Fractional Brownian Motion: Weak and Strong Approximations and Projections 9781119610335, 1119610338, 9781119610342, 1119610346, 9781786302601

Table of contents :
Content: Cover
Half-Title Page
Title Page
Copyright Page
Contents
Notations
Introduction
1. Projection of fBm on the Space of Martingales
1.1. fBm and its integral representations
1.2. Formulation of the main problem
1.3. The lower bound for the distance between fBm and Gaussian martingales
1.4. The existence of minimizing function for the principal functional
1.5. An example of the principal functional with infinite set of minimizing functions
1.6. Uniqueness of the minimizing function for functional with the Molchan kernel and H . 1 2, 1
1.7. Representation of the minimizing function 1.7.1. Auxiliary results1.7.2. Main properties of the minimizing function
1.8. Approximation of a discrete-time fBm by martingales
1.8.1. Description of the discrete-time model
1.8.2. Iterative minimization of the squared distance using alternating
1.8.3. Implementation of the alternating minimization algorithm
1.8.4. Computation of the minimizing function
1.9. Exercises
2. Distance Between fBm and Subclasses of Gaussian Martingales
2.1. fBm and Wiener integrals with power functions
2.1.1. fBm and Wiener integrals with constant integrands 2.1.2. fBm and Wiener integrals involving power integrands with a positive exponent2.1.3. fBm and integrands a(s) with a(s)s-a non-decreasing
2.1.4. fBm and Gaussian martingales involving power integrands with a negative exponent
2.1.5. fBm and the integrands a(s) = a0sa + a1sa+1
2.1.6. fBm and Wiener integrals involving integrands k1 + k2sa
2.2. The comparison of distances between fBm and subspaces of Gaussian martingales
2.2.1. Summary of the results concerning the values of the distances
2.2.2. The comparison of distances 2.2.3. The comparison of upper and lower bounds for the constant cH2.3. Distance between fBm and class of "similar" functions
2.3.1. Lower bounds for the distance
2.3.2. Evaluation
2.4. Distance between fBm and Gaussian martingales in the integral norm
2.5. Distance between fBm with Mandelbrot-Van Ness kernel and Gaussian martingales
2.5.1. Constant function as an integrand
2.5.2. Power function as an integrand
2.5.3. Comparison of Molchan and Mandelbrot-Van Ness kernels
2.6. fBm with the Molchan kernel and H . (0, 1 2), in relation to Gaussian martingales 2.7. Distance between the Wiener process and integrals with respect to fBm2.7.1. Wiener integration with respect to fBm
2.7.2. Wiener process and integrals of power functions with respect to fBm
2.8. Exercises
3. Approximation of fBm by Various Classes of Stochastic Processes
3.1. Approximation of fBm by uniformly convergent series of Lebesgue integrals
3.2. Approximation of fBm by semimartingales
3.2.1. Construction and convergence of approximations
3.2.2. Approximation of an integral with respect to fBm by integrals with respect to semimartingales

Citation preview

Fractional Brownian Motion

Series Editor Nikolaos Limnios

Fractional Brownian Motion Approximations and Projections

Oksana Banna Yuliya Mishura Kostiantyn Ralchenko Sergiy Shklyar

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019931686 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-260-1

Contents

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Chapter 1. Projection of fBm on the Space of Martingales . . . . . .

1

1.1. fBm and its integral representations . . . . . . . . . . . 1.2. Formulation of the main problem . . . . . . . . . . . . 1.3. The lower bound for the distance between fBm and Gaussian martingales . . . . . . . . . . . . . . . . . . . 1.4. The existence of minimizing function for the principal functional . . . . . . . . . . . . . . . . . . . . . . . 1.5. An example of the principal functional with infinite set of minimizing functions . . . . . . . . . . . . . . . . . . . 1.6. Uniqueness of the minimizing function  for functional with the Molchan kernel and H ∈ 12 , 1 . . . . . . . . . . . 1.7. Representation of the minimizing function . . . . . . . 1.7.1. Auxiliary results . . . . . . . . . . . . . . . . . . . . 1.7.2. Main properties of the minimizing function . . . . 1.8. Approximation of a discrete-time fBm by martingales 1.8.1. Description of the discrete-time model . . . . . . . 1.8.2. Iterative minimization of the squared distance using alternating minimization method . . . . . . . . . . . 1.8.3. Implementation of the alternating minimization algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.4. Computation of the minimizing function . . . . . . 1.9. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

2 5

. . . . . .

