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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.

Normal Approximations with Malliavin Calculus

Author: Ivan Nourdin

Publisher: Cambridge University Press

Published: 2012-05-10

Total Pages: 255

ISBN-13: 1107017777

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This book shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus.

Book Synopsis Normal Approximations with Malliavin Calculus by : Ivan Nourdin

Download or read book Normal Approximations with Malliavin Calculus written by Ivan Nourdin and published by Cambridge University Press. This book was released on 2012-05-10 with total page 255 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus.

Brownian Motion

Author: Peter Mörters

Publisher: Cambridge University Press

Published: 2010-03-25

Total Pages:

ISBN-13: 1139486578

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This eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes.

Book Synopsis Brownian Motion by : Peter Mörters

Download or read book Brownian Motion written by Peter Mörters and published by Cambridge University Press. This book was released on 2010-03-25 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes.

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.

Brownian Motion

Author: Mark A. McKibben

Publisher: Nova Science Publishers

Published: 2015

Total Pages: 0

ISBN-13: 9781634836821

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The fields of study in which random fluctuations arise and cannot be ignored are as disparate and numerous as there are synonyms for the word "noise." In the nearly two centuries following the discovery of what has come to be known as Brownian motion, named in homage to botanist Robert Brown, scientists, engineers, financial analysts, mathematicians, and literary authors have posited theories, created models, and composed literary works which have accounted for environmental noise. This volume offers a glimpse into the ways in which Brownian motion has crept into a myriad of fields of study through fifteen distinct chapters written by mathematicians, physicists, and other scholars. The intent is to especially highlight the vastness of scholarly work that explains various facets of Nature made possible by one scientist's curiosity sparked by observing sporadic movement of specks of pollen under a microscope in a 19th century laboratory.

Book Synopsis Brownian Motion by : Mark A. McKibben

Download or read book Brownian Motion written by Mark A. McKibben and published by Nova Science Publishers. This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The fields of study in which random fluctuations arise and cannot be ignored are as disparate and numerous as there are synonyms for the word "noise." In the nearly two centuries following the discovery of what has come to be known as Brownian motion, named in homage to botanist Robert Brown, scientists, engineers, financial analysts, mathematicians, and literary authors have posited theories, created models, and composed literary works which have accounted for environmental noise. This volume offers a glimpse into the ways in which Brownian motion has crept into a myriad of fields of study through fifteen distinct chapters written by mathematicians, physicists, and other scholars. The intent is to especially highlight the vastness of scholarly work that explains various facets of Nature made possible by one scientist's curiosity sparked by observing sporadic movement of specks of pollen under a microscope in a 19th century laboratory.

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 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.

Stochastic Analysis of Mixed Fractional Gaussian Processes

Author: Yuliya Mishura

Publisher: Elsevier

Published: 2018-05-26

Total Pages: 210

ISBN-13: 0081023634

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Stochastic Analysis of Mixed Fractional Gaussian Processes presents the main tools necessary to characterize Gaussian processes. The book focuses on the particular case of the linear combination of independent fractional and sub-fractional Brownian motions with different Hurst indices. Stochastic integration with respect to these processes is considered, as is the study of the existence and uniqueness of solutions of related SDE's. Applications in finance and statistics are also explored, with each chapter supplying a number of exercises to illustrate key concepts. Presents both mixed fractional and sub-fractional Brownian motions Provides an accessible description for mixed fractional gaussian processes that is ideal for Master's and PhD students Includes different Hurst indices

Book Synopsis Stochastic Analysis of Mixed Fractional Gaussian Processes by : Yuliya Mishura

Download or read book Stochastic Analysis of Mixed Fractional Gaussian Processes written by Yuliya Mishura and published by Elsevier. This book was released on 2018-05-26 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic Analysis of Mixed Fractional Gaussian Processes presents the main tools necessary to characterize Gaussian processes. The book focuses on the particular case of the linear combination of independent fractional and sub-fractional Brownian motions with different Hurst indices. Stochastic integration with respect to these processes is considered, as is the study of the existence and uniqueness of solutions of related SDE's. Applications in finance and statistics are also explored, with each chapter supplying a number of exercises to illustrate key concepts. Presents both mixed fractional and sub-fractional Brownian motions Provides an accessible description for mixed fractional gaussian processes that is ideal for Master's and PhD students Includes different Hurst indices

