Harold Kushner - Böcker

The book presents a thorough development of the modern theory of stochastic approximation or recursive stochastic algorithms for both constrained and unconstrained problems. There is a complete development of both probability one and weak convergence methods for very general noise processes. The proofs of convergence use the ODE method, the most powerful to date, with which the asymptotic behavior is characterized by the limit behavior of a mean ODE. The assumptions and proof methods are designed to cover the needs of recent applications. The development proceeds from simple to complex problems, allowing the underlying ideas to be more easily understood. Rate of convergence, iterate averaging, high-dimensional problems, stability-ODE methods, two time scale, asynchronous and decentralized algorithms, general correlated and state-dependent noise, perturbed test function methods, and large devitations methods, are covered. Many motivational examples from learning theory, ergodic cost problems for discrete event systems, wireless communications, adaptive control, signal processing, and elsewhere, illustrate the application of the theory.This second edition is a thorough revision, although the main features and the structure remain unchanged. It contains many additional applications and results, and more detailed discussion. Harold J. Kushner is a University Professor and Professor of Applied Mathematics at Brown University. He has written numerous books and articles on virtually all aspects of stochastic systems theory, and has received various awards including the IEEE Control Systems Field Award.

Changes in the second edition. The second edition differs from the first in that there is a full development of problems where the variance of the diffusion term and the jump distribution can be controlled. Also, a great deal of new material concerning deterministic problems has been added, including very efficient algorithms for a class of problems of wide current interest. This book is concerned with numerical methods for stochastic control and optimal stochastic control problems. The random process models of the controlled or uncontrolled stochastic systems are either diffusions or jump diffusions. Stochastic control is a very active area of research and new problem formulations and sometimes surprising applications appear regu­ larly. We have chosen forms of the models which cover the great bulk of the formulations of the continuous time stochastic control problems which have appeared to date. The standard formats are covered, but much emphasis is given to the newer and less well known formulations. The controlled process might be either stopped or absorbed on leaving a constraint set or upon first hitting a target set, or it might be reflected or "projected" from the boundary of a constraining set. In some of the more recent applications of the reflecting boundary problem, for example the so-called heavy traffic approximation problems, the directions of reflection are actually discontin­ uous. In general, the control might be representable as a bounded function or it might be of the so-called impulsive or singular control types.

This book provides a thorough development of the powerful methods of heavy traffic analysis and approximations with applications to a wide variety of stochastic (e.g. queueing and communication) networks, for both controlled and uncontrolled systems.The approximating models are reflected stochastic differential equations. The analytical and numerical methods yield considerable simplifications and insights and good approximations to both path properties and optimal controls under broad conditions on the data and structure.The general theory is developed, with possibly state dependent parameters, and specialized to many different cases of practical interest.Control problems in telecommunications and applications to scheduling, admissions control, polling, and elsewhere are treated. The necessary probability background is reviewed, including a detailed survey of reflected stochastic differential equations, weak convergence theory, methods for characterizing limit processes, and ergodic problems.

Full of the ideas and affirmations on which bestselling author Harold Kushner has based his life, WHO NEEDS GOD will help readers bring depth and order to their lives through spirituality. It is a book for anyone who has ever stepped back and thought, 'there must be more to life than this?' Here Kushner looks at the nourishment our souls crave, in much the same way that our bodies need the right foods, sunshine and exercise. He maintains that without some kind of spiritual nourishment, our souls remain stunted and underdeveloped. The spiritual life that was so prominent for people in the past is less accessible to us today - because the world is so noisy and full of distractions, because we are so dazzled by our power and success and because in the new millennium religion is often represented by people we no longer trust. In this thought provoking book Kushner shows us that the presence of God in our lives can make a difference and that modern people are capable of pray.

The book deals with several closely related topics concerning approxima­ tions and perturbations of random processes and their applications to some important and fascinating classes of problems in the analysis and design of stochastic control systems and nonlinear filters. The basic mathematical methods which are used and developed are those of the theory of weak con­ vergence. The techniques are quite powerful for getting weak convergence or functional limit theorems for broad classes of problems and many of the techniques are new. The original need for some of the techniques which are developed here arose in connection with our study of the particular applica­ tions in this book, and related problems of approximation in control theory, but it will be clear that they have numerous applications elsewhere in weak convergence and process approximation theory. The book is a continuation of the author's long term interest in problems of the approximation of stochastic processes and its applications to problems arising in control and communication theory and related areas. In fact, the techniques used here can be fruitfully applied to many other areas. The basic random processes of interest can be described by solutions to either (multiple time scale) Ito differential equations driven by wide band or state dependent wide band noise or which are singularly perturbed. They might be controlled or not, and their state values might be fully observable or not (e. g. , as in the nonlinear filtering problem).

