the malliavin calculus and related topics probability and its applications

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The Malliavin Calculus And Related Topics

Author : David Nualart
ISBN : 9781475724370
Genre : Mathematics
File Size : 47. 58 MB
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The origin of this book lies in an invitation to give a series of lectures on Malliavin calculus at the Probability Seminar of Venezuela, in April 1985. The contents of these lectures were published in Spanish in [176]. Later these notes were completed and improved in two courses on Malliavin cal culus given at the University of California at Irvine in 1986 and at Ecole Polytechnique Federale de Lausanne in 1989. The contents of these courses correspond to the material presented in Chapters 1 and 2 of this book. Chapter 3 deals with the anticipating stochastic calculus and it was de veloped from our collaboration with Moshe Zakai and Etienne Pardoux. The series of lectures given at the Eighth Chilean Winter School in Prob ability and Statistics, at Santiago de Chile, in July 1989, allowed us to write a pedagogical approach to the anticipating calculus which is the basis of Chapter 3. Chapter 4 deals with the nonlinear transformations of the Wiener measure and their applications to the study of the Markov property for solutions to stochastic differential equations with boundary conditions.

Generalized Functionals Of Brownian Motion And Their Applications

Author : Nasir Uddin Ahmed
ISBN : 9789814366373
Genre : Brownian motion processes
File Size : 30. 43 MB
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This invaluable research monograph presents a unified and fascinating theory of generalized functionals of Brownian motion and other fundamental processes such as fractional Brownian motion and Levy process OCo covering the classical WienerOCoIto class including the generalized functionals of Hida as special cases, among others. It presents a thorough and comprehensive treatment of the WienerOCoSobolev spaces and their duals, as well as Malliavin calculus with their applications. The presentation is lucid and logical, and is based on a solid foundation of analysis and topology. The monograph develops the notions of compactness and weak compactness on these abstract Fock spaces and their duals, clearly demonstrating their nontrivial applications to stochastic differential equations in finite and infinite dimensional Hilbert spaces, optimization and optimal control problems. Readers will find the book an interesting and easy read as materials are presented in a systematic manner with a complete analysis of classical and generalized functionals of scalar Brownian motion, Gaussian random fields and their vector versions in the increasing order of generality. It starts with abstract Fourier analysis on the Wiener measure space where a striking similarity of the celebrated RieszOCoFischer theorem for separable Hilbert spaces and the space of WienerOCoIto functionals is drawn out, thus providing a clear insight into the subject.

The Malliavin Calculus

Author : Denis R. Bell
ISBN : 9780486152059
Genre : Mathematics
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This introductory text presents detailed accounts of the different forms of the theory developed by Stroock and Bismut, discussions of the relationship between these two approaches, and a variety of applications. 1987 edition.

Equations Involving Malliavin Calculus Operators

Author : Tijana Levajković
ISBN : 9783319656786
Genre : Mathematics
File Size : 44. 31 MB
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This book provides a comprehensive and unified introduction to stochastic differential equations and related optimal control problems. The material is new and the presentation is reader-friendly. A major contribution of the book is the development of generalized Malliavin calculus in the framework of white noise analysis, based on chaos expansion representation of stochastic processes and its application for solving several classes of stochastic differential equations with singular data involving the main operators of Malliavin calculus. In addition, applications in optimal control and numerical approximations are discussed. The book is divided into four chapters. The first, entitled White Noise Analysis and Chaos Expansions, includes notation and provides the reader with the theoretical background needed to understand the subsequent chapters. In particular, we introduce spaces of random variables and stochastic processes, and consider processes that have finite variance on classical and fractional Gaussian white noise probability spaces. We also present processes with infinite variance, particularly Kondratiev stochastic distributions. We introduce the Wick and ordinary multiplication of the processes and state where these operations are well defined. In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator D, the Skorokhod integral δ and the Ornstein-Uhlenbeck operator R are introduced in terms of chaos expansions. The main properties of the operators, which are known in the literature for the square integrable processes, are proven using the chaos expansion approach and extended for generalized and test stochastic processes. Moreover, we discuss fractional versions of these operators. Chapter 3, Equations involving Malliavin Calculus operators, is devoted to the study of several types of stochastic differential equations that involve the operators of Malliavin calculus, introduced in the previous chapter. In particular, we describe the range of the operators D, δ and R. Finally, in Chapter 4, Applications and Numerical Approximations are discussed. Specifically, we consider the stochastic linear quadratic optimal control problem with different forms of noise disturbances, operator differential algebraic equations arising in fluid dynamics, stationary equations and fractional versions of the equations studied – applications never covered in the extant literature. Moreover, numerical validations of the method are provided for specific problems.

