Stochastic Optimal Control

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Stochastic Optimal Control In Infinite Dimension

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Author : Giorgio Fabbri
language : en
Publisher: Springer
Release Date : 2017-06-22

Stochastic Optimal Control In Infinite Dimension written by Giorgio Fabbri and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-06-22 with Mathematics categories.

Providing an introduction to stochastic optimal control in infinite dimension, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in infinite-dimensional stochastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of stochastic PDEs, and in PDEs in infinite dimension. Readers from other fields who want to learn the basic theory will also find it useful. The prerequisites are: standard functional analysis, the theory of semigroups of operators and its use in the study of PDEs, some knowledge of the dynamic programming approach to stochastic optimal control problems in finite dimension, and the basics of stochastic analysis and stochastic equations in infinite-dimensional spaces.

Optimal Control And Estimation

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Author : Robert F. Stengel
language : en
Publisher: Courier Corporation
Release Date : 2012-10-16

Optimal Control And Estimation written by Robert F. Stengel and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-10-16 with Mathematics categories.

Graduate-level text provides introduction to optimal control theory for stochastic systems, emphasizing application of basic concepts to real problems. "Invaluable as a reference for those already familiar with the subject." — Automatica.

Optimal Stochastic Control Stochastic Target Problems And Backward Sde

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Author : Nizar Touzi
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-09-25

Optimal Stochastic Control Stochastic Target Problems And Backward Sde written by Nizar Touzi and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-25 with Mathematics categories.

This book collects some recent developments in stochastic control theory with applications to financial mathematics. We first address standard stochastic control problems from the viewpoint of the recently developed weak dynamic programming principle. A special emphasis is put on the regularity issues and, in particular, on the behavior of the value function near the boundary. We then provide a quick review of the main tools from viscosity solutions which allow to overcome all regularity problems. We next address the class of stochastic target problems which extends in a nontrivial way the standard stochastic control problems. Here the theory of viscosity solutions plays a crucial role in the derivation of the dynamic programming equation as the infinitesimal counterpart of the corresponding geometric dynamic programming equation. The various developments of this theory have been stimulated by applications in finance and by relevant connections with geometric flows. Namely, the second order extension was motivated by illiquidity modeling, and the controlled loss version was introduced following the problem of quantile hedging. The third part specializes to an overview of Backward stochastic differential equations, and their extensions to the quadratic case.​

Stochastic Control In Insurance

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Author : Hanspeter Schmidli
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-11-20

Stochastic Control In Insurance written by Hanspeter Schmidli and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-11-20 with Business & Economics categories.

Yet again, here is a Springer volume that offers readers something completely new. Until now, solved examples of the application of stochastic control to actuarial problems could only be found in journals. Not any more: this is the first book to systematically present these methods in one volume. The author starts with a short introduction to stochastic control techniques, then applies the principles to several problems. These examples show how verification theorems and existence theorems may be proved, and that the non-diffusion case is simpler than the diffusion case. Schmidli’s brilliant text also includes a number of appendices, a vital resource for those in both academic and professional settings.

Stochastic Controls

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Author : Jiongmin Yong
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Stochastic Controls written by Jiongmin Yong and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

As is well known, Pontryagin's maximum principle and Bellman's dynamic programming are the two principal and most commonly used approaches in solving stochastic optimal control problems. * An interesting phenomenon one can observe from the literature is that these two approaches have been developed separately and independently. Since both methods are used to investigate the same problems, a natural question one will ask is the fol lowing: (Q) What is the relationship betwccn the maximum principlc and dy namic programming in stochastic optimal controls? There did exist some researches (prior to the 1980s) on the relationship between these two. Nevertheless, the results usually werestated in heuristic terms and proved under rather restrictive assumptions, which were not satisfied in most cases. In the statement of a Pontryagin-type maximum principle there is an adjoint equation, which is an ordinary differential equation (ODE) in the (finite-dimensional) deterministic case and a stochastic differential equation (SDE) in the stochastic case. The system consisting of the adjoint equa tion, the original state equation, and the maximum condition is referred to as an (extended) Hamiltonian system. On the other hand, in Bellman's dynamic programming, there is a partial differential equation (PDE), of first order in the (finite-dimensional) deterministic case and of second or der in the stochastic case. This is known as a Hamilton-Jacobi-Bellman (HJB) equation.

Stochastic Control Theory

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Author : Makiko Nisio
language : en
Publisher: Springer
Release Date : 2014-11-27

Stochastic Control Theory written by Makiko Nisio and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-11-27 with Mathematics categories.

