Stochastic Controls: Hamiltonian Systems and HJB Equations: 43 - Brossura

Yong, Jiongmin; Zhou, Xun Yu

Brossura

ISBN 10: 1461271541 ISBN 13: 9781461271543

Casa editrice: Springer, 2012

Riassunto

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.

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Recensione

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

Contenuti

1. Basic Stochastic Calculus.- 1. Probability.- 1.1. Probability spaces.- 1.2. Random variables.- 1.3. Conditional expectation.- 1.4. Convergence of probabilities.- 2. Stochastic Processes.- 2.1. General considerations.- 2.2. Brownian motions.- 3. Stopping Times.- 4. Martingales.- 5. Itô’s Integral.- 5.1. Nondifferentiability of Brownian motion.- 5.2. Definition of Itô’s integral and basic properties.- 5.3. Itô’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.- 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. Noncloseness 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.- 3. Maximum Principle and Stochastic Hamiltonian Systems.- 1. Introduction.- 2. The Deterministic Case Revisited.- 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 completion of the proof.- 5. Sufficient Conditions of Optimality.- 6. Problems with State 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.- 4. Dynamic Programming and HJB Equations.- 1. Introduction.- 2. The Deterministic Case Revisited.- 3. The Stochastic Principle of Optimality and the HJB Equation.- 3.1. A stochastic framework for dynamic programming.- 3.2. Principle of optimality.- 3.3. The HJB equation.- 4. Other Properties of the Value Function.- 4.1. Continuous dependence on parameters.- 4.2. Semiconcavity.- 5. Viscosity 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 Remarks.- 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 Feynman-Kac formula.- 3.4. Adjoint variable and value function: Nonsmooth case.- 3.5. Verification 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 Verification Theorems.- 5.1. Smooth case.- 5.2. Nonsmooth case.- 6. Optimal Feedback Controls.- 7. Historical Remarks.- 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 Riccati Equations.- 7. Global Solvability of Stochastic Riccati Equations.- 7.1. Existence: The 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.- 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.

Dettagli dell?edizione

• Editore: Springer
• Data di pubblicazione: 2012
• ISBN 10: 1461271541
• ISBN 13: 9781461271543
• Rilegatura: Copertina flessibile
• Lingua: Inglese
• Numero di pagine: 464
• Contatto del produttore: non disponibile