8

. . . . . .

10

. . . . . .

12

. . . . . .

. . . . . .

17 21 21 28 31 31

. . . . . .

33

. . . . . . . . . . . . . . . . . .

43 44 50

. . . . . .

. . . . . .

. . . . . .

. . . . . .

vi

Fractional Brownian Motion

Chapter 2. Distance Between fBm and Subclasses of Gaussian Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. fBm and Wiener integrals with power functions . . . . . . 2.1.1. fBm and Wiener integrals with constant integrands . 2.1.2. fBm and Wiener integrals involving power integrands with a positive exponent . . . . . . . . . . . . . . . . . . . . . 2.1.3. fBm and integrands a(s) with a(s)s−α non-decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4. fBm and Gaussian martingales involving power integrands with a negative exponent . . . . . . . . . . . . . . 2.1.5. fBm and the integrands a(s) = a0 sα + a1 sα+1 . . . . . 2.1.6. fBm and Wiener integrals involving integrands k1 + k2 sα . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The comparison of distances between fBm and subspaces of Gaussian martingales . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Summary of the results concerning the values of the distances . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. The comparison of distances . . . . . . . . . . . . . . . 2.2.3. The comparison of upper and lower bounds for the constant cH . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Distance between fBm and class of “similar” functions . . 2.3.1. Lower bounds for the distance . . . . . . . . . . . . . . 2.3.2. Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Distance between fBm and Gaussian martingales in the integral norm . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Distance between fBm with Mandelbrot–Van Ness kernel and Gaussian martingales . . . . . . . . . . . . . . . . . 2.5.1. Constant function as an integrand . . . . . . . . . . . . 2.5.2. Power function as an integrand . . . . . . . . . . . . . . 2.5.3. Comparison of Molchan and Mandelbrot–Van Ness kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. fBm with the Molchan kernel and H ∈ 0, 12 , in relation to Gaussian martingales . . . . . . . . . . . . . . . . 2.7. Distance between the Wiener process and integrals with respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1. Wiener integration with respect to fBm . . . . . . . . 2.7.2. Wiener process and integrals of power functions with respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

. . . . . . . .

54 54

. . . .

57

. . . .

65

. . . . . . . .

67 80

. . . .

84

. . . .

109

. . . . . . . .

109 112

. . . .

. . . .

113 118 121 124

. . . .

127

. . . . . . . . . . . .

129 129 131

. . . .

132

. . . .

133

. . . . . . . .

138 138

. . . . . . . .

140 150

. . . .

. . . .

Contents

Chapter 3. Approximation of fBm by Various Classes of Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Approximation of fBm by uniformly convergent series of Lebesgue integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Approximation of fBm by semimartingales . . . . . . . . . 3.2.1. Construction and convergence of approximations . . . 3.2.2. Approximation of an integral with respect to fBm by integrals with respect to semimartingales . . . . . . . . . 3.3. Approximation of fBm by absolutely continuous processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Approximation of multifractional Brownian motion by absolutely continuous processes . . . . . . . . . . . . . . . . . . 3.4.1. Definition and examples . . . . . . . . . . . . . . . . . . 3.4.2. Hölder continuity . . . . . . . . . . . . . . . . . . . . . . 3.4.3. Construction and convergence of approximations . . . 3.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

153

. . . . . . . . . . . .

153 157 157

. . . .

159

. . . .

164

. . . . .

. . . . .

