Contents
Preface
Chapter 1 Preliminaries 1
1.1 Probability and Random Variables 1
1.1.1 Probability Spaces 1
1.1.2 Convergence of Probabilities 4
1.2 Stochastic Processes 5
1.2.1 Continuous Time Martingales 7
1.2.2 Stochastic Integration 8
1.3 The Basic Theory of FBSDEs 9
1.3.1 A Black-Scholes Formula in Finance 11
1.3.2 Formulations of Stochastic Optimal Control Problems 12
Bibliography 13
Chapter 2 Singular Optimal Controls of Stochastic Recursive Systems and H-J-B Inequality 14
2.1 Introduction 14
2.2 Formulation of the Problem 18
2.3 Dynamic Programming Principle 21
2.4 Example 51
2.5 Appendix 52
Bibliography 55
Chapter 3 Stochastic Verifi　cation Theorem of Forward-Backward Controlled Systems for Viscosity Solutions 60
3.1 Introduction 60
3.2 Super-differentials, Sub-diffierentials, and Viscosity Solutions 65
3.3 Stochastic Verifi　cation Theorem for Forward-Backward Controlled Systems 67
3.4 Optimal Feedback Controls 71
Bibliography 73
Chapter 4 Maximum Principle for Forward-Backward Doubly Stochastic Control Systems and Applications 74
4.1 Introduction 75
4.2 Statement of the Problem 79
4.3 Variational Equations and Variational Inequalities 81
4.4 The Maximum Principle in Global Form 94
4.5 Applications to Optimal Control Problems of Stochastic PDEs 96
4.6 Linear Quadratic Nonzero Sum Doubly Stochastic Di.erential Games 100
Bibliography 104
Chapter 5 Stochastic Maximum Principle for Near-Optimal Control of FBSDEs 107
5.1 Introduction 108
5.2 Formulation of the Optimal Control Problem and Basic Assumptions 110
5.3 Main Results 112
5.3.1 Necessary Condition of Near-Optimality 112
5.3.2 Sufficient Condition of Near-Optimality 116
5.4 Examples 121
5.5 Concluding Remarks 124
5.6 Appendix 124
Bibliography 147
Chapter 6 Near Optimal Control of Stochastic Recursive Systems via Viscosity Solution 150
6.1 Introduction 150
6.2 Preliminaries and Notations 152
6.3 Main Results 157
6.4 Conclusions 166
Bibliography 169
Chapter 7 Asymptotic Properties of Coupled Forward-Backward Stochastic Di.erential Equations 172
7.1 Introduction 172
7.2 Preliminaries 174
7.3 Regularity of the solution of FBSDEs 176
7.4 Main Results 190
7.4.1 Convergence of distributions 190
7.4.2 Large deviation principle 197
Bibliography 201