The Markov chain approximation methods are widely used for the numerical solution of nonlinear stochastic control problems in continuous time. This book extends the methods to stochastic systems with delays. The book is the first on the subject and will be of great interest to all those who work with stochastic delay equations and whose main interest is either in the use of the algorithms or in the mathematics. An excellent resource for graduate students, researchers, and practitioners, the work may be used as a graduate-level textbook for a special topics course or seminar on numerical methods in stochastic control.

takes n+1 n n n n its values in some Euclidean space, Y is a random variable, and the “step n size” > 0 is small and might go to zero as n??.

Changes in the second edition. The second edition differs from the first in that there is a full development of problems where the variance of the diffusion term and the jump distribution can be controlled. Also, a great deal of new material concerning deterministic problems has been added, including very efficient algorithms for a class of problems of wide current interest. This book is concerned with numerical methods for stochastic control and optimal stochastic control problems. The random process models of the controlled or uncontrolled stochastic systems are either diffusions or jump diffusions. Stochastic control is a very active area of research and new problem formulations and sometimes surprising applications appear regu­ larly. We have chosen forms of the models which cover the great bulk of the formulations of the continuous time stochastic control problems which have appeared to date. The standard formats are covered, but much emphasis is given to the newer and less well known formulations. The controlled process might be either stopped or absorbed on leaving a constraint set or upon first hitting a target set, or it might be reflected or "projected" from the boundary of a constraining set. In some of the more recent applications of the reflecting boundary problem, for example the so-called heavy traffic approximation problems, the directions of reflection are actually discontin­ uous. In general, the control might be representable as a bounded function or it might be of the so-called impulsive or singular control types.

The aim of this book is the development of the heavy traffic approach to the modeling and analysis of queueing networks, both controlled and uncontrolled, and many applications to computer, communications, and manufacturing systems. The methods exploit the multiscale structure of the physical problem to get approximating models that have the form of reflected diffusion processes, either controlled or uncontrolled. These ap­ proximating models have the basic structure of the original problem, but are significantly simpler. Much of inessential detail is eliminated (or "av­ eraged out"). They greatly simplify analysis, design, and optimization and yield good approximations to problems that would otherwise be intractable, under broad conditions. Queueing-type processes are ubiquitous occurrences in operations re­ search, and in communications and computer systems. Indeed, it is hard to avoid them in modern technology. The subject is now about 100 years old. and there is an enormous literature. Impressive techniques, many based on Markov chain and ergodic theory, have been developed to han­ dle a great variety of models. A sampling of the numerous books includes [6, 8, 18, 27, 33, 46, 81, 86, 132, 133, 220, 243]. But the models of interest are growing fast in the face of the demands of new applications, particularly in communications and computer systems.

The book deals with several closely related topics concerning approxima­ tions and perturbations of random processes and their applications to some important and fascinating classes of problems in the analysis and design of stochastic control systems and nonlinear filters. The basic mathematical methods which are used and developed are those of the theory of weak con­ vergence. The techniques are quite powerful for getting weak convergence or functional limit theorems for broad classes of problems and many of the techniques are new. The original need for some of the techniques which are developed here arose in connection with our study of the particular applica­ tions in this book, and related problems of approximation in control theory, but it will be clear that they have numerous applications elsewhere in weak convergence and process approximation theory. The book is a continuation of the author's long term interest in problems of the approximation of stochastic processes and its applications to problems arising in control and communication theory and related areas. In fact, the techniques used here can be fruitfully applied to many other areas. The basic random processes of interest can be described by solutions to either (multiple time scale) Ito differential equations driven by wide band or state dependent wide band noise or which are singularly perturbed. They might be controlled or not, and their state values might be fully observable or not (e. g. , as in the nonlinear filtering problem).