171 171 173 175 180

Appendix 1. Auxiliary Results from Mathematical, Functional and Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

181

Appendix 2. Evaluation of the Chebyshev Center of a Set of Points in the Euclidean Space . . . . . . . . . . . . . . . . . . . . . .

205

Appendix 3. Simulation of fBm . . . . . . . . . . . . . . . . . . . . . . .

239

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257

Index

265

. . . . .

. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Notations

A A∗

The transpose of the matrix A The conjugate transpose (Hermitian transpose) of the matrix A

ai•

The transpose of the ith row of the matrix A = (aij )

B(α, β)

The beta function

B

H

B(R) C λ ([a, b])

Fractional Brownian motion with Hurst parameter H Borel σ-algebra on R Space of Hölder continuous functions f : [a, b] → R with Hölder exponent λ ∈ (0, 1] equipped with the norm f λ = sup |f (t)| + sup t∈[a,b]

α Da+ f α f Db−

s,t∈[a,b] s=t

|f (t) − f (s)| |s − t|

Riemann–Liouville left-sided fractional derivative of order α Riemann–Liouville right-sided fractional derivative of order α

fa+ (x) gb− (x)

= (f (x) − f (a+))1(a,b) (x) = (g(x) − g(b−))1(a,b) (x)

α f Ia+

Riemann–Liouville left-sided fractional integral of order α

λ

x

Fractional Brownian Motion α Ib− f

Riemann–Liouville right-sided fractional integral of order α Space of measurable p-integrable functions f : [a, b] → R, p > 0, equipped with the norm   p1 T p f Lp ([0,T ]) = |f (s)| ds

Lp ([a, b])

0

LH 2 ([0, T ])

Space of functions f : [0, T ] → R such that  T T 2H−2 |f (s)| |f (u)| |u − s| du ds < ∞ 0

0

equipped with the norm  = f LH 2 ([0,T ])

× M(K)

 T 0

T 0

H(2H − 1)

|f (s)| |f (u)| |u − s|

2H−2

The space of Gaussian martingales t of the form Mt = 0 a(s)dWs , where a ∈ K ⊂ L2 ([0, T ]) The set of minimizing functions for the functional f on L2 ([0, 1]) of the form f (x) = supt∈[0,1]   2 1/2 t z(t, s) − x(s) ds 0 The standard normal distribution

Mf

N (0, 1) N

The set of natural numbers, i.e. the positive integers

R R+

The set of real numbers = [0, ∞)

W1β [0, T ]

The space of measurable functions f : [0, T ] → R such that

f 1,β =

sup

0≤s 1/2, while short memory is inherent in H < 1/2. The combination of these properties is useful in modeling the processes occurring in devices that provide cellular and other types of communication, in physical and biological systems and in finance and insurance. Thus, the fBm itself deserves special attention. We will not discuss all the aspects of fBm here, and recommend the books [BIA 08, KUB 17, MIS 08, MIS 17, MIS 18, NOU 12, NUA 03, SAM 06] for more detail concerning various fractional processes. Now note that the absence of semimartingale and Markov properties always causes the study of the possibility of the approximation of fBm by simpler processes, in a suitable metric. Without claiming a comprehensive review of the available results, we list the following studies: approximation of fBm by the continuous processes of bounded variation was studied in [AND 06, RAL 11b], approximating wavelets were considered in [AYA 03], weak convergence to fBm in the schemes of series of various sequences of processes was discussed in [GOR 78, NIE 04, TAQ 75] and some other studies, and summarized in [MIS 08]. The paper [MUR 11] contains a presentation of fBm in terms of an infinite-dimensional Ornstein–Uhlenbeck

xiv

Fractional Brownian Motion

process. The approximation of fBm by semimartingales is proposed in [DUN 11]. The article [RAL 11a] investigates smooth approximations for the so-called multifractional Brownian motion, a generalization of fBm to the case of time-varying Hurst index. Approximation of fBm using the Karhunen theorem and using various decompositions into series over functional bases is also investigated in great detail. There is also such a question, which, in fact, served as the main incentive for writing this book: is it possible to approximate an fBm by martingales, in a reasonably chosen metric? If not, is it possible to find a projection of fBm on the class of martingales and the distance between fBm and this projection? Such a seemingly simple and easily formulated question actually led to, in our opinion, quite unexpected, non-standard and interesting results that we decided to offer them to the attention of the reader. Metric, which was proposed, has the following form: ρH (M ) := sup

t∈[0,T ]



