Stochastic Control of Hereditary Systems and Applications
This research monograph develops the Hamilton-Jacobi-Bellman (HJB) theory through dynamic programming principle for a class of optimal control problems for stochastic hereditary differential systems. It is driven by a standard Brownian motion and with a b
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Stochastic Modelling and Applied Probability (Formerly: Applications of Mathematics)
Stochastic Optimization Stochastic Control Stochastic Models in Life Sciences
Edited by
Advisory Board
59 B. Rozovski˘ı G. Grimmett D. Dawson D. Geman I. Karatzas F. Kelly Y. Le Jan B. Øksendal G. Papanicolaou E. Pardoux
Mou-Hsiung Chang
Stochastic Control of Hereditary Systems and Applications
Mou-Hsiung Chang 4300 S. Miami Blvd. U.S. Army Research Office Durham, NC 27703-9142 USA [email protected]
Man aging Editors B. Rozovski˘ı Division of Applied Mathematics 182 George St. Providence, RI 02912 USA [email protected]
G. Grimmett Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB UK [email protected]
ISBN: 978-0-387-75805-3 e-ISBN: 978-0-387-75816-9 DOI: 10.1007/978-0-387-75816-9 ISSN: 0172-4568 Stochastic Modelling and Applied Probability Library of Congress Control Number: 2007941276 Mathematics Subject Classification (2000): 93E20, 34K50, 90C15 c 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
This book is dedicated to my wife, Yuen-Man Chang.
Preface
This research monograph develops the Hamilton-Jacobi-Bellman (HJB) theory via the dynamic programming principle for a class of optimal control problems for stochastic hereditary differential equations (SHDEs) driven by a standard Brownian motion and with a bounded or an unbounded but fading memory. These equations represent a class of infinite-dimensional stochastic systems that become increasingly important and have wide range of applications in physics, chemistry, biology, engineering, and economics/finance. The wide applicability of these systems is due to the fact that the reaction of realworld systems to exogenous effects/signals is never “instantaneous” and it needs some time, time that can be translated into a mathematical language by some delay terms. Therefore, to describe these delayed effects, the drift and diffusion coefficients of these stochastic equations depend not only on the current state but also explicitly on the past history of the state variable. The theory developed herein extends the finite-dimensional HJB theory of controlled diffusion processes to its infinite-dimensional counterpart for controlled SHDEs in which a certain infinite-dimensio
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