隨機控制（簡體書）

本書是一部非常系統、完整的研究生教程，也是一部學習控制理論的很好的補充材料。書中有一些原創性的論題是別的圖書沒能包括的，是學習擴散過程中Pontryagin最大原理和Bellman 動態規劃原理之間關係的不二選擇。本書權威性強，其中的材料可以運用在數學、工程和金融的隨機控制和動態優化課程中。目次：基本隨機微積分；隨機優化控制問題；最大原理和隨機哈密頓體系；動態規劃和HJB方程；最大原理和動態規劃之間的關係；線性二次優化控制問題；向後隨機微分方程。讀者對象：應用數學、經濟金融和工程等相關專業的學生、老師和從業人員。

Preface
Notation
Assumption Index
Problem Index
Chapter 1. Basic Stochastic Calculus
1. Probability
1.1. Probability spaces
1.2. Random variables
1.3. Conditional expectation
1.4. Convcrgence of probabilities
2. Stochastic Processes
2.1. General considerations
2.2. Brownian motions
3. Stopping Times
4. Martingales
5. ItS's Integral
5.1. Nondifferentiability of Brownian motion
5.2. Definition of Ites integral and basic properties
5.3. ItS's formula
5.4. Martingale representation theorems
6. Stochastic Differential Equations
6.1. Strong solutions
6.2. Weak solutions
6.3. Linear SDEs
6.4. Other types of SDEs
Chapter 2. Stochastic Optimal Control Problems
1. Introduction
2. Deterministic Cases Revisited
3. Examples of Stochastic Control Problems
3. 1. Production planning
3.2. Investment vs. consumption
3.3. Reinsurance and dividend management
3.4. Technology diffusion
3.5. Queueing systems in heavy traffic
4. Formulations of Stochastic Optimal Control Problems
4.1. Strong formulation
4.2. Weak formulation
5. Existence of Optimal Controls
5.1. A deterministic result
5.2. Existence under strong formulation
5.3. Existence under weak formulation
6. Reachable Sets of Stochastic Control Systems
6.1. Nonconvexity of the reachable sets
6.2. Nonclnseness of the reachable sets
7. Other Stochastic Control Models
7.1. Random duration
7.2. Optimal stopping
7.3. Singular and impulse controls
7.4. Risk-sensitive controls
7.5. Ergodic controls
7.6. Partially observable systems
8. Historical Remarks
Chapter 3. Maximum Principle and Stochastic
Hamiitonian Systems
1. Introduction
2. The Deterministic Case Rcvisited
3. Statement of the Stochastic Maximum Principle
3.1. Adjoint equations
3.2. The maximum principle and stochastic
Hamiltonian systems
3.3. A worked-out example
4. A Proof of the Maximum Principle
4.1. A moment estimate
4.2. Taylor expansions
4.3. Duality analysis and complction of thc proof
5. Sufficient Conditions of Optimality
6. Problems with Statc Constraints
6.1. Formulation of the problem and the maximum principle
6.2. Some preliminary lemmas
6.3. A proof of Theorem 6.1
7. Historical Remarks
Chapter 4. Dynamic Programming and HJB Equations
1. Introduction
2. The Deterministic Casc Revisited
3. The Stochastic Principle of Optimality and the HJB Equation
3.1. A stochastic framework for dynamic programming
3.2. Principlc of optimality
3.3. The HJB cquation
4. Other Properties of the Value Function
4.1. Continuous dependence on parameters
4.2. Semiconcavity
5. Viseo~ity Solutions
5.1. Definitions
5.2. Some properties
6. Uniqueness of Viscosity Solutions
6.1. A uniqueness theorem
6.2. Proofs of Lemmas 6.6 and 6.7
7. Historical Rcmarks
Chapter 5. The Relationship Between the Maximum
Principle and Dynamic Programming
1. Introduction
2. Classical Hamilton-Jacobi Theory
3. Relationship for Deterministic Systems
3.1. Adjoint variable and value function： Smooth case
3.2. Economic interpretation
3.3. Methods of characteristics and the Fcynman Kac formula
3.4. Adjoint variable and value function： Nonsmooth case
3.5. Vcrification theorems
4. Relationship for Stochastic Systems
4.1. Smooth case
4.2. Nonsmooth case： Differentials in the spatial variable
4.3. Nonsmooth case： Differentials in the time variable
5. Stochastic Vcrification Theorems
5.1. Smooth case
5.2. Nonsmooth case
6. Optimal Fccdback Controls
7. Historical Remarks
Chapter 6. Linear Quadratic Optimal Control Problems
1. Introduction
2. The Deterministic LQ Problems Revisited
2.1. Formulation
2.2. A minimization problem of a quadratic functional
2.3. A linear Hamiltonian system
2.4. The Riccati equation and feedback optimal control
3. FormuLation of Stochastic LQ Problems
3.1. Statement of the problems
3.2. Examples
4. Finiteness and Solvability
5. A Necessary Condition and a Hamiltonian System
6. Stochastic Riceati Equations
7. GLobal Solvability of Stochastic Riccati EQuations
7.1. Existence： Thc standard case
7.2. Existence： The case C = 0， S = 0， and Q，G >_ 0
7.3. Existence： The one-dimensional case
8. A Mean-variance Portfolio Selection Problem
9. Historical Remarks
Chapter 7. Backward Stochastic Differential Equations
1. Introduction
2. Linear Backward Stochastic Differential EQuations
3. Nonlinear Backward Stochastic Differential Equations
3.1. BSDEs in finite deterministic durations： Method of
contraction mapping
3.2. BSDEs in random durations： Method of continuation
4. Feynman-Kac-Type Formulae
4.1. Representation via SDEs
4.2. Representation via BSDEs
5. Forward-Backward Stochastic Differential Equations
5.1. General formulation and nonsolvability
5.2. The four-step scheme， a heuristic derivation
5.3. Several solvable classes of FBSDEs
6. Option Pricing Problems
6.1. European call options and the Black-Scholes formula
6.2. Other options
7. Historical Remarks
References
Index