Selected Aspects of Fractional Brownian Motion

Selected Aspects of Fractional Brownian Motion

Author: Ivan Nourdin

Publisher: Springer Science & Business Media

Published: 2013-01-17

Total Pages: 133

ISBN-13: 884702823X

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Fractional Brownian motion (fBm) is a stochastic process which deviates significantly from Brownian motion and semimartingales, and others classically used in probability theory. As a centered Gaussian process, it is characterized by the stationarity of its increments and a medium- or long-memory property which is in sharp contrast with martingales and Markov processes. FBm has become a popular choice for applications where classical processes cannot model these non-trivial properties; for instance long memory, which is also known as persistence, is of fundamental importance for financial data and in internet traffic. The mathematical theory of fBm is currently being developed vigorously by a number of stochastic analysts, in various directions, using complementary and sometimes competing tools. This book is concerned with several aspects of fBm, including the stochastic integration with respect to it, the study of its supremum and its appearance as limit of partial sums involving stationary sequences, to name but a few. The book is addressed to researchers and graduate students in probability and mathematical statistics. With very few exceptions (where precise references are given), every stated result is proved.


Book Synopsis Selected Aspects of Fractional Brownian Motion by : Ivan Nourdin

Download or read book Selected Aspects of Fractional Brownian Motion written by Ivan Nourdin and published by Springer Science & Business Media. This book was released on 2013-01-17 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractional Brownian motion (fBm) is a stochastic process which deviates significantly from Brownian motion and semimartingales, and others classically used in probability theory. As a centered Gaussian process, it is characterized by the stationarity of its increments and a medium- or long-memory property which is in sharp contrast with martingales and Markov processes. FBm has become a popular choice for applications where classical processes cannot model these non-trivial properties; for instance long memory, which is also known as persistence, is of fundamental importance for financial data and in internet traffic. The mathematical theory of fBm is currently being developed vigorously by a number of stochastic analysts, in various directions, using complementary and sometimes competing tools. This book is concerned with several aspects of fBm, including the stochastic integration with respect to it, the study of its supremum and its appearance as limit of partial sums involving stationary sequences, to name but a few. The book is addressed to researchers and graduate students in probability and mathematical statistics. With very few exceptions (where precise references are given), every stated result is proved.

Fractional Brownian Motion

Fractional Brownian Motion

Author: Oksana Banna

Publisher: John Wiley & Sons

Published: 2019-04-30

Total Pages: 288

ISBN-13: 1786302608

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This monograph studies the relationships between fractional Brownian motion (fBm) and other processes of more simple form. In particular, this book solves the problem of the projection of fBm onto the space of Gaussian martingales that can be represented as Wiener integrals with respect to a Wiener process. It is proved that there exists a unique martingale closest to fBm in the uniform integral norm. Numerical results concerning the approximation problem are given. The upper bounds of distances from fBm to the different subspaces of Gaussian martingales are evaluated and the numerical calculations are involved. The approximations of fBm by a uniformly convergent series of Lebesgue integrals, semimartingales and absolutely continuous processes are presented. As auxiliary but interesting results, the bounds from below and from above for the coefficient appearing in the representation of fBm via the Wiener process are established and some new inequalities for Gamma functions, and even for trigonometric functions, are obtained.


Book Synopsis Fractional Brownian Motion by : Oksana Banna

Download or read book Fractional Brownian Motion written by Oksana Banna and published by John Wiley & Sons. This book was released on 2019-04-30 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph studies the relationships between fractional Brownian motion (fBm) and other processes of more simple form. In particular, this book solves the problem of the projection of fBm onto the space of Gaussian martingales that can be represented as Wiener integrals with respect to a Wiener process. It is proved that there exists a unique martingale closest to fBm in the uniform integral norm. Numerical results concerning the approximation problem are given. The upper bounds of distances from fBm to the different subspaces of Gaussian martingales are evaluated and the numerical calculations are involved. The approximations of fBm by a uniformly convergent series of Lebesgue integrals, semimartingales and absolutely continuous processes are presented. As auxiliary but interesting results, the bounds from below and from above for the coefficient appearing in the representation of fBm via the Wiener process are established and some new inequalities for Gamma functions, and even for trigonometric functions, are obtained.

Stochastic Calculus for Fractional Brownian Motion and Applications

Stochastic Calculus for Fractional Brownian Motion and Applications

Author: Francesca Biagini

Publisher: Springer Science & Business Media

Published: 2008-02-17

Total Pages: 331

ISBN-13: 1846287979

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The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance.