E(BtH − Mt )2

1/2

,

where M = {Mt , t ∈ [0, T ]} is a martingale adapted to the filtration generated by B H . So, we consider the distance in the space L∞ ([0, T ]; L2 (Ω)). The first problem, considered in this book, is the minimization of ρH (M ) over the class of adapted martingales. Chapter 1 is fully devoted to this problem. We perform the following procedures step by step: introducing the so-called Molchan representation of fBm via Volterra kernel and the underlying Wiener process; proving that minimum is achieved within the class of martingales of t the form Mt = 0 a(s)dWs , where W is the underlying Wiener process and a is a non-random function from L2 ([0, T ]). As a result, the minimization problem becomes analytical. Since it is essentially minimax problem, we used a convex analysis to establish the existence and uniqueness of minimizing function a. The existence follows from the convexity of the distance. However, the proof of the uniqueness essentially relies on self-similarity of fBm. If some other Gaussian process is considered instead of fBm, the minimum of the distance may be attained for multiple functions a. Then, we propose an original probabilistic representation of the minimizing function a and establish several properties of this function. However, its analytical representation is unknown; therefore, the problem is to find its values numerically. In this connection, we considered a discrete-time counterpart of the minimization problem and reduced it, via iterative minimization using alternating minimization method, to the calculation of the Chebyshev center. It allows us to draw the plots of the minimizing function, as well as the plot of square distance between fBm and the space of adapted Gaussian martingales as a function of the Hurst index.

Introduction

xv

So, since the problem of finding a minimizing function in the whole class L2 ([0, T ]) turned out to be one that requires a numerical solution, and it is necessary to use fairly advanced methods, we then tried to minimize the distance of an fBm to the subclasses of martingales corresponding to simpler functions, in order to obtain an analytical solution, or numerical, but with simpler methods, without using tools of convex analysis. Since the Volterra kernel in the Molchan representation of fBm consists of power functions, it is natural to consider various subclasses of L2 ([0, T ]) consisting of power functions and their combinations. Even in this case, the problem of minimization is not easy and allows an explicit solution only in some cases, many of which are discussed in detail in Chapter 2. Somewhat unexpected, however, for some reason, natural, is the fact that the normalizing constant in the Volterra kernel, which usually does not play any role and is even often omitted, comes to the fore in calculations and, so to speak, directs the result. Moreover, in the course of calculations, interesting new relations were obtained for gamma functions and their combinations, and even a new upper bound for the cardinal sine function sinx x was produced. Chapter 3 is devoted to the approximations of fBm by various processes of comparatively simple structure. In particular, we represent fBm as a uniformly convergent series of Lebesgue integrals, describe the semimartingale approximation of fBm and propose a construction of absolutely continuous processes that converge to fBm in certain Besov-type spaces. Special attention is given to the approximation of pathwise stochastic integrals with respect to fBm. In the last section of this chapter, we study smooth approximations of multifractional Brownian motion. Appendix 1 contains the necessary auxiliary facts from mathematical, functional and stochastic analyses, especially from the theory of gamma functions, elements of convex analysis, the Garsia–Rodemich–Rumsey inequality, basics of martingales and semimartingales and introduction to stochastic integration with respect to an fBm. Appendix 2 explains how to evaluate the Chebyshev center, together with pseudocode. In Appendix 3, we describe several techniques of fBm simulation. In particular, we consider in detail the Cholesky decomposition of the covariance matrix, the Hosking method (also known as the Durbin–Levinson algorithm) and the very efficient method of exact simulation via circulant embedding and fast Fourier transform. A more detailed description of the book’s content by section is at the beginning of each chapter. The results presented in this book are based on the authors’ papers [BAN 08, BAN 11, BAN 15, DOR 13, MIS 09, RAL 10, RAL 11a, RAL 11b, RAL 12, SHK 14] as well as on the results from [DAV 87, DIE 02, DUN 11, HOS 84, SHE 15, WOO 94].

