## ＜電子ブック＞Stochastic Controls : Hamiltonian Systems and HJB Equations

責任表示 by Jiongmin Yong, Xun Yu Zhou Yong, Jiongmin Zhou, Xun Yu SpringerLink (Online service) English (英語) Springer New York Imprint: Springer 1999- New York, NY, United States シリーズ Applications of Mathematics, Stochastic Modelling and Applied Probability ; 43 As is well known, Pontryagin's maximum principle and Bellman's dynamic programming are the two principal and most commonly used approaches in solving stochastic optimal control problems. * An interest...ing phenomenon one can observe from the literature is that these two approaches have been developed separately and independently. Since both methods are used to investigate the same problems, a natural question one will ask is the fol lowing: (Q) What is the relationship betwccn the maximum principlc and dy namic programming in stochastic optimal controls? There did exist some researches (prior to the 1980s) on the relationship between these two. Nevertheless, the results usually werestated in heuristic terms and proved under rather restrictive assumptions, which were not satisfied in most cases. In the statement of a Pontryagin-type maximum principle there is an adjoint equation, which is an ordinary differential equation (ODE) in the (finite-dimensional) deterministic case and a stochastic differential equation (SDE) in the stochastic case. The system consisting of the adjoint equa tion, the original state equation, and the maximum condition is referred to as an (extended) Hamiltonian system. On the other hand, in Bellman's dynamic programming, there is a partial differential equation (PDE), of first order in the (finite-dimensional) deterministic case and of second or der in the stochastic case. This is known as a Hamilton-Jacobi-Bellman (HJB) equation.続きを見る 1. Basic Stochastic Calculus1. Probability2. Stochastic Processes3. Stopping Times4. Martingales5. Itô's Integral6. Stochastic Differential Equations2. Stochastic Optimal Control Problems1. Introduction2. Deterministic Cases Revisited3. Examples of Stochastic Control Problems4. Formulations of Stochastic Optimal Control Problems5. Existence of Optimal Controls6. Reachable Sets of Stochastic Control Systems7. Other Stochastic Control Models8. Historical Remarks3. Maximum Principle and Stochastic Hamiltonian Systems1. Introduction2. The Deterministic Case Revisited3. Statement of the Stochastic Maximum Principle4. A Proof of the Maximum Principle5. Sufficient Conditions of Optimality6. Problems with State Constraints7. Historical Remarks4. Dynamic Programming and HJB Equations1. Introduction2. The Deterministic Case Revisited3. The Stochastic Principle of Optimality and the HJB Equation4. Other Properties of the Value Function5. Viscosity Solutions6. Uniqueness of Viscosity Solutions7. Historical Remarks5. The Relationship Between the Maximum Principle and Dynamic Programming1. Introduction2. Classical Hamilton-Jacobi Theory3. Relationship for Deterministic Systems4. Relationship for Stochastic Systems5. Stochastic Verification Theorems6. Optimal Feedback Controls7. Historical Remarks6. Linear Quadratic Optimal Control Problems1. Introduction2. The Deterministic LQ Problems Revisited3. Formulation of Stochastic LQ Problems4. Finiteness and Solvability5. A Necessary Condition and a Hamiltonian System6. Stochastic Riccati Equations7. Global Solvability of Stochastic Riccati Equations8. A Mean-variance Portfolio Selection Problem9. Historical Remarks7. Backward Stochastic Differential Equations1. Introduction2. Linear Backward Stochastic Differential Equations3. Nonlinear Backward Stochastic Differential Equations4. Feynman-Kac-Type Formulae5. Forward-Backward Stochastic Differential Equations6. Option Pricing Problems7. Historical RemarksReferences.続きを見る Full text available from SpringerLink ebooks - Mathematics and Statistics (Archive)

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レコードID 3646345 QA273.A1-274.9 QA274-274.9 519.2 Mathematics. Distribution (Probability theory). Mathematics. Probability Theory and Stochastic Processes. ssj0001298591 9781461271543(print) 9781461214663 2020.06.27 2020.06.28