Book Synopsis Stochastic Calculus for Fractional Brownian Motion and Applications by : Francesca Biagini

Download or read book Stochastic Calculus for Fractional Brownian Motion and Applications written by Francesca Biagini and published by Springer Science & Business Media. This book was released on 2008-02-17 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance.

Stochastic Calculus for Fractional Brownian Motion and Related Processes

Stochastic Calculus for Fractional Brownian Motion and Related Processes

Author: Yuliya Mishura

Publisher: Springer

Published: 2008-04-12

Total Pages: 411

ISBN-13: 3540758739

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This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.


Book Synopsis Stochastic Calculus for Fractional Brownian Motion and Related Processes by : Yuliya Mishura

Download or read book Stochastic Calculus for Fractional Brownian Motion and Related Processes written by Yuliya Mishura and published by Springer. This book was released on 2008-04-12 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.

Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion

Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion

Author: Corinne Berzin

Publisher: Springer

Published: 2014-10-15

Total Pages: 169

ISBN-13: 3319078755

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This book is devoted to a number of stochastic models that display scale invariance. It primarily focuses on three issues: probabilistic properties, statistical estimation and simulation of the processes considered. It will be of interest to probability specialists, who will find here an uncomplicated presentation of statistics tools and to those statisticians who wants to tackle the most recent theories in probability in order to develop Central Limit Theorems in this context; both groups will also benefit from the section on simulation. Algorithms are described in great detail, with a focus on procedures that is not usually found in mathematical treatises. The models studied are fractional Brownian motions and processes that derive from them through stochastic differential equations. Concerning the proofs of the limit theorems, the “Fourth Moment Theorem” is systematically used, as it produces rapid and helpful proofs that can serve as models for the future. Readers will also find elegant and new proofs for almost sure convergence. The use of diffusion models driven by fractional noise has been popular for more than two decades now. This popularity is due both to the mathematics itself and to its fields of application. With regard to the latter, fractional models are useful for modeling real-life events such as value assets in financial markets, chaos in quantum physics, river flows through time, irregular images, weather events and contaminant diffusio n problems.


Book Synopsis Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion by : Corinne Berzin

Download or read book Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion written by Corinne Berzin and published by Springer. This book was released on 2014-10-15 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to a number of stochastic models that display scale invariance. It primarily focuses on three issues: probabilistic properties, statistical estimation and simulation of the processes considered. It will be of interest to probability specialists, who will find here an uncomplicated presentation of statistics tools and to those statisticians who wants to tackle the most recent theories in probability in order to develop Central Limit Theorems in this context; both groups will also benefit from the section on simulation. Algorithms are described in great detail, with a focus on procedures that is not usually found in mathematical treatises. The models studied are fractional Brownian motions and processes that derive from them through stochastic differential equations. Concerning the proofs of the limit theorems, the “Fourth Moment Theorem” is systematically used, as it produces rapid and helpful proofs that can serve as models for the future. Readers will also find elegant and new proofs for almost sure convergence. The use of diffusion models driven by fractional noise has been popular for more than two decades now. This popularity is due both to the mathematics itself and to its fields of application. With regard to the latter, fractional models are useful for modeling real-life events such as value assets in financial markets, chaos in quantum physics, river flows through time, irregular images, weather events and contaminant diffusio n problems.

Stochastic Calculus and Differential Equations for Physics and Finance

Stochastic Calculus and Differential Equations for Physics and Finance

Author: Joseph L. McCauley

Publisher: Cambridge University Press

Published: 2013-02-21

Total Pages: 219

ISBN-13: 0521763401

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Provides graduate students and practitioners in physics and economics with a better understanding of stochastic processes.


Book Synopsis Stochastic Calculus and Differential Equations for Physics and Finance by : Joseph L. McCauley

Download or read book Stochastic Calculus and Differential Equations for Physics and Finance written by Joseph L. McCauley and published by Cambridge University Press. This book was released on 2013-02-21 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides graduate students and practitioners in physics and economics with a better understanding of stochastic processes.

