Overview

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 interesting 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.

Full Product Details

Author:   Jiongmin Yong ,  Xun Yu Zhou
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1999
Volume:   43
Dimensions:   Width: 15.50cm , Height: 2.40cm , Length: 23.50cm
Weight:   0.706kg
ISBN:  

9781461271543

ISBN 10:   1461271541
Pages:   439
Publication Date:   27 September 2012
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

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. Feynman—Kac-Type Formulae.- 5. Forward—Backward Stochastic Differential Equations.- 6. Option Pricing Problems.- 7. Historical Remarks.- References.

Reviews

From the reviews: SIAM REVIEW The presentation of this book is systematic and self-contained...Summing up, this book is a very good addition to the control literature, with original features not found in other reference books. Certain parts could be used as basic material for a graduate (or postgraduate) course...This book is highly recommended to anyone who wishes to study the relationship between Pontryagin's maximum principle and Bellman's dynamic programming principle applied to diffusion processes. MATHEMATICS REVIEW This is an authoratative book which should be of interest to researchers in stochastic control, mathematical finance, probability theory, and applied mathematics. Material out of this book could also be used in graduate courses on stochastic control and dynamic optimization in mathematics, engineering, and finance curricula. Tamer Basar, Math. Review

From the reviews: SIAM REVIEW The presentation of this book is systematic and self-contained...Summing up, this book is a very good addition to the control literature, with original features not found in other reference books. Certain parts could be used as basic material for a graduate (or postgraduate) course...This book is highly recommended to anyone who wishes to study the relationship between Pontryagin's maximum principle and Bellman's dynamic programming principle applied to diffusion processes. MATHEMATICS REVIEW This is an authoratative book which should be of interest to researchers in stochastic control, mathematical finance, probability theory, and applied mathematics. Material out of this book could also be used in graduate courses on stochastic control and dynamic optimization in mathematics, engineering, and finance curricula. Tamer Basar, Math. Review

"From the reviews: SIAM REVIEW ""The presentation of this book is systematic and self-contained…Summing up, this book is a very good addition to the control literature, with original features not found in other reference books. Certain parts could be used as basic material for a graduate (or postgraduate) course…This book is highly recommended to anyone who wishes to study the relationship between Pontryagin’s maximum principle and Bellman’s dynamic programming principle applied to diffusion processes."" MATHEMATICS REVIEW This is an authoratative book which should be of interest to researchers in stochastic control, mathematical finance, probability theory, and applied mathematics. Material out of this book could also be used in graduate courses on stochastic control and dynamic optimization in mathematics, engineering, and finance curricula. Tamer Basar, Math. Review"

From the reviews: SIAM REVIEW The presentation of this book is systematic and self-contained...Summing up, this book is a very good addition to the control literature, with original features not found in other reference books. Certain parts could be used as basic material for a graduate (or postgraduate) course...This book is highly recommended to anyone who wishes to study the relationship between Pontryagin's maximum principle and Bellman's dynamic programming principle applied to diffusion processes. MATHEMATICS REVIEW This is an authoratative book which should be of interest to researchers in stochastic control, mathematical finance, probability theory, and applied mathematics. Material out of this book could also be used in graduate courses on stochastic control and dynamic optimization in mathematics, engineering, and finance curricula. Tamer Basar, Math. Review

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