This book offers a systematic introduction to the optimal stochastic control theory via the dynamic programming principle, which is a powerful tool to analyze control problems. First we consider completely observable control problems with finite horizons. Using a time discretization we construct a nonlinear semigroup related to the dynamic programming principle (DPP), whose generator provides the Hamilton–Jacobi–Bellman (HJB) equation, and we characterize the value function via the nonlinear semigroup, besides the viscosity solution theory. When we control not only the dynamics of a system but also the terminal time of its evolution, control-stopping problems arise. This problem is treated in the same frameworks, via the nonlinear semigroup. Its results are applicable to the American option price problem. Zero-sum two-player time-homogeneous stochastic differential games and viscosity solutions of the Isaacs equations arising from such games are studied via a nonlinear semigroup related to DPP (the min-max principle, to be precise). Using semi-discretization arguments, we construct the nonlinear semigroups whose generators provide lower and upper Isaacs equations. Concerning partially observable control problems, we refer to stochastic parabolic equations driven by colored Wiener noises, in particular, the Zakai equation. The existence and uniqueness of solutions and regularities as well as Itô's formula are stated. A control problem for the Zakai equations has a nonlinear semigroup whose generator provides the HJB equation on a Banach space. The value function turns out to be a unique viscosity solution for the HJB equation under mild conditions. This edition provides a more generalized treatment of the topic than does the earlier book Lectures on Stochastic Control Theory (ISI Lecture Notes 9), where time-homogeneous cases are dealt with. Here, for finite time-horizon control problems, DPP was formulated as a one-parameter nonlinear semigroup, whose generator provides the HJB equation, by using a time-discretization method. The semigroup corresponds to the value function and is characterized as the envelope of Markovian transition semigroups of responses for constant control processes. Besides finite time-horizon controls, the book discusses control-stopping problems in the same frameworks.

Stochastic Control In Discrete And Continuous Time

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Author : Atle Seierstad
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-07-03

Stochastic Control In Discrete And Continuous Time written by Atle Seierstad and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-03 with Mathematics categories.

This book contains an introduction to three topics in stochastic control: discrete time stochastic control, i. e. , stochastic dynamic programming (Chapter 1), piecewise - terministic control problems (Chapter 3), and control of Ito diffusions (Chapter 4). The chapters include treatments of optimal stopping problems. An Appendix - calls material from elementary probability theory and gives heuristic explanations of certain more advanced tools in probability theory. The book will hopefully be of interest to students in several ?elds: economics, engineering, operations research, ?nance, business, mathematics. In economics and business administration, graduate students should readily be able to read it, and the mathematical level can be suitable for advanced undergraduates in mathem- ics and science. The prerequisites for reading the book are only a calculus course and a course in elementary probability. (Certain technical comments may demand a slightly better background. ) As this book perhaps (and hopefully) will be read by readers with widely diff- ing backgrounds, some general advice may be useful: Don’t be put off if paragraphs, comments, or remarks contain material of a seemingly more technical nature that you don’t understand. Just skip such material and continue reading, it will surely not be needed in order to understand the main ideas and results. The presentation avoids the use of measure theory.

Numerical Methods For Stochastic Control Problems In Continuous Time

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Author : Harold Kushner
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-27

Numerical Methods For Stochastic Control Problems In Continuous Time written by Harold Kushner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-27 with Mathematics categories.

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.

Stochastic Optimal Transportation

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Author : Toshio Mikami
language : en
Publisher: Springer Nature
Release Date : 2021-06-15

Stochastic Optimal Transportation written by Toshio Mikami and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-15 with Mathematics categories.

In this book, the optimal transportation problem (OT) is described as a variational problem for absolutely continuous stochastic processes with fixed initial and terminal distributions. Also described is Schrödinger’s problem, which is originally a variational problem for one-step random walks with fixed initial and terminal distributions. The stochastic optimal transportation problem (SOT) is then introduced as a generalization of the OT, i.e., as a variational problem for semimartingales with fixed initial and terminal distributions. An interpretation of the SOT is also stated as a generalization of Schrödinger’s problem. After the brief introduction above, the fundamental results on the SOT are described: duality theorem, a sufficient condition for the problem to be finite, forward–backward stochastic differential equations (SDE) for the minimizer, and so on. The recent development of the superposition principle plays a crucial role in the SOT. A systematic method is introduced to consider two problems: one with fixed initial and terminal distributions and one with fixed marginal distributions for all times. By the zero-noise limit of the SOT, the probabilistic proofs to Monge’s problem with a quadratic cost and the duality theorem for the OT are described. Also described are the Lipschitz continuity and the semiconcavity of Schrödinger’s problem in marginal distributions and random variables with given marginals, respectively. As well, there is an explanation of the regularity result for the solution to Schrödinger’s functional equation when the space of Borel probability measures is endowed with a strong or a weak topology, and it is shown that Schrödinger’s problem can be considered a class of mean field games. The construction of stochastic processes with given marginals, called the marginal problem for stochastic processes, is discussed as an application of the SOT and the OT.

Stochastic Optimal Control And The U S Financial Debt Crisis

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Author : Jerome L. Stein
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-03-30

Stochastic Optimal Control And The U S Financial Debt Crisis written by Jerome L. Stein and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-03-30 with Business & Economics categories.

Stochastic Optimal Control (SOC)—a mathematical theory concerned with minimizing a cost (or maximizing a payout) pertaining to a controlled dynamic process under uncertainty—has proven incredibly helpful to understanding and predicting debt crises and evaluating proposed financial regulation and risk management. Stochastic Optimal Control and the U.S. Financial Debt Crisis analyzes SOC in relation to the 2008 U.S. financial crisis, and offers a detailed framework depicting why such a methodology is best suited for reducing financial risk and addressing key regulatory issues. Topics discussed include the inadequacies of the current approaches underlying financial regulations, the use of SOC to explain debt crises and superiority over existing approaches to regulation, and the domestic and international applications of SOC to financial crises. Principles in this book will appeal to economists, mathematicians, and researchers interested in the U.S. financial debt crisis and optimal risk management.