. COMPLETION OF THE PROOF OF THEOREM 1 87
7. THE CASE OF A PROCESS OF INDEPENDENT RANDOM VARIABLES 87
2. THE MAXIMUM PRINCIPLE FOR MARKOV CONTROLS 88
1. STATEMENT OF THE MAXIMUM PRINCIPLE 88
2. AUXILIARY RESULTS 90
3. PROOF OF THEOREM 2 92
4. A SIMPLE PROBLEM OF OPTIMAL CONTROL 94
3. THE METHOD OF DYNAMIC PROGRAMMING AND ITS CONNECTION WITH THE MAXIMUM
PRINCIPLE 97
1. THE BASIC IDEA OF DYNAMIC PROGRAMMING 97
2. THE BELLMAN VALUE FUNCTION 98
3. THE BELLMAN FUNCTIONAL EQUATION 99
4. PROOF OF THEOREM 4 100
5. THE CONNECTION BETWEEN THE BELLMAN EQUATION AND THE MAXIMUM PRINCIPLE
104
4. CONSTRUCTION OF MARKOV CONTROLS 107
1. THE PROBLEM OF CONSTRUCTING MARKOV CONTROLS 107
2. THE LINEAR CONVEX MODEL 107
3. A LEMMA ON MARKOV DEPENDENCY 108
4. PROOF OF THEOREM 6 109
5. MARKOV PROGRAMMES IN THE GALE MODEL 110
6. TWO LEMMAS 111
7. PROOF OF THEOREM 7 112
IMAGE 5
VIII CONTENTS
5. MARKOV PRICES IN MODELS OF ECONOMIC DYNAMICS 113
1. MARKOV PRICES IN THE GALE MODEL 113
2. THE STOCHASTIC ANALOGUE OF THE MODEL OF THE DYNAMIC DISTRIBUTION OF
RESOURCES 115
COMMENTS ON CHAPTER 3 117
4 OPTIMAL ECONOMIC PLANNING OVER AN INFINITE HORIZON: WEAK TURNPIKE
THEOREMS
1. THE STATIONARY INFINITE HORIZON MODEL 119
1. PRELIMINARY COMMENTS 119
2. DEFINITION OF THE STATIONARY MODEL 120
3. STATIONARY PROGRAMMES AND PRICES 122
4. ASSUMPTIONS FOR STATIONARY MODELS 123
2. THE TURNPIKE AND ITS SUPPORTING PRICES 123
1. THE EXISTENCE OF TURNPIKE PROGRAMMES 123
2. STATIONARY GENERALIZED PRICES SUPPORTING THE TURNPIKE 124
3. STATIONARY PRICES SUPPORTING THE TURNPIKE 126
3. UNIFORM STRICT CONCAVITY AND UNIFORM CONTINUITY CONDITIONS FOR
UTILITY FUNCTIONALS 128
1. DEFINITION OF CONCAVITY CONDITIONS 128
2. INTEGRAL FUNCTIONALS POSSESSING PROPERTIES (F.I) AND (F.2) 129
3. NONINTEGRAL FUNCTIONALS POSSESSING PROPERTIES (F.I) AND (F.2) 131 4.
UNIFORM CONTINUITY CONDITIONS 132
4. WEAK TURNPIKE THEOREMS 133
1. PRELIMINARY COMMENTS 133
2. STATEMENT AND DISCUSSION OF RESULTS 134
3. PROOF OF THEOREMS 4 AND 5 136
4. THE TURNPIKE THEOREM FOR GOOD PROGRAMMES 138
5. THE EXISTENCE OF AN OPTIMAL PROGRAMME OVER AN INFINITE HORIZON 139
1. STATEMENT OF THE EXISTENCE THEOREM AND CONSTRUCTION OF THE BROCK H
FUNCTIONAL 139
2. TRANSITION FROM THE H FUNCTIONAL TO THE SUM Y.F, 41
COMMENTS ON CHAPTER 4 142
5 APPROXIMATION OF PROGRAMMES-AND STRONG TURNPIKE THEOREMS
1. APPROXIMATION OF PROGRAMMES 143
1. THE CANONICAL APPROXIMATION SCHEME 143
2. A SUFFICIENT CONDITION FOR THE CANONICAL APPROXIMATION TO BE A
PROGRAMME 144
3. SOME ESTIMATES 145
4. FORWARD A-APPROXIMATION 146
5. BACKWARD A-APPROXIMATION 148
IMAGE 6
CONTENTS IX
2. WINTER PROGRAMMES 149
1. ASSUMPTIONS AND STATEMENT OF THE EXISTENCE RESULT 149
2. SKETCH OF THE PROOF 150
3. FORMAL DESCRIPTION OF THE PROGRAMME C 152
4. X, IS A PROGRAMME 152
5. UTILITY ESTIMATES FOR 153
3. THE STRONG TURNPIKE THEOREM 155
1. STATEMENT OF THE RESULT 155
2. THE TURNPIKE THEOREM FOR THE PSEUDOMETRIC P, 156
3. ESTIMATION OF 2 P, 157
4. PROOF OF LEMMA 5 158
4. THE MODEL WITH HISTORY BEGINNING AT 0 158
1. DESCRIPTION OF THE MODEL 158
2. THE MODEL M WITH HISTORY BEGINNING AT - OO AND CORRESPONDING TO THE
MODEL M 159
3. EXISTENCE OF MAJORIZING M-PROGRAMMES 159
4. THE TURNPIKE 160
5. RESULTS CONCERNING PRICES 161
COMMENTS ON CHAPTER 5 162
APPENDICES
