In this thesis, we address a convex stochastic optimal control problem in mathematical finance, with the goal of minimizing a general quadratic loss function of the wealth at close of trade. We study this problem in the setting of an Ito process market model, in which the underlying filtration to which the market parameters are adapted is the joint filtration of the driving Brownian motion for the market model, together with the filtration of an independent finite-state Markov chain which models occasional changes in "regime states'', that is our model allows for "regime switching'' among a finite number of regime states. Other aspects of the problem that we address in this thesis are: (1) The portfolio vector of holdings in the risky assets is confined to a given closed and convex constraint set; (2) There is a "state constraint'' in the form of a stipulated almost-sure lower bound on the wealth at close of trade. The combination of constraints represented by (1) and (2) makes the optimization problem quite challenging. The powerful and effective method of {\em auxiliary markets}, of Cvitanic and Karatzas [Ann. Appl. Prob., v.2, 767-818, 1992] for dealing with convex portfolio constraints, does not appear to extend to problems with regime-switching, while the more recent approach of Donnelly and Heunis [SIAM Jour. Control Optimiz., v.50, 2431-2461, 2012], which deals with both regime-switching and the convex portfolio constraints (1), is nevertheless confounded when one adds state constraints of the form (2) to the problem. The reason for this is clear: state constraints of the form (2) typically involve "singular'' Lagrange multipliers which fall well outside the scope of the "well-behaved'' Lagrange multipliers, manifested either as random variables or stochastic processes, which suffice when one is dealing only with portfolio constraints such as (1) above. In these circumstances we resort to an "abstract'' duality approach of Rockafellar and Moreau, which has been applied with considerable success to finite-dimensional problems of stochastic mathematical programming in which singular Lagrange multipliers also naturally arise. The main goal of this thesis is to adapt and extend the Rockafellar-Moreau approach to the stochastic optimal control problem summarized above. We find that this is indeed possible, although some considerable effort is required in view of the infinite dimensionality of the problem. We construct an appropriate space of Lagrange multipliers, synthesize a dual optimization problem, establish optimality relations which give necessary and sufficient conditions for the given optimization problem and its dual to each have a solution with zero duality gap, and use the optimality relations to synthesize an optimal portfolio in terms of the Lagrange multipliers.

Issues in Robotics and Automation / 2013 Edition is a ScholarlyEditions™ book that delivers timely, authoritative, and comprehensive information about Computing Information and Control. The editors have built Issues in Robotics and Automation: 2013 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Computing Information and Control in this book to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in Robotics and Automation: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.

Yes, it makes a lot of sense. This paper studies how to design simple loss functions for central banks, as parsimonious approximations to social welfare. We show, both analytically and quantitatively, that simple loss functions should feature a high weight on measures of economic activity, sometimes even larger than the weight on inflation. Two main factors drive our result. First, stabilizing economic activity also stabilizes other welfare relevant variables. Second, the estimated model features mitigated inflation distortions due to a low elasticity of substitution between monopolistic goods and a low interest rate sensitivity of demand. The result holds up in the presence of measurement errors, with large shocks that generate a trade-off between stabilizing inflation and resource utilization, and also when ensuring a low probability of hitting the zero lower bound on interest rates.

A rigorous introduction to optimal control theory, with an emphasis on applications in economics. This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. The theory of ordinary differential equations (ODEs) is the backbone of the theory developed in the book, and chapter 2 offers a detailed review of basic concepts in the theory of ODEs, including the solution of systems of linear ODEs, state-space analysis, potential functions, and stability analysis. Following this, the book covers the main results of optimal control theory, in particular necessary and sufficient optimality conditions; game theory, with an emphasis on differential games; and the application of control-theoretic concepts to the design of economic mechanisms. Appendixes provide a mathematical review and full solutions to all end-of-chapter problems. The material is presented at three levels: single-person decision making; games, in which a group of decision makers interact strategically; and mechanism design, which is concerned with a designer's creation of an environment in which players interact to maximize the designer's objective. The book focuses on applications; the problems are an integral part of the text. It is intended for use as a textbook or reference for graduate students, teachers, and researchers interested in applications of control theory beyond its classical use in economic growth. The book will also appeal to readers interested in a modeling approach to certain practical problems involving dynamic continuous-time models.

This monograph collects in one place the basic definitions, a careful description of the model, and discussion of how convex optimization can be used in multi-period trading, all in a common notation and framework.

Surveys the theory and history of the alternating direction method of multipliers, and discusses its applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others.

Recently Geometric Programming has been applied to study a variety of problems in the analysis and design of communication systems from information theory and queuing theory to signal processing and network protocols. Geometric Programming for Communication Systems begins its comprehensive treatment of the subject by providing an in-depth tutorial on the theory, algorithms, and modeling methods of Geometric Programming. It then gives a systematic survey of the applications of Geometric Programming to the study of communication systems. It collects in one place various published results in this area, which are currently scattered in several books and many research papers, as well as to date unpublished results. Geometric Programming for Communication Systems is intended for researchers and students who wish to have a comprehensive starting point for understanding the theory and applications of geometric programming in communication systems.