Stochastic Calculus via Regularizations

Author: Francesco Russo

Publisher: Springer Nature

Published: 2022-11-15

Total Pages: 656

ISBN-13: 3031094468

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The book constitutes an introduction to stochastic calculus, stochastic differential equations and related topics such as Malliavin calculus. On the other hand it focuses on the techniques of stochastic integration and calculus via regularization initiated by the authors. The definitions relies on a smoothing procedure of the integrator process, they generalize the usual Itô and Stratonovich integrals for Brownian motion but the integrator could also not be a semimartingale and the integrand is allowed to be anticipating. The resulting calculus requires a simple formalism: nevertheless it entails pathwise techniques even though it takes into account randomness. It allows connecting different types of pathwise and non pathwise integrals such as Young, fractional, Skorohod integrals, enlargement of filtration and rough paths. The covariation, but also high order variations, play a fundamental role in the calculus via regularization, which can also be applied for irregular integrators. A large class of Gaussian processes, various generalizations of semimartingales such that Dirichlet and weak Dirichlet processes are revisited. Stochastic calculus via regularization has been successfully used in applications, for instance in robust finance and on modeling vortex filaments in turbulence. The book is addressed to PhD students and researchers in stochastic analysis and applications to various fields.

Book Synopsis Stochastic Calculus via Regularizations by : Francesco Russo

Download or read book Stochastic Calculus via Regularizations written by Francesco Russo and published by Springer Nature. This book was released on 2022-11-15 with total page 656 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book constitutes an introduction to stochastic calculus, stochastic differential equations and related topics such as Malliavin calculus. On the other hand it focuses on the techniques of stochastic integration and calculus via regularization initiated by the authors. The definitions relies on a smoothing procedure of the integrator process, they generalize the usual Itô and Stratonovich integrals for Brownian motion but the integrator could also not be a semimartingale and the integrand is allowed to be anticipating. The resulting calculus requires a simple formalism: nevertheless it entails pathwise techniques even though it takes into account randomness. It allows connecting different types of pathwise and non pathwise integrals such as Young, fractional, Skorohod integrals, enlargement of filtration and rough paths. The covariation, but also high order variations, play a fundamental role in the calculus via regularization, which can also be applied for irregular integrators. A large class of Gaussian processes, various generalizations of semimartingales such that Dirichlet and weak Dirichlet processes are revisited. Stochastic calculus via regularization has been successfully used in applications, for instance in robust finance and on modeling vortex filaments in turbulence. The book is addressed to PhD students and researchers in stochastic analysis and applications to various fields.

Ambit Stochastics

Author: Ole E. Barndorff-Nielsen

Publisher: Springer

Published: 2018-11-01

Total Pages: 402

ISBN-13: 3319941291

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Drawing on advanced probability theory, Ambit Stochastics is used to model stochastic processes which depend on both time and space. This monograph, the first on the subject, provides a reference for this burgeoning field, complete with the applications that have driven its development. Unique to Ambit Stochastics are ambit sets, which allow the delimitation of space-time to a zone of interest, and ambit fields, which are particularly well-adapted to modelling stochastic volatility or intermittency. These attributes lend themselves notably to applications in the statistical theory of turbulence and financial econometrics. In addition to the theory and applications of Ambit Stochastics, the book also contains new theory on the simulation of ambit fields and a comprehensive stochastic integration theory for Volterra processes in a non-semimartingale context. Written by pioneers in the subject, this book will appeal to researchers and graduate students interested in empirical stochastic modelling.

Book Synopsis Ambit Stochastics by : Ole E. Barndorff-Nielsen

Download or read book Ambit Stochastics written by Ole E. Barndorff-Nielsen and published by Springer. This book was released on 2018-11-01 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Drawing on advanced probability theory, Ambit Stochastics is used to model stochastic processes which depend on both time and space. This monograph, the first on the subject, provides a reference for this burgeoning field, complete with the applications that have driven its development. Unique to Ambit Stochastics are ambit sets, which allow the delimitation of space-time to a zone of interest, and ambit fields, which are particularly well-adapted to modelling stochastic volatility or intermittency. These attributes lend themselves notably to applications in the statistical theory of turbulence and financial econometrics. In addition to the theory and applications of Ambit Stochastics, the book also contains new theory on the simulation of ambit fields and a comprehensive stochastic integration theory for Volterra processes in a non-semimartingale context. Written by pioneers in the subject, this book will appeal to researchers and graduate students interested in empirical stochastic modelling.