This book is an introduction to optimal stochastic control for continuous time Markov processes and the theory of viscosity solutions. It covers dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. New chapters in this second edition introduce the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets and two-controller, zero-sum differential games.

This book is devoted to the systematic exposition of the contemporary theory of controlled Markov processes with discrete time parameter or in another termi nology multistage Markovian decision processes. We discuss the applications of this theory to various concrete problems. Particular attention is paid to mathe matical models of economic planning, taking account of stochastic factors. The authors strove to construct the exposition in such a way that a reader interested in the applications can get through the book with a minimal mathe matical apparatus. On the other hand, a mathematician will find, in the appropriate chapters, a rigorous theory of general control models, based on advanced measure theory, analytic set theory, measurable selection theorems, and so forth. We have abstained from the manner of presentation of many mathematical monographs, in which one presents immediately the most general situation and only then discusses simpler special cases and examples. Wishing to separate out difficulties, we introduce new concepts and ideas in the simplest setting, where they already begin to work. Thus, before considering control problems on an infinite time interval, we investigate in detail the case of the finite interval. Here we first study in detail models with finite state and action spaces-a case not requiring a departure from the realm of elementary mathematics, and at the same time illustrating the most important principles of the theory.

This book is concerned with a class of discrete-time stochastic control processes known as controlled Markov processes (CMP's), also known as Markov decision processes or Markov dynamic programs. Starting in the mid-1950swith Richard Bellman, many contributions to CMP's have been made, and applications to engineering, statistics and operations research, among other areas, have also been developed. The purpose of this book is to present some recent developments on the theory of adaptive CMP's, i. e. , CMP's that depend on unknown parameters. Thus at each decision time, the controller or decision-maker must estimate the true parameter values, and then adapt the control actions to the estimated values. We do not intend to describe all aspects of stochastic adaptive control; rather, the selection of material reflects our own research interests. The prerequisite for this book is a knowledgeof real analysis and prob ability theory at the level of, say, Ash (1972) or Royden (1968), but no previous knowledge of control or decision processes is required. The pre sentation, on the other hand, is meant to beself-contained,in the sensethat whenever a result from analysisor probability is used, it is usually stated in full and references are supplied for further discussion, if necessary. Several appendices are provided for this purpose. The material is divided into six chapters. Chapter 1 contains the basic definitions about the stochastic control problems we are interested in; a brief description of some applications is also provided.

Stochastic control is a very active area of research. This monograph, written by two leading authorities in the field, has been updated to reflect the latest developments. It covers effective numerical methods for stochastic control problems in continuous time on two levels, that of practice and that of mathematical development. It is broadly accessible for graduate students and researchers.

The theory of Markov decision processes focuses on controlled Markov chains in discrete time. The authors establish the theory for general state and action spaces and at the same time show its application by means of numerous examples, mostly taken from the fields of finance and operations research. By using a structural approach many technicalities (concerning measure theory) are avoided. They cover problems with finite and infinite horizons, as well as partially observable Markov decision processes, piecewise deterministic Markov decision processes and stopping problems. The book presents Markov decision processes in action and includes various state-of-the-art applications with a particular view towards finance. It is useful for upper-level undergraduates, Master's students and researchers in both applied probability and finance, and provides exercises (without solutions).

Markov processes are processes that have limited memory. In particular, their dependence on the past is only through the previous state. They are used to model the behavior of many systems including communications systems, transportation networks, image segmentation and analysis, biological systems and DNA sequence analysis, random atomic motion and diffusion in physics, social mobility, population studies, epidemiology, animal and insect migration, queueing systems, resource management, dams, financial engineering, actuarial science, and decision systems. Covering a wide range of areas of application of Markov processes, this second edition is revised to highlight the most important aspects as well as the most recent trends and applications of Markov processes. The author spent over 16 years in the industry before returning to academia, and he has applied many of the principles covered in this book in multiple research projects. Therefore, this is an applications-oriented book that also includes enough theory to provide a solid ground in the subject for the reader. - Presents both the theory and applications of the different aspects of Markov processes - Includes numerous solved examples as well as detailed diagrams that make it easier to understand the principle being presented - Discusses different applications of hidden Markov models, such as DNA sequence analysis and speech analysis.

The theory of controlled processes is one of the most recent mathematical theories to show very important applications in modern engineering, parti cularly for constructing automatic control systems, as well as for problems of economic control. However, actual systems subject to control do not admit a strictly deterministic analysis in view of random factors of various kinds which influence their behavior. Such factors include, for example, random noise occurring in the electrical system, variations in the supply and demand of commodities, fluctuations in the labor force in economics, and random failures of components on an automated line. The theory of con trolled processes takes the random nature of the behavior of a system into account. In such cases it is natural, when choosing a control strategy, to proceed from the average expected result, taking note of all the possible variants of the behavior of a controlled system. An extensive literature is devoted to various economic and engineering systems of control (some of these works are listed in the Bibliography). is no text which adequately covers the general However, as of now there mathematical theory of controlled processes. The authors ofthis monograph have attempted to fill this gap. In this volume the general theory of discrete-parameter (time) controlled processes (Chapter 1) and those with continuous-time (Chapter 2), as well as the theory of controlled stochastic differential equations (Chapter 3), are presented.