Selected Aspects Of Fractional Brownian Motion

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Selected Aspects of Fractional Brownian Motion

Ivan Nourdin
Selected Aspects of Fractional Brownian Motion Book PDF

Author : Ivan Nourdin
Publisher : Springer Science & Business Media
Page : 133 pages
File Size : 40,7 Mb
Release : 2013-01-17
Category : Mathematics
ISBN : 9788847028234

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Selected Aspects of Fractional Brownian Motion by Ivan Nourdin Pdf

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

Oksana Banna,Yuliya Mishura,Kostiantyn Ralchenko,Sergiy Shklyar
Fractional Brownian Motion Book PDF

Author : Oksana Banna,Yuliya Mishura,Kostiantyn Ralchenko,Sergiy Shklyar
Publisher : John Wiley & Sons
Page : 288 pages
File Size : 46,7 Mb
Release : 2019-04-30
Category : Mathematics
ISBN : 9781786302601

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Fractional Brownian Motion by Oksana Banna,Yuliya Mishura,Kostiantyn Ralchenko,Sergiy Shklyar Pdf

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

Francesca Biagini,Yaozhong Hu,Bernt Øksendal,Tusheng Zhang
Stochastic Calculus for Fractional Brownian Motion and Applications Book PDF

Author : Francesca Biagini,Yaozhong Hu,Bernt Øksendal,Tusheng Zhang
Publisher : Springer Science & Business Media
Page : 331 pages
File Size : 48,6 Mb
Release : 2008-02-17
Category : Mathematics
ISBN : 9781846287978

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Stochastic Calculus for Fractional Brownian Motion and Applications by Francesca Biagini,Yaozhong Hu,Bernt Øksendal,Tusheng Zhang Pdf

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.

Normal Approximations with Malliavin Calculus

Ivan Nourdin,Giovanni Peccati
Normal Approximations with Malliavin Calculus Book PDF

Author : Ivan Nourdin,Giovanni Peccati
Publisher : Cambridge University Press
Page : 255 pages
File Size : 51,7 Mb
Release : 2012-05-10
Category : Mathematics
ISBN : 9781107017771

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Normal Approximations with Malliavin Calculus by Ivan Nourdin,Giovanni Peccati Pdf

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

Stochastic Analysis of Mixed Fractional Gaussian Processes

Yuliya Mishura,Mounir Zili
Stochastic Analysis of Mixed Fractional Gaussian Processes Book PDF

Author : Yuliya Mishura,Mounir Zili
Publisher : Elsevier
Page : 210 pages
File Size : 45,8 Mb
Release : 2018-05-26
Category : Mathematics
ISBN : 9780081023631

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Stochastic Analysis of Mixed Fractional Gaussian Processes by Yuliya Mishura,Mounir Zili Pdf

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

Francesco Russo,Pierre Vallois
Stochastic Calculus via Regularizations Book PDF

Author : Francesco Russo,Pierre Vallois
Publisher : Springer Nature
Page : 656 pages
File Size : 55,7 Mb
Release : 2022-11-15
Category : Mathematics
ISBN : 9783031094460

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Stochastic Calculus via Regularizations by Francesco Russo,Pierre Vallois Pdf

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.

Brownian Motion

Peter Mörters,Yuval Peres
Brownian Motion Book PDF

Author : Peter Mörters,Yuval Peres
Publisher : Cambridge University Press
Page : 128 pages
File Size : 42,5 Mb
Release : 2010-03-25
Category : Mathematics
ISBN : 9781139486576

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Brownian Motion by Peter Mörters,Yuval Peres Pdf

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.

Ambit Stochastics

Ole E. Barndorff-Nielsen,Fred Espen Benth,Almut E. D. Veraart
Ambit Stochastics Book PDF

Author : Ole E. Barndorff-Nielsen,Fred Espen Benth,Almut E. D. Veraart
Publisher : Springer
Page : 402 pages
File Size : 40,9 Mb
Release : 2018-11-01
Category : Mathematics
ISBN : 9783319941295

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Ambit Stochastics by Ole E. Barndorff-Nielsen,Fred Espen Benth,Almut E. D. Veraart Pdf

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.

Long-Range Dependence and Self-Similarity

Vladas Pipiras,Murad S. Taqqu
Long Range Dependence and Self Similarity Book PDF

Author : Vladas Pipiras,Murad S. Taqqu
Publisher : Cambridge University Press
Page : 693 pages
File Size : 42,5 Mb
Release : 2017-04-18
Category : Business & Economics
ISBN : 9781107039469

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Long-Range Dependence and Self-Similarity by Vladas Pipiras,Murad S. Taqqu Pdf

A modern and rigorous introduction to long-range dependence and self-similarity, complemented by numerous more specialized up-to-date topics in this research area.

