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Stochastic Controls

Hamiltonian Systems and HJB Equations

  • Book
  • © 1999

Overview

Authors:
  1. Jiongmin Yong

    1. Department of Mathematics, Fudan University, Shanghai, China

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    You can also search for this author in PubMed   Google Scholar

  2. Xun Yu Zhou

    1. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong

    View author publications

    You can also search for this author in PubMed   Google Scholar

Part of the book series: Stochastic Modelling and Applied Probability (SMAP, volume 43)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xxii
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  2. Basic Stochastic Calculus

    • Jiongmin Yong, Xun Yu Zhou
    Pages 1-50

  3. Stochastic Optimal Control Problems

    • Jiongmin Yong, Xun Yu Zhou
    Pages 51-100

  4. Maximum Principle and Stochastic Hamiltonian Systems

    • Jiongmin Yong, Xun Yu Zhou
    Pages 101-156

  5. Dynamic Programming and HJB Equations

    • Jiongmin Yong, Xun Yu Zhou
    Pages 157-215

  6. The Relationship Between the Maximum Principle and Dynamic Programming

    • Jiongmin Yong, Xun Yu Zhou
    Pages 217-280

  7. Linear Quadratic Optimal Control Problems

    • Jiongmin Yong, Xun Yu Zhou
    Pages 281-344

  8. Backward Stochastic Differential Equations

    • Jiongmin Yong, Xun Yu Zhou
    Pages 345-400

  9. Back Matter

    Pages 401-439
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Keywords

About this book

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.

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

Authors and Affiliations

  • Department of Mathematics, Fudan University, Shanghai, China

    Jiongmin Yong

  • Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong

    Xun Yu Zhou

Bibliographic Information

  • Book Title: Stochastic Controls

  • Book Subtitle: Hamiltonian Systems and HJB Equations

  • Authors: Jiongmin Yong, Xun Yu Zhou

  • Series Title: Stochastic Modelling and Applied Probability

  • DOI: https://doi.org/10.1007/978-1-4612-1466-3

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1999

  • Hardcover ISBN: 978-0-387-98723-1Published: 22 June 1999

  • Softcover ISBN: 978-1-4612-7154-3Published: 27 September 2012

  • eBook ISBN: 978-1-4612-1466-3Published: 06 December 2012

  • Series ISSN: 0172-4568

  • Series E-ISSN: 2197-439X

  • Edition Number: 1

  • Number of Pages: XXII, 439

  • Topics: Probability Theory and Stochastic Processes

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