Stochastic control by functional analysis methods:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam [u.a.]
North-Holland Publ.
1982
|
| Schriftenreihe: | Studies in mathematics and its applications
11 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000006137&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XV, 410 S. |
| ISBN: | 044486329X |
Internformat
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| 100 | 1 | |a Bensoussan, Alain |d 1940- |e Verfasser |0 (DE-588)13295947X |4 aut | |
| 245 | 1 | 0 | |a Stochastic control by functional analysis methods |c Alain Bensoussan |
| 264 | 1 | |a Amsterdam [u.a.] |b North-Holland Publ. |c 1982 | |
| 300 | |a XV, 410 S. | ||
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| 490 | 1 | |a Studies in mathematics and its applications |v 11 | |
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Datensatz im Suchindex
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| adam_text | xi
CONTENTS
CHAPTER I. STOCHASTIC CALCULUS AND STOCHASTIC DIFFERENTIAL
EQUATIONS 1
INTRODUCTION 1
1. PRELIMINARIES 2
1.1. Random variables 2
1.2. Expectation Conditional expectation 5
1.3. Stochastic processes 7
1.4. Martingales 9
2. STOCHASTIC INTEGRALS 10
2.1. Wiener process 10
2.2. Construction of the stochastic integral 12
2.3. Stochastic process defined by a stochastic integral 16
2.4. Extension of the stochastic integral 18
3. ITO s FORMULA 20
4. STOCHASTIC DIFFERENTIAL EQUATIONS 32
4.1. Setting of the problem 32
4.2. Lipschitz case 33
5. GIRSANOV TRANSFORMATION 37
5.1. Fundamental lemma 37
5.2. Girsanov s Theorem 42
5.3. Application to the concept of weak solution of a
stochastic differential equation 46
CHAPTER II. PARTIAL DIFFERENTIAL EQUATIONS 51
INTRODUCTION 51
1. FUNCTIONAL SPACES 52
1.1. Sobolev spaces 52
1.2. Concept of trace 54
1.3. Green s formula 57
Xii CONTENTS
2. THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS 59
2.1. The basic existence and uniqueness results 59
2.2. Variational techniques 64
3. PARABOLIC EQUATIONS 75
3.1. Functional spaces 75
3.2. Variational formulation 78
3.3. Regularity 82
3.3.1. Regularity with respect to time 82
3.3.2. Regularity with respect to space variables 84
3.3.3. Other regularity results 87
3.4. The Cauchy problem 94
CHAPTER HI. MARTINGALE PROBLEM 101
INTRODUCTION 101
1. PROPERTIES OF CONTINUOUS MARTINGALES 101
1.1. Square integrable continuous martingales 101
1.2. Stochastic integrals 104
1.3. A representation theorem for continuous martingales 105
2. DEFINITION OF THE MARTINGALE PROBLEM 107
2.1. Setting of the problem 107
2.2. Properties of the martingale problem 108
2.3. Some a priori estimates 112
3. EXISTENCE AND UNIQUENESS OF THE SOLUTION OF THE MARTINGALE
PROBLEM 117
3.1. Uniqueness 117
3.2. Existence 122
4. INTERPRETATION OF THE SOLUTION OF P.D.E. 125
4.1. Elliptic equations 125
4.2. Parabolic equations 127
5. SEMI GROUPS 129
5.1. Semi group of diffusions 129
5.2. Stopped diffusions 133
CONTENTS xiii
CHAPTER IV. STOCHASTIC CONTROL WITH COMPLETE INFORMATION 139
INTRODUCTION 139
1. SETTING OF THE PROBLEM 139
1.1. Notation. Assumptions 139
1.2. Controlled martingale 140
2. THE EQUATION OF DYNAMIC PROGRAMMING 142
Orientation 142
2.1. Notation 142
2.2. Study of the H.J.B. equation 143
3. SOLUTION OF THE STOCHASTIC CONTROL PROBLEM 150
4. EVOLUTION PROBLEMS 156
4.1. Parabolic equations 156
4.2. Stochastic control problem 158
4.3. Martingale property 160
5. SEMI GROUP FORMULATION 161
5.1. A property of the equation u 161
5.2. The problem of semi group envelope 164
5.3. Interpretation of the discretized problem 170
5.4. A regularity result 178
CHAPTER V. FILTERING AND PREDICTION FOR LINEAR S.D.E. 191
INTRODUCTION 191
1. SETTING OF THE PROBLEM 191
1.1. State equation 191
1.2. The observation process 195
1.3. Statement of the problem 197
2. CHARACTERIZATION OF THE BEST ESTIMATE 199
2.1. Main result 199
2.2. Quadratic optimization problems 203
3. RECURSIVITY KALMAN FILTER 208
3.1. A general system 208
3.2. Recursive formulas 210
3.3. Kalman filter 213
3.4. The innovation process 216
4. PREDICTION 219
xiv CONTENTS
CHAPTER VI. VARIATIONS. METHODS IN STOCHASTIC CONTROL 221
INTRODUCTION 221
1. MODEL WITH ADDITIVE NOISE 221
1.1. Setting of the model 221
1.2. A density result 224
