Stochastic Control of Hereditary Systems and Applications / Edition 1

by Mou-Hsiung Chang

Overview

ThisresearchmonographdevelopstheHamilton-Jacobi-Bellman(HJB)theory viathedynamicprogrammingprincipleforaclassofoptimalcontrolproblems for shastic hereditary differential 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 inflnite-dimensional shastic systems 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 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 coeficients of these shastic 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 inflnite-dimensional counterpart for c- trolledSHDEsinwhichacertaininflnite-dimensionalBanachspaceorHilbert space is critically involved in order to account for the bounded or unbounded memory. Another type of inflnite-dimensional HJB theory that is not treated in this monograph but arises from real-world application problems can often be modeled by controlled shastic partial differential equations. Although they are both inflnite dimensional in nature and are both in the infancy of their developments, the SHDE exhibits many characteristics that are not in common with shastic partial differential equations. Consequently, the HJB theory for controlled SHDEs is parallel to and cannot betreated as a subset of the theory developed for controlled shastic partial differential equations.

Product Details

ISBN-13:	9780387758053
Publisher:	Springer New York
Publication date:	01/23/2008
Series:	Stochastic Modelling and Applied Probability , #59
Edition description:	2008
Pages:	406
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

and Summary.- Shastic Hereditary Differential Equations.- Shastic Calculus.- Optimal Classical Control.- Optimal Stopping.- Discrete Approximations.- Option Pricing.- Hereditary Portfolio Optimization.

Editorial Reviews

From the reviews:

"A large class of models from physics, chemistry … etc., is described by shastic 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 shastic 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 shastic 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)

From the Publisher