xvi

Fractional Brownian Motion

It is assumed that the reader is familiar with the basic concepts of mathematical analysis and the theory of random processes, but we tried to make the book self-contained, and therefore most of the necessary information is included in the text. This book will be of interest to a wide audience of readers; it is comprehensible to graduate students and even senior students, useful to specialists in both stochastics and convex analysis, and to everyone interested in fractional processes and their applications. We are grateful to everyone who contributed to the creation of this book, especially to Georgiy Shevchenko, who is the author of the results concerning the probabilistic representation of the minimizing function. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko, Sergiy Shklyar January 2019

1 Projection of fBm on the Space of Martingales

Consider the fractional Brownian motion (fBm) with Hurst index H ∈ (0, 1). Its definition and properties will be considered in more detail in section 1.1; however, let us mention immediately that fBm is a Gaussian process and anyhow not a martingale or even a semimartingale for H = 12 . Hence, a natural question arises: what is the distance between fBm and the space of Gaussian martingales in an appropriate metric and how do we determine the projection of fBm on the space of Gaussian martingales? Why is it not reasonable to consider non-Gaussian martingales? In this chapter, we will answer this and other related questions. The chapter is organized as follows. In section 1.1, we give the main properties of fBm, including its integral representations. In section 1.2, we formulate the minimizing problem simplifying it at the same time. In section 1.3, we strictly propose a positive lower bound for the distance between fBm and the space of Gaussian martingales. Sections 1.4 and 1.5 are devoted to the general problem of minimization of the functional f on L2 ([0, 1]) that has the following form:  f (x) = sup t∈[0,1]

0

t

z(t, s) − x(s)

2

1/2 ds

[1.1]

with arbitrary kernel z(t, s) satisfying condition (A) for any t ∈ [0, 1] the kernel z(t, ·) ∈ L2 ([0, t]) and  t z(t, s)2 ds < ∞. sup t∈[0,1]

0

Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

[1.2]

2

Fractional Brownian Motion

We shall call the functional f the principal functional . It is proved in section 1.4 that the principal functional f is convex, continuous and unbounded on infinity, consequently the minimum is reached. Section 1.5 gives an example of the kernel z(t, s) where a minimizing function for the principal functional is not unique (moreover, being convex, the set of minimizing functions is infinite). Sections 1.6–1.8 are devoted to the problem of minimization of principal functional f with the kernel z corresponding to fBm, i.e. with the kernel z from [1.7]. It is proved in section 1.6 that in this case, the minimizing function for the principal functional is unique. In section 1.7 it is proved that the minimizing function has a special form, namely a probabilistic representation, and many properties of the minimizing function have been established. Since we have no explicit analytical representation of the minimizing function, in section 1.8 we provide the discrete-time counterpart of the minimization problem and give the results explaining how to calculate the minimizing function numerically via evaluation of the Chebyshev center, illustrating the numerics with a couple of plots. 1.1. fBm and its integral representations In this section, we define fBm and collect some of its main properties. We refer to the books [BIA 08, MIS 08, MIS 18, NOU 12] for the detailed presentation of this topic. Let (Ω, F, P) be a complete probability space with a filtration {Ft }t≥0 satisfying the standard assumptions. Definition 1.1.– An fBm with associated Hurst index H ∈ (0, 1) is a Gaussian process B H = {BtH , Ft , t ≥ 0}, such that 1) EBtH = 0, t ≥ 0,   2) EBtH BsH = 12 t2H + s2H − |t − s|2H , s, t ≥ 0. The following statements can be derived directly from the above definition. 1) If H = 12 , then an fBm is a standard Wiener process.