Malliavin Calculus And Its Applications

Author : David Nualart
ISBN : 9780821847794
Genre : Mathematics
File Size : 66. 55 MB
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The Malliavin calculus was developed to provide a probabilistic proof of Hormander's hypoellipticity theorem. The theory has expanded to encompass other significant applications. The main application of the Malliavin calculus is to establish the regularity of the probability distribution of functionals of an underlying Gaussian process. In this way, one can prove the existence and smoothness of the density for solutions of various stochastic differential equations. More recently, applications of the Malliavin calculus in areas such as stochastic calculus for fractional Brownian motion, central limit theorems for multiple stochastic integrals, and mathematical finance have emerged. The first part of the book covers the basic results of the Malliavin calculus. The middle part establishes the existence and smoothness results that then lead to the proof of Hormander's hypoellipticity theorem. The last part discusses the recent developments for Brownian motion, central limit theorems, and mathematical finance.

Stochastic Analysis And Related Topics

Author : Hayri Korezlioglu
ISBN : 9783540391869
Genre : Mathematics
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The Silvri Workshop was divided into a short summer school and a working conference, producing lectures and research papers on recent developments in stochastic analysis on Wiener space. The topics treated in the lectures relate to the Malliavin calculus, the Skorohod integral and nonlinear functionals of white noise. Most of the research papers are applications of these subjects. This volume addresses researchers and graduate students in stochastic processes and theoretical physics.

Malliavin Calculus And Stochastic Analysis

Author : Frederi Viens
ISBN : 9781461459064
Genre : Mathematics
File Size : 43. 62 MB
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The stochastic calculus of variations of Paul Malliavin (1925 - 2010), known today as the Malliavin Calculus, has found many applications, within and beyond the core mathematical discipline. Stochastic analysis provides a fruitful interpretation of this calculus, particularly as described by David Nualart and the scores of mathematicians he influences and with whom he collaborates. Many of these, including leading stochastic analysts and junior researchers, presented their cutting-edge research at an international conference in honor of David Nualart's career, on March 19-21, 2011, at the University of Kansas, USA. These scholars and other top-level mathematicians have kindly contributed research articles for this refereed volume.

Internationale Mathematische Nachrichten

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ISBN : UCAL:B4591150
Genre : Mathematics
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Issues for Dec. 1952- include section: Nachrichten der Österreichischen Mathematischen Gesellschaft.

Loeb Measures In Practice Recent Advances

Author : Nigel Cutland
ISBN : 3540413847
Genre : Business & Economics
File Size : 74. 15 MB
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This expanded version of the 1997 European Mathematical Society Lectures given by the author in Helsinki, begins with a self-contained introduction to nonstandard analysis (NSA) and the construction of Loeb Measures, which are rich measures discovered in 1975 by Peter Loeb, using techniques from NSA. Subsequent chapters sketch a range of recent applications of Loeb measures due to the author and his collaborators, in such diverse fields as (stochastic) fluid mechanics, stochastic calculus of variations ("Malliavin" calculus) and the mathematical finance theory. The exposition is designed for a general audience, and no previous knowledge of either NSA or the various fields of applications is assumed.

Bulletin Of The American Mathematical Society

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ISBN : UOM:49015003188605
Genre : Mathematics
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