Malliavin Calculus And Its Applications
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Malliavin Calculus And Its Applications
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Author : David Nualart
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
Malliavin Calculus And Its Applications written by David Nualart and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.
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.
Malliavin Calculus
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Author : Marta Sanz Solé
language : en
Publisher: EPFL Press
Release Date : 2005-01-01
Malliavin Calculus written by Marta Sanz Solé and has been published by EPFL Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-01-01 with Mathematics categories.
Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics. This book presents applications of Malliavin calculus to the analysis of probability laws of solutions to stochastic partial differential equations driven by Gaussian noises that are white in time and coloured in space. The first five chapters introduce the calculus itself b.
The Malliavin Calculus
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Author : Denis R. Bell
language : en
Publisher: Courier Corporation
Release Date : 2012-12-03
The Malliavin Calculus written by Denis R. Bell and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-03 with Mathematics categories.
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.
Malliavin Calculus
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Author : Jamie Flux
language : en
Publisher: Independently Published
Release Date : 2024-12-05
Malliavin Calculus written by Jamie Flux and has been published by Independently Published this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-05 with Mathematics categories.
Embark on a transformative journey through the forefront of stochastic calculus with this comprehensive and authoritative exploration of Malliavin Calculus and its myriad applications. This monumental work, spanning 66 meticulously crafted chapters, delves into advanced mathematical concepts and innovative methodologies that challenge conventional boundaries and pioneer new horizons in mathematics and programming. Key Features: In-Depth Theoretical Frameworks: Gain a profound understanding of the Wiener space, Malliavin derivative, and Skorokhod integral, including their definitions, properties, and intricate relationships within the stochastic calculus landscape. Cutting-Edge Applications: Explore the application of Malliavin Calculus across diverse fields such as quantitative finance, machine learning, quantum stochastic calculus, stochastic control theory, and robotics. Each chapter provides original theoretical developments that bridge the gap between abstract concepts and practical implementation. Advanced Topics in Stochastic Analysis: Delve into specialized subjects like fractional Brownian motion, stochastic partial differential equations, non-commutative Malliavin Calculus, and stochastic topology, pushing the boundaries of traditional analysis and opening avenues for future research. Interdisciplinary Perspectives: Benefit from the integration of interdisciplinary approaches that connect mathematics with fields like biology, physics, economics, and engineering, demonstrating the universal applicability of Malliavin Calculus. Examples of Groundbreaking Content: In the chapter on Fractional Brownian Motion and Malliavin Calculus, discover pioneering techniques for addressing the challenges posed by the non-Markovian nature of fractional Brownian motion. Learn how these methods open new pathways in modeling and analyzing systems with memory effects, impacting fields such as financial mathematics and signal processing. Stochastic Neural Networks and Deep Learning offers an in-depth investigation into how stochastic differential equations model neural dynamics. Uncover innovative methodologies that enhance the robustness and interpretability of deep learning models, crucial for advancing artificial intelligence and machine learning applications. Explore Non-Commutative Malliavin Calculus, where the complexities of defining derivatives and integrals in non-commutative settings are unraveled. This chapter provides advanced theoretical developments with significant implications for quantum field theory and quantum information, equipping readers to tackle complex quantum stochastic systems. The chapter on Stochastic Optimal Transportation introduces the emerging field that connects Malliavin Calculus with optimal transport theory. Investigate transport maps and coupling of stochastic processes, with applications that extend to economics and fluid dynamics.
Analysis Of Variations For Self Similar Processes
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Author : Ciprian Tudor
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-08-13
Analysis Of Variations For Self Similar Processes written by Ciprian Tudor 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 2013-08-13 with Mathematics categories.
Self-similar processes are stochastic processes that are invariant in distribution under suitable time scaling, and are a subject intensively studied in the last few decades. This book presents the basic properties of these processes and focuses on the study of their variation using stochastic analysis. While self-similar processes, and especially fractional Brownian motion, have been discussed in several books, some new classes have recently emerged in the scientific literature. Some of them are extensions of fractional Brownian motion (bifractional Brownian motion, subtractional Brownian motion, Hermite processes), while others are solutions to the partial differential equations driven by fractional noises. In this monograph the author discusses the basic properties of these new classes of self-similar processes and their interrelationship. At the same time a new approach (based on stochastic calculus, especially Malliavin calculus) to studying the behavior of the variations of self-similar processes has been developed over the last decade. This work surveys these recent techniques and findings on limit theorems and Malliavin calculus.
Malliavin Calculus With Applications To Stochastic Partial Differential Equations
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Author : Marta Sanz-Sole
language : en
Publisher: CRC Press
Release Date : 2005-08-17
Malliavin Calculus With Applications To Stochastic Partial Differential Equations written by Marta Sanz-Sole and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-08-17 with Mathematics categories.
Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics. This book present
Differentiable Measures And The Malliavin Calculus
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Author : Vladimir Igorevich Bogachev
language : en
Publisher: American Mathematical Soc.
Release Date : 2010-07-21
Differentiable Measures And The Malliavin Calculus written by Vladimir Igorevich Bogachev and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-21 with Mathematics categories.
This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject.
Backward Stochastic Differential Equations
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Author : N El Karoui
language : en
Publisher: CRC Press
Release Date : 1997-01-17
Backward Stochastic Differential Equations written by N El Karoui and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-17 with Mathematics categories.
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.
Dirichlet Forms And Analysis On Wiener Space
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Author : Nicolas Bouleau
language : en
Publisher: Walter de Gruyter
Release Date : 2010-10-13
Dirichlet Forms And Analysis On Wiener Space written by Nicolas Bouleau and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-13 with Mathematics categories.
The subject of this book is analysis on Wiener space by means of Dirichlet forms and Malliavin calculus. There are already several literature on this topic, but this book has some different viewpoints. First the authors review the theory of Dirichlet forms, but they observe only functional analytic, potential theoretical and algebraic properties. They do not mention the relation with Markov processes or stochastic calculus as discussed in usual books (e.g. Fukushima’s book). Even on analytic properties, instead of mentioning the Beuring-Deny formula, they discuss “carré du champ” operators introduced by Meyer and Bakry very carefully. Although they discuss when this “carré du champ” operator exists in general situation, the conditions they gave are rather hard to verify, and so they verify them in the case of Ornstein-Uhlenbeck operator in Wiener space later. (It should be noticed that one can easily show the existence of “carré du champ” operator in this case by using Shigekawa’s H-derivative.) In the part on Malliavin calculus, the authors mainly discuss the absolute continuity of the probability law of Wiener functionals. The Dirichlet form corresponds to the first derivative only, and so it is not easy to consider higher order derivatives in this framework. This is the reason why they discuss only the first step of Malliavin calculus. On the other hand, they succeeded to deal with some delicate problems (the absolute continuity of the probability law of the solution to stochastic differential equations with Lipschitz continuous coefficients, the domain of stochastic integrals (Itô-Ramer-Skorokhod integrals), etc.). This book focuses on the abstract structure of Dirichlet forms and Malliavin calculus rather than their applications. However, the authors give a lot of exercises and references and they may help the reader to study other topics which are not discussed in this book. Zentralblatt Math, Reviewer: S.Kusuoka (Hongo)
Malliavin Calculus For L Vy Processes With Applications To Finance
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Author : Giulia Di Nunno
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-10-08
Malliavin Calculus For L Vy Processes With Applications To Finance written by Giulia Di Nunno 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 2008-10-08 with Mathematics categories.
This book is an introduction to Malliavin calculus as a generalization of the classical non-anticipating Ito calculus to an anticipating setting. It presents the development of the theory and its use in new fields of application.