Theory and Statistical Applications of Stochastic Processes

Yuliya Mishura,Georgiy Shevchenko
Theory and Statistical Applications of Stochastic Processes Book PDF

Author : Yuliya Mishura,Georgiy Shevchenko
Publisher : John Wiley & Sons
Page : 400 pages
File Size : 52,6 Mb
Release : 2018-01-04
Category : Mathematics
ISBN : 9781786300508

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Theory and Statistical Applications of Stochastic Processes by Yuliya Mishura,Georgiy Shevchenko Pdf

This book is concerned with the theory of stochastic processes and the theoretical aspects of statistics for stochastic processes. It combines classic topics such as construction of stochastic processes, associated filtrations, processes with independent increments, Gaussian processes, martingales, Markov properties, continuity and related properties of trajectories with contemporary subjects: integration with respect to Gaussian processes, Itȏ integration, stochastic analysis, stochastic differential equations, fractional Brownian motion and parameter estimation in diffusion models.

Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach

Bijnan Bandyopadhyay,Shyam Kamal
Stabilization and Control of Fractional Order Systems A Sliding Mode Approach Book PDF

Author : Bijnan Bandyopadhyay,Shyam Kamal
Publisher : Springer
Page : 226 pages
File Size : 41,9 Mb
Release : 2014-07-22
Category : Technology & Engineering
ISBN : 9783319086217

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Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach by Bijnan Bandyopadhyay,Shyam Kamal Pdf

In the last two decades fractional differential equations have been used more frequently in physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology, electro chemistry and many others. It opens a new and more realistic way to capture memory dependent phenomena and irregularities inside the systems by using more sophisticated mathematical analysis. This monograph is based on the authors’ work on stabilization and control design for continuous and discrete fractional order systems. The initial two chapters and some parts of the third chapter are written in tutorial fashion, presenting all the basic concepts of fractional order system and a brief overview of sliding mode control of fractional order systems. The other parts contain deal with robust finite time stability of fractional order systems, integral sliding mode control of fractional order systems, co-operative control of multi-agent systems modeled as fractional differential equation, robust stabilization of discrete fractional order systems, high performance control using soft variable structure control and contraction analysis by integer and fractional order infinitesimal variations.

Stochastic Calculus for Fractional Brownian Motion and Related Processes

Yuliya Mishura
Stochastic Calculus for Fractional Brownian Motion and Related Processes Book PDF

Author : Yuliya Mishura
Publisher : Springer
Page : 398 pages
File Size : 51,6 Mb
Release : 2008-04-12
Category : Mathematics
ISBN : 9783540758730

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Stochastic Calculus for Fractional Brownian Motion and Related Processes by Yuliya Mishura Pdf

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.

Fractal Geometry

Kenneth Falconer
Fractal Geometry Book PDF

Author : Kenneth Falconer
Publisher : John Wiley & Sons
Page : 457 pages
File Size : 41,8 Mb
Release : 2013-12-31
Category : Mathematics
ISBN : 9781118762868

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Fractal Geometry by Kenneth Falconer Pdf

The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines. Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences. Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals Carefully explains each topic using illustrative examples and diagrams Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics Features a wide range of exercises, enabling readers to consolidate their understanding Supported by a website with solutions to exercises and additional material http://www.wileyeurope.com/fractal Leads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)

Basic Theory

Anatoly Kochubei,Yuri Luchko
Basic Theory Book PDF

Author : Anatoly Kochubei,Yuri Luchko
Publisher : Walter de Gruyter GmbH & Co KG
Page : 489 pages
File Size : 52,9 Mb
Release : 2019-02-19
Category : Mathematics
ISBN : 9783110571622

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Basic Theory by Anatoly Kochubei,Yuri Luchko Pdf

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This first volume collects authoritative chapters covering the mathematical theory of fractional calculus, including fractional-order operators, integral transforms and equations, special functions, calculus of variations, and probabilistic and other aspects.

Chaos, Complexity And Transport - Proceedings Of The Cct '15

Gwenn Boedec,Christophe Eloy,Xavier Leoncini
Chaos Complexity And Transport Proceedings Of The Cct 15 Book PDF

Author : Gwenn Boedec,Christophe Eloy,Xavier Leoncini
Publisher : World Scientific
Page : 296 pages
File Size : 41,5 Mb
Release : 2017-01-05
Category : Science
ISBN : 9789813202757

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Chaos, Complexity And Transport - Proceedings Of The Cct '15 by Gwenn Boedec,Christophe Eloy,Xavier Leoncini Pdf

The main goal is to offer to readers a panorama of recent progress in nonlinear physics, complexity and transport with attractive chapters readable by a broad audience. It allows to gain an insight into these active fields of research and notably promotes the interdisciplinary studies from mathematics to experimental physics. To reach this aim, the book collects a selection of contributions to the third edition of the CCT conference (Marseilles, 1-5 June 2015).