This book unifies the maximum principle and dynamic programming, the two most commonly used approaches in solving optimal control problems. The author shows that the viscosity solution theory provides the unifying framework.

This monograph unifies the two key approaches in solving optimal control problems. The book will be of interest to researchers and graduate students in applied probability, control engineering, and econometrics.

Autorentext

Jingrui Sun received his PhD in Mathematics from the University of Science and Technology of China in 2015. From 2015 to 2017, he was a Postdoctoral Fellow at the Hong Kong Polytechnic University and then a Research Fellow at the National University of Singapore. From 2017 to 2018, he was a Visiting Assistant Professor at the University of Central Florida, USA. Since the spring of 2019, he has been an Assistant Professor at the Southern University of Science and Technology, China. Dr. Sun has broad interests in the area of control theory and its applications. Aside from his primary research on stochastic optimal control and differential games, he is exploring forward and backward stochastic differential equations, stochastic analysis, and mathematical finance. Jiongmin Yong received his PhD from Purdue University in 1986 and is currently a Professor of Mathematics at the University of Central Florida, USA. His main research interests include stochastic control, stochastic differential equations, and optimal control of partial differential equations. Professor Yong has co-authored the following influential books: "Stochastic Control: Hamiltonian Systems and HJB Equations" (with X. Y. Zhou, Springer 1999), "Forward-Backward Stochastic Differential Equations and Their Applications" (with J. Ma, Springer 1999), and "Optimal Control Theory for Infinite-Dimensional Systems" (with X. Li, Birkhauser 1995). His current interests include time-inconsistent stochastic control problems.

Inhalt

1. Basic Stochastic Calculus.- 1. Probability.- 2. Stochastic Processes.- 3. Stopping Times.- 4. Martingales.- 5. Itô's Integral.- 6. Stochastic Differential Equations.- 2. Stochastic Optimal Control Problems.- 1. Introduction.- 2. Deterministic Cases Revisited.- 3. Examples of Stochastic Control Problems.- 4. Formulations of Stochastic Optimal Control Problems.- 5. Existence of Optimal Controls.- 6. Reachable Sets of Stochastic Control Systems.- 7. Other Stochastic Control Models.- 8. Historical Remarks.- 3. Maximum Principle and Stochastic Hamiltonian Systems.- 1. Introduction.- 2. The Deterministic Case Revisited.- 3. Statement of the Stochastic Maximum Principle.- 4. A Proof of the Maximum Principle.- 5. Sufficient Conditions of Optimality.- 6. Problems with State Constraints.- 7. Historical Remarks.- 4. Dynamic Programming and HJB Equations.- 1. Introduction.- 2. The Deterministic Case Revisited.- 3. The Stochastic Principle of Optimality and the HJB Equation.- 4. Other Propertiesof the Value Function.- 5. Viscosity Solutions.- 6. Uniqueness of Viscosity Solutions.- 7. Historical Remarks.- 5. The Relationship Between the Maximum Principle and Dynamic Programming.- 1. Introduction.- 2. Classical Hamilton-Jacobi Theory.- 3. Relationship for Deterministic Systems.- 4. Relationship for Stochastic Systems.- 5. Stochastic Verification Theorems.- 6. Optimal Feedback Controls.- 7. Historical Remarks.- 6. Linear Quadratic Optimal Control Problems.- 1. Introduction.- 2. The Deterministic LQ Problems Revisited.- 3. Formulation of Stochastic LQ Problems.- 4. Finiteness and Solvability.- 5. A Necessary Condition and a Hamiltonian System.- 6. Stochastic Riccati Equations.- 7. Global Solvability of Stochastic Riccati Equations.- 8. A Mean-variance Portfolio Selection Problem.- 9. Historical Remarks.- 7. Backward Stochastic Differential Equations.- 1. Introduction.- 2. Linear Backward Stochastic Differential Equations.- 3. Nonlinear Backward Stochastic Differential Equations.- 4. FeynmanKac-Type Formulae.- 5. ForwardBackward Stochastic Differential Equations.- 6. Option Pricing Problems.- 7. Historical Remarks.- References.