Stochastic Analysis of Mixed Fractional Gaussian Processes

By Yuliya Mishura and Mounir Zili

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

• Language: English
• Publisher: Elsevier Science
• Release date: May 26, 2018
• ISBN: 9780081023631

Yuliya Mishura

Yuliya Mishura is Professor and Head of the Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Ukraine. Her research interests include stochastic analysis, theory of stochastic processes, stochastic differential equations, numerical schemes, financial mathematics, risk processes, statistics of stochastic processes, and models with long-range dependence.

Book preview

Stochastic Analysis of Mixed Fractional Gaussian Processes

Yuliya Mishura

Mounir Zili

Series Editor

Nikolaos Limnios

Table of Contents

Cover image

Title page

Copyright

Preface

Introduction

1: Gaussian Processes

Abstract

1.1 Some preliminaries

1.2 Gaussian variables and vectors

1.3 Gaussian processes

1.4 Exercises

2: Fractional and Sub-fractional Brownian Motions

Abstract

2.1 Fractional Brownian motion

2.2 Sub-fractional Brownian motion

2.3 Long- and short-range dependence of fBm and sfBm

2.4 Moving average representation of fBm and sfBm

2.5 Spectral representation of fBm and sfBm

2.6 Asymptotic growth of a Gaussian self-similar process with application to sub-fractional Brownian motion

2.7 Wiener integration with respect to sub-fractional Brownian motion

2.8 Compact interval representations of fractional processes

2.9 Girsanov theorem for sub-fractional Brownian motion

2.10 Comparison of fractional and sub-fractional processes via Slepian’s lemma

2.11 Exercises

3: Mixed Fractional and Mixed Sub-fractional Brownian Motions

Abstract

3.1 The main properties of mixed fractional and mixed sub-fractional Brownian motions

3.2 The behavior of the increments of mfBm and msfBm

3.3 Invertibility of the covariance matrix of mfBm and msfBm

3.4 Some properties of the mfBm’s and msfBm’s sample paths

3.5 A series expansion of mixed sub-fractional Brownian motion

3.6 Study of the semi-martingale property of the mixed processes

3.7 Mixed sub-fractional colored-white heat equation

3.8 Exercises

Appendix

A.1 Elements of calculus. Some functional classes and properties of functions

A.2 Convex, concave functions and some related questions

A.3 Elements of linear, metric, normed and Hilbert spaces

A.4 Elements of fractional calculus. Hardy–Littlewood theorem

A.5 Hausdorff and capacitarian dimensions

A.6 On the total variation of a signed measure

A.7 Elements of matrix analysis

A.8 Elements of stochastic processes

A.9 Two Fernique’s theorems

A.10 Weak convergence of stochastic processes

Bibliography

Index

Copyright

First published 2018 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

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MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

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The rights of Yuliya Mishura and Mounir Zili to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

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A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-245-8

Printed and bound in the UK and US

Preface

Yuliya Mishura; Mounir Zili March 2018

This book reflects upon the current trends that arise in both the theoretical study and the practical modeling of phenomena that exhibit certain properties of fractionality. That is, there is a tendency to transition from one type of fractionality to several types that are present at the same time. In this connection, there is a need to construct models that include different types of fractionality. Various approaches to the construction of such models can be considered; however, in our opinion, one of the simplest approaches, suitable for applications in engineering as well as in economics and finance, is the modeling of complex fractional properties using the so-called mixed models, or, more simply, linear combinations of some elementary fractional processes. In this case, we can adequately model different levels of fractionality as well as not complicate the model too much. On the basis of these arguments, we have devoted this book to the mixed fractional Gaussian processes, that is, the linear combinations of fractional Brownian motions and sub-fractional Brownian motions. The book is a textbook as well as a monograph, because, on the one hand, we tried to make it useful for beginners and included the necessary information from analysis, the theory of random processes, the theory of Gaussian processes, fractional calculus, Hausdorff and capacitarian dimensions, etc.; and on the other hand, we wanted to make it useful for researchers, for whom we presented our latest theoretical achievements in this field.