 H d 2) An fBm is self-similar with the self-similarity parameter H, i.e. Bct =  H H d c Bt for any c > 0. Here, = means that all finite-dimensional distributions of both processes coincide. 3) An fBm has stationary increments that is implied by the form of its incremental covariance:  2 E BtH − BsH = (t − s)2H . [1.3]

Projection of fBm on the Space of Martingales

3

4) The increments of an fBm are independent only in the case H = 1/2. They are negatively correlated for H ∈ (0, 1/2) and positively correlated for H ∈ (1/2, 1). Due to the Kolmogorov continuity theorem, property [1.3] implies that an fBm has a continuous modification. Moreover, this modification is γ-Hölder continuous on each finite interval for any γ ∈ (0, H). It is also well-known that an fBm is not a process of bounded variation. If H = 12 , then it is neither a semimartingale nor a Markov process. An fBm can be represented as an integral of a deterministic kernel with respect to the standard Wiener process in several ways. We start with the Molchan representation (or Volterra-type representation) of fBm B H = {BtH , Ft , t ≥ 0} via the Wiener process on a finite interval (see, for example, [NOR 99b, NUA 03]). It states that a Wiener process W = {Wt , Ft , t ≥ 0} exists, such that for any t ≥ 0  BtH

t

= 0

z(t, s) dWs ,

[1.4]

where the Molchan kernel is defined by  z(t, s) = cH tH−1/2 s1/2−H (t − s)H−1/2   − H − 12 s1/2−H



t

u

H−3/2

(u − s)

H−1/2

 du 10 s, EWs (Wt − Ws ) = E Wt − Ws Ws = 0. Hence, by the Cauchy–Schwarz inequality, 

2 t − W s E (Wt − Ws )2 = |t − s| . |ρ(t) − ρ(s)| ≤ E W Therefore, ρ(t) = t = W



t 0

t 0

θ(s) ds with |θ(s)| ≤ 1, and

θ(s) dWs +

 t 0

1 − θ2 (s) dZs ,

where Z = {Zt , Ft , t ∈ [0, T ]} is a Wiener process independent of W . Then,  E

 −

BtH

0



t

=E 

0 t

=E 0



t

= 0

=t

2H

t

s a(s) dW

2 

z(t, s) dWs −

0

 z(t, s) dWs −

z(t, s) − a(s)θ(s) 

−2

t 0

t

t

0

2

 a(s)θ(s) dWs − 2 a(s)θ(s) dWs 

ds +

t 0

a(s)θ(s)z(t, s) ds +

0

t

 a(s) 1 − θ2 (s) dZs



t

+ 0

2

  a2 (s) 1 − θ2 (s) ds

  a2 (s) 1 − θ2 (s) ds



t 0

a2 (s) ds.

[1.13]

Since z(t, s) > 0, we see that the minimum in [1.13] is reached at the function a(s)θ(s) = |a(s)|, i.e. θ(s) = sgn a(s).



Corollary 1.1.– Since the proof of Lemma 1.1 is valid for any non-negative kernel z, satisfying condition [1.2], from now on, considering the non-negative kernel z, we can restrict ourselves to Gaussian martingales of the form Mt = t a(s) dWs , where W is the underlying Wiener process, and function a is non0 negative.

8

Fractional Brownian Motion

1.3. The lower bound for the distance between fBm and Gaussian martingales Denote M(K) the space of the Gaussian martingales of the form t Mt = 0 a(s)dWs , where a ∈ K ⊂ L2 ([0, T ]). In the following theorem, using stochastic considerations, we establish a non-zero lower bound for the distance between fBm with the Molchan kernel and the space of all Gaussian martingales, i.e. the space M(L2 ([0, T ])). All processes are considered on the fixed interval [0, T ]. Theorem 1.2.– The value   ρT := ρ2 B H , M(L2 ([0, T ])) =

