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Author: Wendell H. Fleming
Publisher: Edizioni della Normale
ISBN: 9788876422508
Category : Mathematics
Languages : en
Pages : 0
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Book Description
These notes are based on a series of lectures delivered at the Scuola Normale Superiore in March 1986. They are intended to explore some connections between the theory of control of Markov stochastic processes and certain classes of nonlinear evolution equations. These connections arise by considering the dynamic programming equation associated with a stochastic control problem. Particular attention is given to controlled Markov diffusion processes on finite dimensional Euclidean space. In that case, the dynamic programming equation is a nonlinear partial differential equation of second order elliptic or parabolic type. For deterministic control the dynamic programming equation reduces to first order. From the viewpoint of nonlinear evolution equations, the interest is in whether one can find some stochastic control problem for which the given evolution equation is the dynamic programming equation. Classical solutions to first order or degenerate second order elliptic/parabolic equations with given boundary Cauchy data do not usually exist. One must instead consider generalized solutions. Viscosity solutions methods have substantially extended the theory.
Author: Wendell H. Fleming
Publisher: Edizioni della Normale
ISBN: 9788876422508
Category : Mathematics
Languages : en
Pages : 0
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Book Description
These notes are based on a series of lectures delivered at the Scuola Normale Superiore in March 1986. They are intended to explore some connections between the theory of control of Markov stochastic processes and certain classes of nonlinear evolution equations. These connections arise by considering the dynamic programming equation associated with a stochastic control problem. Particular attention is given to controlled Markov diffusion processes on finite dimensional Euclidean space. In that case, the dynamic programming equation is a nonlinear partial differential equation of second order elliptic or parabolic type. For deterministic control the dynamic programming equation reduces to first order. From the viewpoint of nonlinear evolution equations, the interest is in whether one can find some stochastic control problem for which the given evolution equation is the dynamic programming equation. Classical solutions to first order or degenerate second order elliptic/parabolic equations with given boundary Cauchy data do not usually exist. One must instead consider generalized solutions. Viscosity solutions methods have substantially extended the theory.
Author: Wendell H. Fleming
Publisher: Springer Science & Business Media
ISBN: 0387310711
Category : Mathematics
Languages : en
Pages : 436
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Book Description
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.
Author: Wendell Helms Fleming
Publisher:
ISBN:
Category : Differential games
Languages : en
Pages : 80
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Book Description
Author: Wendell Helms Fleming
Publisher:
ISBN: 9786610461998
Category : Markov processes
Languages : en
Pages : 428
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Book Description
This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance.; In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.
Author: Wendell Helms Fleming
Publisher:
ISBN:
Category : Differential games
Languages : en
Pages : 0
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Book Description
Author: Tatsien Li
Publisher: World Scientific
ISBN: 9814546429
Category :
Languages : en
Pages : 286
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Book Description
This volume contains 30 research papers presenting the recent development and trend on the following subjects: nonlinear hyperbolic equations (systems); nonlinear parabolic equations (systems); infinite-dimensional dynamical systems; applications (free boundary problems, phase transitions, etc.).
Author: N. U. Ahmed
Publisher: Springer Nature
ISBN: 3031372603
Category : Mathematics
Languages : en
Pages : 236
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Book Description
This book offers the first comprehensive presentation of measure-valued solutions for nonlinear deterministic and stochastic evolution equations on infinite dimensional Banach spaces. Unlike traditional solutions, measure-valued solutions allow for a much broader class of abstract evolution equations to be addressed, providing a broader approach. The book presents extensive results on the existence of measure-valued solutions for differential equations that have no solutions in the usual sense. It covers a range of topics, including evolution equations with continuous/discontinuous vector fields, neutral evolution equations subject to vector measures as impulsive forces, stochastic evolution equations, and optimal control of evolution equations. The optimal control problems considered cover the existence of solutions, necessary conditions of optimality, and more, significantly complementing the existing literature. This book will be of great interest to researchers in functional analysis, partial differential equations, dynamic systems and their optimal control, and their applications, advancing previous research and providing a foundation for further exploration of the field.
Author: C.M. Dafermos
Publisher: Elsevier
ISBN: 008046565X
Category : Mathematics
Languages : en
Pages : 653
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Book Description
The material collected in this volume reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear parabolic PDE's. Articles will highlight the present as well as expected future directions of development of the field with particular emphasis on applications. The article by Ambrosio and Savaré discusses the most recent development in the theory of gradient flow of probability measures. After an introduction reviewing the properties of the Wasserstein space and corresponding subdifferential calculus, applications are given to evolutionary partial differential equations. The contribution of Herrero provides a description of some mathematical approaches developed to account for quantitative as well as qualitative aspects of chemotaxis. Particular attention is paid to the limits of cell's capability to measure external cues on the one hand, and to provide an overall description of aggregation models for the slim mold Dictyostelium discoideum on the other. The chapter written by Masmoudi deals with a rather different topic - examples of singular limits in hydrodynamics. This is nowadays a well-studied issue given the amount of new results based on the development of the existence theory for rather general systems of equations in hydrodynamics. The paper by DeLellis addreses the most recent results for the transport equations with regard to possible applications in the theory of hyperbolic systems of conservation laws. Emphasis is put on the development of the theory in the case when the governing field is only a BV function. The chapter by Rein represents a comprehensive survey of results on the Poisson-Vlasov system in astrophysics. The question of global stability of steady states is addressed in detail. The contribution of Soner is devoted to different representations of non-linear parabolic equations in terms of Markov processes. After a brief introduction on the linear theory, a class of non-linear equations is investigated, with applications to stochastic control and differential games. The chapter written by Zuazua presents some of the recent progresses done on the problem of controllabilty of partial differential equations. The applications include the linear wave and heat equations,parabolic equations with coefficients of low regularity, and some fluid-structure interaction models. - Volume 1 focuses on the abstract theory of evolution - Volume 2 considers more concrete probelms relating to specific applications - Volume 3 reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear PDEs
Author: N El Karoui
Publisher: CRC Press
ISBN: 9780582307339
Category : Mathematics
Languages : en
Pages : 236
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Book Description
This book presents the texts of seminars presented during the years 1995 and 1996 at the Université Paris VI and is the first attempt to present a survey on this subject. Starting from the classical conditions for existence and unicity of a solution in the most simple case-which requires more than basic stochartic calculus-several refinements on the hypotheses are introduced to obtain more general results.
Author: Jan A Van Casteren
Publisher: World Scientific
ISBN: 9814464171
Category : Mathematics
Languages : en
Pages : 825
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Book Description
The book provides a systemic treatment of time-dependent strong Markov processes with values in a Polish space. It describes its generators and the link with stochastic differential equations in infinite dimensions. In a unifying way, where the square gradient operator is employed, new results for backward stochastic differential equations and long-time behavior are discussed in depth. The book also establishes a link between propagators or evolution families with the Feller property and time-inhomogeneous Markov processes. This mathematical material finds its applications in several branches of the scientific world, among which are mathematical physics, hedging models in financial mathematics, and population models.