The book consists of three main chapters and an Appendix. Chapter 1 is devoted to general information from the theory of Gaussian processes. We have taken only the necessary information from this vast theory. In addition to the general definitions, the main issues that are presented here are: a variant of the theorem on normal correlation with necessary conclusions; continuity and Hölder properties of Gaussian processes; uniform bounds and maximal inequalities and, finally, equivalence and singularity of measures related to Gaussian processes. We tried to provide standard facts with detailed comments on the possible properties of the principal characteristics of Gaussian processes, such as incremental distance, properties of covariance matrix, linear independence of the values of Gaussian processes and the corresponding properties of the induced measure, in order to make the study transparent and show the development of the Gaussian property over time. Chapter 2 is devoted to fractional and sub-fractional Brownian motions. For a sub-fractional Brownian motion, existence and the main properties are studied, together with various representations including moving average, spectral and compact interval representations. Also discussed are long- and short-range dependence of fBm and sfBm, Wiener integration with respect to these processes, asymptotic growth of trajectories and the Girsanov theorem for a sub-fractional Brownian motion. Chapter 3 is the most extensive, because it is devoted to the main subject of this book, namely mixed fractional and mixed sub-fractional Brownian motions. Non-degeneracy of such combinations, properties of their trajectories, series expansions, some computer simulations and semi-martingale and non-semimartingale properties are studied. The chapter concludes with a study of the properties of a mild solution of mixed sub-fractional colored-white heat equation. All main chapters are supplied with exercises, implementation of which will help the reader to better understand the proposed theory. The Introduction contains an extended description of the previous research devoted to fractional and sub-fractional Gaussian processes. The Appendix contains the necessary preliminary information from algebra, calculus and stochastics.

To summarize, this book will be useful to specialists in the field of fractional processes, as well as to practitioners, students, graduate students and readers interested in applications of fractional random processes.

Introduction

Fractional Gaussian processes are considered widely for the following two reasons: (1) the theory of fractional Gaussian processes is quite interesting in itself and (2) it has extensive applications. Fractional Brownian motion (fBm) seems to be the simplest fractional Gaussian process. It was introduced in the pioneer paper [KOL 40], and has since been widely studied and has numerous applications in the fields of economy, finance and engineering. For example, in the opinion of engineers, fBm has been successfully applied in fields of engineering such as characterization of texture in bone radiographs, image processing and segmentation, medical image analysis and network traffic analysis. Its peculiarity is the presence of the so-called memory, that is, the absence of the Markov property, which manifests itself in all fairly complex dynamic systems. For all these reasons, the theory of fBm was considered to the smallest detail. We mention here only the books and some extended papers devoted to fBm and its applications, including [DOU 02, SAM 06, BIA 08, MIS 08, NOU 12, MAN 97], to name a few.

However, fBm alone cannot serve as an adequate model in all spheres of applications, and more complex fractional random processes are needed to model real phenomena. On the one hand, the fractional Brownian motion, which is characterized by a single parameter, namely the Hurst index, cannot serve as a good model where there are several levels of fractionality. Thus, a mixed model can be introduced as a linear combination of different fractional Gaussian processes. The simplest case is a mixed model based on both standard and fractional Brownian motions, which turns out to be more flexible. One of the reasons to consider such a model comes from the modern mathematical finance, where it has become very popular to assume that the underlying random noise consists of two parts, namely the fundamental part, describing the economical background for the stock price, and the trading part, related to the randomness inherent to the stock market. The fundamental part of the noise has a long memory, while the trading part is a white noise. Such mixed fractional Brownian motion was introduced in [CHE 01] to present a stochastic model of the discounted stock price in some arbitrage-free and complete financial markets, and since then it has been sufficiently well studied. The model has the form M = B + αBH, where B is a Brownian motion, BH is an independent fractional Brownian motion with Hurst index H ∈ (0, 1) and α is a non-zero real number. It was shown that M is not a semi-martingale if H ∈ (0, 1/2) ⋃ (1/2, 3/4], and is equivalent in measure to a Brownian motion if H ∈ (3/4, 1). From this point on, the mixed fBm has been comprehensively studied. In particular, the absence of arbitrage in the class of self-similar strategies was established in [AND 06]; the existence, uniqueness and properties of the solution of the stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index H > 1/2 were studied in [MIS 11a, MIS 11b, MIS 12, MEL 15]. In [ELN 03], a modification of the mixed fBm, the so-called fractional mixed fractional Brownian motion (FMFBM), was introduced and the lower-lower class of non-random functions (one of the Lévy classes) for such a process was characterized. In [ZIL 06], a slightly different mixed fractional Brownian motion M = aB + bBH, where a, b are real numbers, was considered, and the fractional differentiability of its sample paths was investigated.

Parameter estimation in the mixed Brownian–fractional Brownian model is provided in [ACH 10, CAI 16, DOZ 15, FIL 08, KOZ 15, XIA 11]. These techniques are described in [KUB 17].

It is possible to say that financial and other models also very often demonstrate the phenomena of multifractionality, where the depths of the memory vary with time; however, such multifractional processes will not be the subject of our consideration. We refer