Selected Aspects of Fractional Brownian Motion

Selected Aspects of Fractional Brownian Motion
Author: Ivan Nourdin
Publisher: Springer Science & Business Media
Total Pages: 133
Release: 2013-01-17
Genre: Mathematics
ISBN: 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.

Stochastic Calculus for Fractional Brownian Motion and Applications

Stochastic Calculus for Fractional Brownian Motion and Applications
Author: Francesca Biagini
Publisher: Springer Science & Business Media
Total Pages: 331
Release: 2008-02-17
Genre: Mathematics
ISBN: 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.

Fractional Brownian Motion

Fractional Brownian Motion
Author: Oksana Banna
Publisher: John Wiley & Sons
Total Pages: 288
Release: 2019-04-30
Genre: Mathematics
ISBN: 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.

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
Total Pages: 411
Release: 2008-01-02
Genre: Mathematics
ISBN: 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.

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
Total Pages: 219
Release: 2013-02-21
Genre: Business & Economics
ISBN: 0521763401

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

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
Total Pages: 195
Release: 2014-10-15
Genre: Mathematics
ISBN: 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.

Normal Approximations with Malliavin Calculus

Normal Approximations with Malliavin Calculus
Author: Ivan Nourdin
Publisher: Cambridge University Press
Total Pages: 255
Release: 2012-05-10
Genre: Mathematics
ISBN: 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.

Stochastic Analysis of Mixed Fractional Gaussian Processes

Stochastic Analysis of Mixed Fractional Gaussian Processes
Author: Yuliya Mishura
Publisher: Elsevier
Total Pages: 212
Release: 2018-05-26
Genre: Mathematics
ISBN: 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

Texturing & Modeling

Texturing & Modeling
Author: David S. Ebert
Publisher: Morgan Kaufmann
Total Pages: 714
Release: 2003
Genre: Computer graphics
ISBN: 1558608486

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The third edition of this classic tutorial and reference on procedural texturing and modeling is thoroughly updated to meet the needs of today's 3D graphics professionals and students. New for this edition are chapters devoted to real-time issues, cellular texturing, geometric instancing, hardware acceleration, futuristic environments, and virtual universes. In addition, the familiar authoritative chapters on which readers have come to rely contain all-new material covering L-systems, particle systems, scene graphs, spot geometry, bump mapping, cloud modeling, and noise improvements. There are many new spectacular color images to enjoy, especially in this edition's full-color format. As in the previous editions, the authors, who are the creators of the methods they discuss, provide extensive, practical explanations of widely accepted techniques as well as insights into designing new ones. New to the third edition are chapters by two well-known contributors: Bill Mark of NVIDIA and John Hart of the University of Illinois at Urbana-Champaign on state-of-the-art topics not covered in former editions. An accompanying Web site (www.texturingandmodeling.com) contains all of the book's sample code in C code segments (all updated to the ANSI C Standard) or in RenderMan shading language, plus files of many magnificent full-color illustrations. No other book on the market contains the breadth of theoretical and practical information necessary for applying procedural methods. More than ever, Texturing & Modeling remains the chosen resource for professionals and advanced students in computer graphics and animation. *New chapters on: procedural real-time shading by Bill Mark, procedural geometric instancing and real-time solid texturing by John Hart, hardware acceleration strategies by David Ebert, cellular texturing by Steven Worley, and procedural planets and virtual universes by Ken Musgrave. *New material on Perlin Noise by Ken Perlin. *Printed in full color throughout. *Companion Web site contains revised sample code and dozens of images.

Fractional Brownian Motion

Fractional Brownian Motion
Author: Oksana Banna
Publisher: John Wiley & Sons
Total Pages: 258
Release: 2019-04-10
Genre: Mathematics
ISBN: 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.