Mou-Hsiung Chang - Böcker

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.

The classical probability theory initiated by Kolmogorov and its quantum counterpart, pioneered by von Neumann, were created at about the same time in the 1930s, but development of the quantum theory has trailed far behind. Although highly appealing, the quantum theory has a steep learning curve, requiring tools from both probability and analysis and a facility for combining the two viewpoints. This book is a systematic, self-contained account of the core of quantum probability and quantum stochastic processes for graduate students and researchers. The only assumed background is knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes. From there, the book introduces additional tools from analysis, and then builds the quantum probability framework needed to support applications to quantum control and quantum information and communication. These include quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups and processes, and large-time asymptotic behavior of quantum Markov semigroups.

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.