Markov Processes And Differential Equations

Author : Mark I. Freidlin
Publisher : Birkhäuser
Page : 155 pages
File Size : 48,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034891912

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Markov Processes and Differential Equations by Mark I. Freidlin Pdf

Probabilistic methods can be applied very successfully to a number of asymptotic problems for second-order linear and non-linear partial differential equations. Due to the close connection between the second order differential operators with a non-negative characteristic form on the one hand and Markov processes on the other, many problems in PDE's can be reformulated as problems for corresponding stochastic processes and vice versa. In the present book four classes of problems are considered: - the Dirichlet problem with a small parameter in higher derivatives for differential equations and systems - the averaging principle for stochastic processes and PDE's - homogenization in PDE's and in stochastic processes - wave front propagation for semilinear differential equations and systems. From the probabilistic point of view, the first two topics concern random perturbations of dynamical systems. The third topic, homog- enization, is a natural problem for stochastic processes as well as for PDE's. Wave fronts in semilinear PDE's are interesting examples of pattern formation in reaction-diffusion equations. The text presents new results in probability theory and their applica- tion to the above problems. Various examples help the reader to understand the effects. Prerequisites are knowledge in probability theory and in partial differential equations.

Author : Rabi Bhattacharya,Edward C. Waymire
Publisher : Springer Nature
Page : 502 pages
File Size : 54,6 Mb
Release : 2023-11-16
Category : Mathematics
ISBN : 9783031332968

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Continuous Parameter Markov Processes and Stochastic Differential Equations by Rabi Bhattacharya,Edward C. Waymire Pdf

This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications. The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples. After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem, used to construct continuous parameter Markov processes. Illustrated with examples, it is a cornerstone of Feller’s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes, and processes with independent increments, or Lévy processes. The greater part of the book is devoted to Itô’s fascinating theory of stochastic differential equations, and to the study of asymptotic properties of diffusions in all dimensions, such as explosion, transience, recurrence, existence of steady states, and the speed of convergence to equilibrium. A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes. Among Special Topics chapters, two study anomalous diffusions: one on skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.

Author : Wendell H. Fleming,Halil Mete Soner
Publisher : Springer Science & Business Media
Page : 436 pages
File Size : 52,7 Mb
Release : 2006-02-04
Category : Mathematics
ISBN : 9780387310718

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Controlled Markov Processes and Viscosity Solutions by Wendell H. Fleming,Halil Mete Soner Pdf

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 : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 43,6 Mb
Release : 2024-06-29
Category : Electronic
ISBN : 9789814464178

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Markov Processes, Feller Semigroups and Evolution Equations by Anonim Pdf

Author : J. A. van Casteren
Publisher : World Scientific
Page : 825 pages
File Size : 49,6 Mb
Release : 2011
Category : Mathematics
ISBN : 9789814322188

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Markov Processes, Feller Semigroups and Evolution Equations by J. A. van Casteren Pdf

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.

Author : Xuerong Mao,Chenggui Yuan
Publisher : Imperial College Press
Page : 430 pages
File Size : 42,6 Mb
Release : 2006
Category : Mathematics
ISBN : 9781860947018

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Stochastic Differential Equations with Markovian Switching by Xuerong Mao,Chenggui Yuan Pdf

This textbook provides the first systematic presentation of the theory of stochastic differential equations with Markovian switching. It presents the basic principles at an introductory level but emphasizes current advanced level research trends. The material takes into account all the features of Ito equations, Markovian switching, interval systems and time-lag. The theory developed is applicable in different and complicated situations in many branches of science and industry.

Author : Daniel T. Gillespie
Publisher : Elsevier
Page : 590 pages
File Size : 51,6 Mb
Release : 1991-12-02
Category : Mathematics
ISBN : 9780080918372

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Markov Processes by Daniel T. Gillespie Pdf

Markov process theory is basically an extension of ordinary calculus to accommodate functions whos time evolutions are not entirely deterministic. It is a subject that is becoming increasingly important for many fields of science. This book develops the single-variable theory of both continuous and jump Markov processes in a way that should appeal especially to physicists and chemists at the senior and graduate level. A self-contained, prgamatic exposition of the needed elements of random variable theory Logically integrated derviations of the Chapman-Kolmogorov equation, the Kramers-Moyal equations, the Fokker-Planck equations, the Langevin equation, the master equations, and the moment equations Detailed exposition of Monte Carlo simulation methods, with plots of many numerical examples Clear treatments of first passages, first exits, and stable state fluctuations and transitions Carefully drawn applications to Brownian motion, molecular diffusion, and chemical kinetics

Author : Sigurd Assing,Wolfgang M. Schmidt
Publisher : Springer
Page : 146 pages
File Size : 45,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540697862

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Continuous Strong Markov Processes in Dimension One by Sigurd Assing,Wolfgang M. Schmidt Pdf

The book presents an in-depth study of arbitrary one-dimensional continuous strong Markov processes using methods of stochastic calculus. Departing from the classical approaches, a unified investigation of regular as well as arbitrary non-regular diffusions is provided. A general construction method for such processes, based on a generalization of the concept of a perfect additive functional, is developed. The intrinsic decomposition of a continuous strong Markov semimartingale is discovered. The book also investigates relations to stochastic differential equations and fundamental examples of irregular diffusions.

