Stochastic Modelling and Applied Probability

Stochastic Controls

Hamiltonian Systems and HJB Equations

Authors: Yong, Jiongmin, Zhou, Xun Yu

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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


Table of contents (7 chapters)

  • Basic Stochastic Calculus

    Yong, Jiongmin (et al.)

    Pages 1-50

  • Stochastic Optimal Control Problems

    Yong, Jiongmin (et al.)

    Pages 51-100

  • Maximum Principle and Stochastic Hamiltonian Systems

    Yong, Jiongmin (et al.)

    Pages 101-156

  • Dynamic Programming and HJB Equations

    Yong, Jiongmin (et al.)

    Pages 157-215

  • The Relationship Between the Maximum Principle and Dynamic Programming

    Yong, Jiongmin (et al.)

    Pages 217-280

Buy this book

eBook £93.50
price for United Kingdom (gross)
  • ISBN 978-1-4612-1466-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover £133.00
price for United Kingdom (gross)
  • ISBN 978-0-387-98723-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover £117.00
price for United Kingdom (gross)
  • ISBN 978-1-4612-7154-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Stochastic Controls
Book Subtitle
Hamiltonian Systems and HJB Equations
Authors
Series Title
Stochastic Modelling and Applied Probability
Series Volume
43
Copyright
1999
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4612-1466-3
DOI
10.1007/978-1-4612-1466-3
Hardcover ISBN
978-0-387-98723-1
Softcover ISBN
978-1-4612-7154-3
Series ISSN
0172-4568
Edition Number
1
Number of Pages
XXII, 439
Topics