Stochastic Control by Functional Analysis Methods

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Front Cover 2
Stochastic Control by Functional Analysis Methods 5
Copyright Page 6
Contents 13
CHAPTER I. STOCHASTIC CALCULUS AND STOCHASTIC DIFFERENTIAL EQUATIONS 19
- INTRODUCTION 19
- 1. PRELIMINARIES 20
- 2. STOCHASTIC INTEGRALS 28
- 3. ITO's FORMULA 38
- 4. STOCHASTIC DIFFERENTIAL EQUATIONS 50
- 5. GIRSANOV TRANSFORMATION 55
CHAPTER II. PARTIAL DIFFERENTIAL EQUATIONS 69
- INTRODUCTION 69
- 1. FUNCTIONAL SPACES 70
- 2. THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS 77

Front Cover 2
Stochastic Control by Functional Analysis Methods 5
Copyright Page 6
Contents 13
CHAPTER I. STOCHASTIC CALCULUS AND STOCHASTIC DIFFERENTIAL EQUATIONS 19
- INTRODUCTION 19
- 1. PRELIMINARIES 20
- 2. STOCHASTIC INTEGRALS 28
- 3. ITO's FORMULA 38
- 4. STOCHASTIC DIFFERENTIAL EQUATIONS 50
- 5. GIRSANOV TRANSFORMATION 55
CHAPTER II. PARTIAL DIFFERENTIAL EQUATIONS 69
- INTRODUCTION 69
- 1. FUNCTIONAL SPACES 70
- 2. THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS 77
- 3. PARABOLIC EQUATIONS 93
CHAPTER III. MARTINGALE PROBLEM 119
- INTRODUCTION 119
- 1. PROPERTIES OF CONTINUOUS MARTINGALES 119
- 2. DEFINITION OF THE MARTINGALE PROBLEM 125
- 3. EXISTENCE AND UNIQUENESS OF THE SOLUTION OF THE MARTINGALE PROBLEM 135
- 4. INTERPRETATION OF THE SOLUTION OF P.D.E. 143
- 5. SEMI GROUPS 147
CHAPTER IV. STOCHASTIC CONTROL WITH COMPLETE INFORMATION 157
- 1. SETTING OF THE PROBLEM 157
- INTRODUCTION 157
- 2. THE EQUATION OF DYNAMIC PROGRAMMING 160
- 3. SOLUTION OF THE STOCHASTIC CONTROL PROBLEM 168
- 4. EVOLUTION PROBLEMS 174
- 5. SEMI GROUP FORMULATION 179
CHAPTER V. FILTERING AND PREDICTION FOR LINEAR S.D.E. 209
- INTRODUCTION 209
- 1. SETTING OF THE PROBLEM 209
- 2. CHARACTERIZATION OF THE BEST ESTIMATE 217
- 3. RECURSIVITY - KALMAN FILTER 226
- 4. PREDICTION 237
CHAPTER VI. VARIATIONAL METHODS IN STOCHASTIC CONTROL 239
- 1. MODEL WITH ADDITIVE NOISE 239
- INTRODUCTION 239
- 2. THE CASE WITH INCOMPLETE OBSERVATION 252
- 3. SEPARATION PRINCIPLE 271
CHAPTER VII. PROBLEMS OF OPTIMAL STOPPING 297
- INTRODUCTION 297
- 1. SETTING OF THE PROBLEM 297
- 2. UNILATERAL PROBLEMS 299
- 3. VARIATIONAL INEQUALITIES 305
- 4. SOLUTION OF THE OPTIMAL STOPPING TIME PROBLEM 319
- 5. SEMI GROUP APPROACH 325
- 6. INTERPRETATION AS A PROBLEM OF OPTIMAL STOPPING 357
CHAPTER VIII. IMPULSIVE CONTROL 373
- INTRODUCTION 373
- 1. SETTING OF THE PROBLEM 374
- 2. QUASI VARIATIONAL INEQUALITIES 378
- 3. SOLUTION OF THE IMPULSIVE CONTROL PROBLEM 386
- 4. SEMI GROUP APPROACH 409
- REFERENCES 417

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