I MEASURABLE SELECTION THEOREMS AND THEIR APPLICATIONS
1. BASIC DEFINITIONS 165
2. LEMMAS NEEDED FOR THE REDUCTION OF THE MEASURABLE SELECTION THEOREM
TO A SPECIAL CASE 166
3. THE SPECIAL CASE OF THE MEASURABLE SELECTION THEOREM 168
4. PROOF OF THE MEASURABLE SELECTION THEOREM 169
5. SOME COROLLARIES OF THE MEASURABLE SELECTION THEOREM 169
II CONDITIONAL DISTRIBUTIONS 1. EXISTENCE THEOREM FOR CONDITIONAL
DISTRIBUTIONS 173
2. REDUCTION OF THE EXISTENCE THEOREM TO A SPECIAL CASE 173
3. LIFTINGS 175
4. PROOF OF THE SPECIAL CASE OF THE EXISTENCE THEOREM 175
5. RANDOM CONVEX SETS AND THE GENERALIZED JENSEN S INEQUALITY 176
III SOME GENERAL RESULTS FROM MEASURE THEORY AND FUNCTIONAL ANALYSIS 1.
THE SEPARATION THEOREM 179
2. THE KUHN-TUCKER THEOREM 179
3. LIUSTERNIK S THEOREM 181
4. L P SPACES 181
5. KOMLOS S THEOREM AND ITS APPLICATION TO OPTIMIZATION PROBLEMS 182 6.
THE YOSIDA HEWITT THEOREM 183
7. MONOTONE CLASS THEOREMS 184
REFERENCES . 187
FURTHER REFERENCES 195
INDEX 203
|
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| id | DE-604.BV000812946 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:19:52Z |
| institution | BVB |
| isbn | 0120620804 |
| language | Undetermined |
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| physical | XXI, 208 S. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Academic Pr. |
| record_format | marc |
| series2 | Economic theory, econometrics, and mathematical economics |
| spelling | Arkin, Vadim I. Verfasser aut Stochastic models of control and economic dynamics V. I. Arkin ; I. V. Evstigneev London u.a. Academic Pr. 1987 XXI, 208 S. txt rdacontent n rdamedia nc rdacarrier Economic theory, econometrics, and mathematical economics EST: Verojatnostnye modeli upravlenija i ėkonomičeskoj dinamiki <engl.> Stochastische Kontrolltheorie (DE-588)4263657-7 gnd rswk-swf Ökonometrie (DE-588)4132280-0 gnd rswk-swf Stochastische Kontrolltheorie (DE-588)4263657-7 s Ökonometrie (DE-588)4132280-0 s DE-604 Estigneev, I. V. Verfasser aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000509853&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arkin, Vadim I. Estigneev, I. V. Stochastic models of control and economic dynamics Stochastische Kontrolltheorie (DE-588)4263657-7 gnd Ökonometrie (DE-588)4132280-0 gnd |
| subject_GND | (DE-588)4263657-7 (DE-588)4132280-0 |
| title | Stochastic models of control and economic dynamics |
| title_auth | Stochastic models of control and economic dynamics |
| title_exact_search | Stochastic models of control and economic dynamics |
| title_full | Stochastic models of control and economic dynamics V. I. Arkin ; I. V. Evstigneev |
| title_fullStr | Stochastic models of control and economic dynamics V. I. Arkin ; I. V. Evstigneev |
| title_full_unstemmed | Stochastic models of control and economic dynamics V. I. Arkin ; I. V. Evstigneev |
| title_short | Stochastic models of control and economic dynamics |
| title_sort | stochastic models of control and economic dynamics |
| topic | Stochastische Kontrolltheorie (DE-588)4263657-7 gnd Ökonometrie (DE-588)4132280-0 gnd |
| topic_facet | Stochastische Kontrolltheorie Ökonometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000509853&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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