Author : Kazuaki Taira
Publisher : Springer
Page : 192 pages
File Size : 43,7 Mb
Release : 2009-06-17
Category : Mathematics
ISBN : 9783642016776

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Boundary Value Problems and Markov Processes by Kazuaki Taira Pdf

This is a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel’ boundary conditions in probability theory. It presents new developments in the theory of singular integrals.

Author : Daniel W. Stroock
Publisher : Princeton University Press
Page : 289 pages
File Size : 50,9 Mb
Release : 2003-05-06
Category : Mathematics
ISBN : 9781400835577

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Markov Processes from K. Itô's Perspective (AM-155) by Daniel W. Stroock Pdf

Kiyosi Itô's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Itô's program. The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Itô interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Itô's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Itô's stochastic integral calculus. In the second half, the author provides a systematic development of Itô's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Itô's theme and ends with an application to the characterization of the paths on which a diffusion is supported. The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes.

Author : Daniel W. Stroock
Publisher : Princeton University Press
Page : 288 pages
File Size : 46,5 Mb
Release : 2003-05-26
Category : Mathematics
ISBN : 9780691115436

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Markov Processes from K. Itô's Perspective by Daniel W. Stroock Pdf

Kiyosi Itô's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Itô's program. The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Itô interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Itô's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Itô's stochastic integral calculus. In the second half, the author provides a systematic development of Itô's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Itô's theme and ends with an application to the characterization of the paths on which a diffusion is supported. The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes.

Author : Rafail Khasminskii
Publisher : Springer Science & Business Media
Page : 353 pages
File Size : 53,8 Mb
Release : 2011-09-20
Category : Mathematics
ISBN : 9783642232800

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Stochastic Stability of Differential Equations by Rafail Khasminskii Pdf

Since the publication of the first edition of the present volume in 1980, the stochastic stability of differential equations has become a very popular subject of research in mathematics and engineering. To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure stability, and for the existence of stationary and periodic solutions of stochastic differential equations have been widely used in the literature. In this updated volume readers will find important new results on the moment Lyapunov exponent, stability index and some other fields, obtained after publication of the first edition, and a significantly expanded bibliography. This volume provides a solid foundation for students in graduate courses in mathematics and its applications. It is also useful for those researchers who would like to learn more about this subject, to start their research in this area or to study the properties of concrete mechanical systems subjected to random perturbations.

Author : Vassili N. Kolokoltsov
Publisher : Cambridge University Press
Page : 394 pages
File Size : 40,6 Mb
Release : 2010-07-15
Category : Mathematics
ISBN : 9781139489737

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Nonlinear Markov Processes and Kinetic Equations by Vassili N. Kolokoltsov Pdf

A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from general vector-valued differential equations and yields a natural link with probability, both in interpreting results and in the tools of analysis. This brilliant book, the first devoted to the area, develops this interplay between probability and analysis. After systematically presenting both analytic and probabilistic techniques, the author uses probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior. The book addresses the most fundamental questions in the theory of nonlinear Markov processes: existence, uniqueness, constructions, approximation schemes, regularity, law of large numbers and probabilistic interpretations. Its careful exposition makes the book accessible to researchers and graduate students in stochastic and functional analysis with applications to mathematical physics and systems biology.

Author : Serguei Primak,Valeri Kontorovich,Vladimir Lyandres
Publisher : John Wiley & Sons
Page : 446 pages
File Size : 47,8 Mb
Release : 2005-01-28
Category : Technology & Engineering
ISBN : 9780470021170

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Stochastic Methods and their Applications to Communications by Serguei Primak,Valeri Kontorovich,Vladimir Lyandres Pdf

Stochastic Methods & their Applications to Communications presents a valuable approach to the modelling, synthesis and numerical simulation of random processes with applications in communications and related fields. The authors provide a detailed account of random processes from an engineering point of view and illustrate the concepts with examples taken from the communications area. The discussions mainly focus on the analysis and synthesis of Markov models of random processes as applied to modelling such phenomena as interference and fading in communications. Encompassing both theory and practice, this original text provides a unified approach to the analysis and generation of continuous, impulsive and mixed random processes based on the Fokker-Planck equation for Markov processes. Presents the cumulated analysis of Markov processes Offers a SDE (Stochastic Differential Equations) approach to the generation of random processes with specified characteristics Includes the modelling of communication channels and interfer ences using SDE Features new results and techniques for the of solution of the generalized Fokker-Planck equation Essential reading for researchers, engineers, and graduate and upper year undergraduate students in the field of communications, signal processing, control, physics and other areas of science, this reference will have wide ranging appeal.

Author : Stewart N. Ethier,Thomas G. Kurtz
Publisher : John Wiley & Sons
Page : 552 pages
File Size : 47,8 Mb
Release : 1986-04-04
Category : Mathematics
ISBN : UCAL:B4405430

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Markov Processes by Stewart N. Ethier,Thomas G. Kurtz Pdf

As a graduate text/reference on Markov Processes and their relationship to operator semigroups, this book presents several different approaches to proving weak approximation theorems for Markov processes, emphasizing the interplay of methods of characterization and approximation.