Texturing and Modeling

Texturing and Modeling

Author: David S. Ebert

Publisher: Academic Press

Published: 2014-05-19

Total Pages: 363

ISBN-13: 1483297020

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Congratulations to Ken Perlin for his 1997 Technical Achievement Award from the Academy of Motion Picture Arts and Science Board of Governors, given in recognition of the development of "Turbulence", Perlin Noise, a technique discussed in this book which is used to produce natural appearing textures on computer-generated surfaces for motion picture visual effects. Dr. Perlin joins Darwyn Peachey (co-developer of RenderMan(R), also discussed in the book) in being honored with this prestigious award. * * Written at a usable level by the developers of the techniques* Serves as a source book for those writing rendering systems, shaders, and animations.* Discusses the design and implementation of noise functions.* Contains procedural modeling of gases, hypertextures, mountains, and landscapes.* Provides a toolbox of specific procedures and basic primitive functions for producing realistic images.* Procedures are presented in C code segments or in Renderman shading language. * 3.5" disk contains the code from within the book for easy implementation


Book Synopsis Texturing and Modeling by : David S. Ebert

Download or read book Texturing and Modeling written by David S. Ebert and published by Academic Press. This book was released on 2014-05-19 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: Congratulations to Ken Perlin for his 1997 Technical Achievement Award from the Academy of Motion Picture Arts and Science Board of Governors, given in recognition of the development of "Turbulence", Perlin Noise, a technique discussed in this book which is used to produce natural appearing textures on computer-generated surfaces for motion picture visual effects. Dr. Perlin joins Darwyn Peachey (co-developer of RenderMan(R), also discussed in the book) in being honored with this prestigious award. * * Written at a usable level by the developers of the techniques* Serves as a source book for those writing rendering systems, shaders, and animations.* Discusses the design and implementation of noise functions.* Contains procedural modeling of gases, hypertextures, mountains, and landscapes.* Provides a toolbox of specific procedures and basic primitive functions for producing realistic images.* Procedures are presented in C code segments or in Renderman shading language. * 3.5" disk contains the code from within the book for easy implementation

The Malliavin Calculus and Related Topics

The Malliavin Calculus and Related Topics

Author: David Nualart

Publisher: Springer Science & Business Media

Published: 2013-12-11

Total Pages: 273

ISBN-13: 1475724373

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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.


Book Synopsis The Malliavin Calculus and Related Topics by : David Nualart

Download or read book The Malliavin Calculus and Related Topics written by David Nualart and published by Springer Science & Business Media. This book was released on 2013-12-11 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Fractional Brownian Motion

Fractional Brownian Motion

Author: Oksana Banna

Publisher: John Wiley & Sons

Published: 2019-04-10

Total Pages: 258

ISBN-13: 1119610338

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This monograph studies the relationships between fractional Brownian motion (fBm) and other processes of more simple form. In particular, this book solves the problem of the projection of fBm onto the space of Gaussian martingales that can be represented as Wiener integrals with respect to a Wiener process. It is proved that there exists a unique martingale closest to fBm in the uniform integral norm. Numerical results concerning the approximation problem are given. The upper bounds of distances from fBm to the different subspaces of Gaussian martingales are evaluated and the numerical calculations are involved. The approximations of fBm by a uniformly convergent series of Lebesgue integrals, semimartingales and absolutely continuous processes are presented. As auxiliary but interesting results, the bounds from below and from above for the coefficient appearing in the representation of fBm via the Wiener process are established and some new inequalities for Gamma functions, and even for trigonometric functions, are obtained.


Book Synopsis Fractional Brownian Motion by : Oksana Banna

Download or read book Fractional Brownian Motion written by Oksana Banna and published by John Wiley & Sons. This book was released on 2019-04-10 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph studies the relationships between fractional Brownian motion (fBm) and other processes of more simple form. In particular, this book solves the problem of the projection of fBm onto the space of Gaussian martingales that can be represented as Wiener integrals with respect to a Wiener process. It is proved that there exists a unique martingale closest to fBm in the uniform integral norm. Numerical results concerning the approximation problem are given. The upper bounds of distances from fBm to the different subspaces of Gaussian martingales are evaluated and the numerical calculations are involved. The approximations of fBm by a uniformly convergent series of Lebesgue integrals, semimartingales and absolutely continuous processes are presented. As auxiliary but interesting results, the bounds from below and from above for the coefficient appearing in the representation of fBm via the Wiener process are established and some new inequalities for Gamma functions, and even for trigonometric functions, are obtained.

Stochastic Calculus for Fractional Brownian Motion and Related Processes

Stochastic Calculus for Fractional Brownian Motion and Related Processes

Author: Yuliya Mishura

Publisher: Springer Science & Business Media

Published: 2008-01-02

Total Pages: 411

ISBN-13: 3540758720

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This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.


Book Synopsis Stochastic Calculus for Fractional Brownian Motion and Related Processes by : Yuliya Mishura

Download or read book Stochastic Calculus for Fractional Brownian Motion and Related Processes written by Yuliya Mishura and published by Springer Science & Business Media. This book was released on 2008-01-02 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.