2   t sup E BtH − a(s) dWs

inf

a∈L2 ([0,T ]) 0≤t≤T

0

admits the following lower bound:  2 1 − t2H − (1 − t)2H ρT ≥ max · T 2H > 0. 0≤t≤1 16t2H

[1.14]

Proof.– Note that our kernel z(t, s) is homogeneous in the following sense: z(t, s) = T H−1/2 z(t/T, s/T ). Therefore,  sup E

0≤t≤T

 BtH

−

0

= T 2H−1 sup E 0≤t≤T

2

t

a(s) dWs



t/T ·T

0

= T 2H sup E 0≤u≤1



t

0≤t≤T

0

(z(t, s) − a(s))2 ds

(z(t/T, s/T ) − a(s/T · T )T 1/2−H )2 ds

u 0

 = sup E

(z(u, v) − a(v · T )T 1/2−H )2 dv,

and consequently ρT can be rewritten via ρ1 , namely ρT = T 2H ρ1 . This implies that we can restrict the consideration to the case of ρT with T = 1. Now we construct a lower bound for 2   t  t  2 max E BtH − z(t, s) − a(s) ds. a(s) dWs = max

0≤t≤1

0

0≤t≤1

0

Projection of fBm on the Space of Martingales

Let 0 < t1 ≤ 1. Consider the random variable

1 0

9

a(s)dWs =: B. Then,

2   t  2 a(s) dWs = max E BtH − E[B | Ft ] max E BtH −

0≤t≤1

0≤t≤1

0

  2  2  . ≥ max E BtH1 − E[B | Ft1 ] , E B1H − B

Now we can use variance partitioning. Obviously, for every square integrable random variable η, we have     E (η − E[η | Bt1 ])2 | Bt1 = E η 2 | Bt1 − (E[η | Bt1 ])2 . Hence, E(η − E[η | Bt1 ])2 = Eη 2 − E(E[η | Bt1 ])2 . We apply the inequality Eη 2 ≥ E(E[η | Bt1 ])2 for η = BtH1 − E[B | Ft1 ] and for η = B1H − B, and obtain  max E

 BtH

0≤t≤1

−

t

0

2 a(s) dWs

   2    2      ≥ max E BtH1 − E B  BtH1 , E E B1H  BtH1 − E B  BtH1

≥

     2    2  1  H E Bt1 − E B  BtH1 . + E E B1H  BtH1 − E B  BtH1 2

Note that for all real numbers P , Q and r, the inequality (P − r)2 (Q − r)2 (P − Q)2 + ≥ 2 2 4

[1.15]

holds true because 2(P − r)2 + 2(Q − r)2 − (P − Q)2 = (P + Q − 2r)2 ≥ 0. Therefore,  max E

0≤t≤1

 BtH

−

0

t

2 a(s) dWs

2  H H  H  H 2 E B B 1 1  H t 1 H 1 = E BtH1 − ≥ E Bt1 − E B1  Bt1  2 Bt1 4 4 E BtH1

10

Fractional Brownian Motion

 2  2H 1 + t2H 1 1 − (1 − t1 ) H = E Bt1 1 − 4 2t2H 1     2H 2 2H 2 1 − t2H 1 + t2H 1 2H 1 − (1 − t1 ) 1 − (1 − t1 ) 1− = t1 = . 4 2t2H 16t2H 1 1 Remark 1.1.– By substituting t = obtain a loose lower bound  2H 2 2 −2 ρT ≥ · T 2H . 16 · 22H

1 2



into maximized expression in [1.14], we

1.4. The existence of minimizing function for the principal functional Recall that in the analytic form, our goal is to find  inf

t

sup

x∈L2 ([0,T ]) t∈[0,T ]

0

z(t, s) − x(s)

2

ds =

inf

x∈L2 ([0,T ])

f (x),

where the functional f = f (x) is defined via [1.1], and a minimizing element x ∈ L2 ([0, T ]) if the infimum is reached. In the original formulation, z is the kernel related to an fBm. However, the solution of this problem is based on the general properties of functionals in a Hilbert space, in particular, functionals defined by kernels satisfying the assumption (A) with inequality [1.2]. Therefore, in this section, we consider arbitrary kernel z satisfying assumption (A), which implies that the functional f = f (x) is well defined for any x ∈ L2 ([0, 1]). From this point on, with a view to simplifying the computations, let us consider in this chapter only the case T = 1. Lemma 1.2.– For any x, y ∈ L2 ([0, 1]), |f (x) − f (y)| ≤ x − yL2 ([0,1]) .

