ThisresearchmonographdevelopstheHamilton-Jacobi-Bellman(HJB)theory viathedynamicprogrammingprincipleforaclassofoptimalcontrolproblems for stochastic hereditary di?erential equations (SHDEs) driven by a standard Brownian motion and with a bounded or an unbounded but fading m- ory. These equations represent a class of in?nite-dimensional stochastic s- tems that become increasingly important and have wide range of applications in physics, chemistry, biology, engineering, and economics/?nance. The wide applicability of these systems is due to the fact that the reaction of re- world systems to exogenous e?ects/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 e?ects, the drift and di?usion coe?cients 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 ?nite-dimensional HJB theory of controlled di?usion processes to its in?nite-dimensional counterpart for c- trolledSHDEsinwhichacertainin?nite-dimensionalBanachspaceorHilbert space is critically involved in order to account for the bounded or unbounded memory. Another type of in?nite-dimensional HJB theory that is not treated in this monograph but arises from real-world application problems can often be modeled by controlled stochastic partial di?erential equations. Although they are both in?nite dimensional in nature and are both in the infancy of their developments, the SHDE exhibits many characteristics that are not in common with stochastic partial di?erential equations. Consequently, the HJB theory for controlled SHDEs is parallel to and cannot betreated as a subset of the theory developed for controlled stochastic partial di?erential equations.

From the reviews:

"A large class of models from physics, chemistry … etc., is described by stochastic hereditary differential equations (SHDEs) driven by a standard Brownian motion. … The monograph is addressed to researchers and advanced graduate students with interest in the theory and applications of optimal control for SHDEs. … The monograph provides a systematic and careful exposition of the fundamental results of the control problems for stochastic hereditary differential systems and represents an essential source of information for anyone who wants to work in the field." (Constantin Tudor, Mathematical Reviews, Issue 2009 e)

“The theme of this research monograph is a set of equations that represent a class of infinite-dimensional stochastic systems. … This monograph can serve as an introduction and/or a research reference for researchers and advanced graduate students with a special interest in theory and applications of optimal control of SHDEs. The monograph is intended to be as self-contained as possible. … Theory developed in this monograph can be extended with additional efforts to hereditary differential equations driven by semimartingales, such as Lévy processes.” (Adriana Horníková, Technometrics, Vol. 52 (2), May, 2010)