1.3. Necessary conditions 227
2. THE CASE WITH INCOMPLETE OBSERVATION 234
2.1. Statement of the problem 234
2.2. Preliminary results 237
2.3. Necessary conditions 240
2.4. The quadratic case. Separation principle 243
3. SEPARATION PRINCIPLE 253
Orientation 253
3.1. The Kaiman filter 254
3.2. A dynamic programming equation 255
3.3. Solution of the stochastic control problem with
incomplete information 269
CHAPTER VII. PROBLEMS OF OPTIMAL STOPPING 279
INTRODUCTION 279
1. SETTING OF THE PROBLEM 279
1.1. Assumptions. Notation 279
1.2. Remarks 281
2. UNILATERAL PROBLEMS 281
2.1. Penalized problem 281
2.2. Limit problem in the regular case 282
3. VARIATIONAL INEQUALITIES 287
Orientation 287
3.1. Limit of the penalized problem 288
3.2. Weakening the coercitivity assumption 290
4. SOLUTION OF THE OPTIMAL STOPPING TIME PROBLEM 301
4.1. The regular case 301
4.2. The non regular case 303
4.3. Interpretation of the penalized problem 305
5. SEMI GROUP APPROACH 307
5.1. Solution of the V.I. as a maximum element 307
5.2. The case of general semi group 315
CONTENTS XV
5.3. Discretization 325
5.4. Case without continuity 335
6. INTERPRETATION AS A PROBLEM OF OPTIMAL STOPPING 339
6.1. Markov process related to the serai group 339
6.2. Problem of optimal stopping 341
6.3. Interpretation of the discretized problem 345
CHAPTER VIII. IMPULSIVE CONTROL 355
INTRODUCTION 355
1. SETTING OF THE PROBLEM 355
1.1. Assumptions. Notation 356
1.2. The model 357
1.3. Some remarks 359
2. QUASI VARIATIONAL INEQUALITIES 360
Orientation 360
2.1. Setting of the problem 360
2.2. Solution of the Q.V.I. 361
3. SOLUTION OF THE IMPULSIVE CONTROL PROBLEM 368
3.1. The main result 368
3.2. Interpretation of the decreasing scheme 337
4. SEMI GROUP APPROACH 391
4.1. Solution of the Q.V.I, as a maximum element 391
4.2. Case of general semi groups 392
4.3. Discretization 395
REFERENCES 399
|
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| id | DE-604.BV000033362 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T07:14:22Z |
| institution | BVB |
| isbn | 044486329X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000006137 |
| oclc_num | 7945040 |
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| owner | DE-12 DE-91 DE-BY-TUM DE-384 DE-703 DE-739 DE-355 DE-BY-UBR DE-824 DE-29T DE-706 DE-188 DE-83 |
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| physical | XV, 410 S. |
| publishDate | 1982 |
| publishDateSearch | 1982 |
| publishDateSort | 1982 |
| publisher | North-Holland Publ. |
| record_format | marc |
| series | Studies in mathematics and its applications |
| series2 | Studies in mathematics and its applications |
| spellingShingle | Bensoussan, Alain 1940- Stochastic control by functional analysis methods Studies in mathematics and its applications Analyse stochastique ram Commande, théorie de la ram Stochastic control theory Steuerung (DE-588)4057472-6 gnd Stochastische Kontrolltheorie (DE-588)4263657-7 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4057472-6 (DE-588)4263657-7 (DE-588)4057630-9 (DE-588)4018916-8 |
| title | Stochastic control by functional analysis methods |
| title_auth | Stochastic control by functional analysis methods |
| title_exact_search | Stochastic control by functional analysis methods |
| title_full | Stochastic control by functional analysis methods Alain Bensoussan |
| title_fullStr | Stochastic control by functional analysis methods Alain Bensoussan |
| title_full_unstemmed | Stochastic control by functional analysis methods Alain Bensoussan |
| title_short | Stochastic control by functional analysis methods |
| title_sort | stochastic control by functional analysis methods |
| topic | Analyse stochastique ram Commande, théorie de la ram Stochastic control theory Steuerung (DE-588)4057472-6 gnd Stochastische Kontrolltheorie (DE-588)4263657-7 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Analyse stochastique Commande, théorie de la Stochastic control theory Steuerung Stochastische Kontrolltheorie Stochastischer Prozess Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000006137&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000646 |
| work_keys_str_mv | AT bensoussanalain stochasticcontrolbyfunctionalanalysismethods |
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