[1.16]

Proof.– Evidently, for any x, y ∈ L2 ([0, 1]) and 0 ≤ t ≤ 1,  0

t

z(t, s) − x(s)

2

1/2 ds

 ≤

t 0

2 x(s) − y(s) ds

 + 0

t

z(t, s) − y(s)

1/2

2

1/2 ds

.

Projection of fBm on the Space of Martingales

11

Therefore,  sup

t

z(t, s) − x(s)

0

t∈[0,1]

 ≤ sup

t 0

t∈[0,1]

2

x(s) − y(s)

1/2 ds

2



1/2 ds

+ sup t∈[0,1]

0

t

2 z(t, s) − y(s) ds

1/2 ,

which is clearly equivalent to the inequality f (x) ≤ x − yL2 ([0,1]) + f (y). Swapping x and y, we establish [1.16] and thus obtain the proof.



Corollary 1.2.– The functional f is continuous on L2 ([0, 1]). Lemma 1.3.– The following inequalities hold for any function x ∈ L2 ([0, 1]): | xL2 ([0,1]) − z(1, ·)L2 ([0,1]) | ≤ f (x) ≤ xL2 ([0,1]) + f (0).

[1.17]

Proof.– The left-hand side of [1.17] immediately follows from the inequalities  f (x) ≥

0

1

2 z(1, s) − x(s) ds

1/2 = z(1, ·) − xL2 ([0,1])

≥ | xL2 ([0,1]) − z(1, ·)L2 ([0,1]) |, 

and the right-hand side of [1.17] follows from [1.16]. Lemma 1.4.– The functional f is convex on L2 ([0, 1]). Proof.– We have to prove that for any x, y ∈ L2 ([0, 1]) and any α ∈ [0, 1],   f αx + (1 − α)y ≤ αf (x) + (1 − α)f (y).

[1.18]

Applying the triangle inequality, we note that for any t ∈ [0, 1] 

t 0

αx(s) + (1 − α)y(s) − z(t, s)

 ≤

0

t

 2 α z(t, s) − x(s) ds

2

 12

ds

 12

 + 0

t

 2 (1 − α) z(t, s) − y(s) ds

 12

,

12

Fractional Brownian Motion

whence  sup t∈[0,1]

t

0

αx(s) + (1 − α)y(s) − z(t, s)

 ≤ α sup t∈[0,1]

t 0

z(t, s) − x(s) 

+ (1 − α) sup t∈[0,1]

t 0

2

ds

2

ds

 12

 12

2 z(t, s) − y(s) ds

 12

, 

and inequality [1.18] follows. Theorem 1.3.– The functional f reaches its minimal value on L2 ([0, 1]).

Proof.– By Corollary 1.2 and Lemma 1.4 the functional f is continuous and convex. By Lemma 1.3, f (x) tends to +∞ as x → ∞. Hence, it follows from Proposition A1.2 that f reaches its minimal value.  1.5. An example of the principal functional with infinite set of minimizing functions We continue to study arbitrary kernel z satisfying assumption (A), which implies that the functional f is well defined for any x ∈ L2 ([0, 1]). Note that the set Mf of minimizing functions for the functional f is convex. In this section, we consider an example of kernel z for which Mf contains more than one point and consequently is infinite. First, we establish the following lower bound for the functional f , which is similar to the particular case, considered in Theorem 1.2. Lemma 1.5.– 1) Let the kernel z of the functional f defined by [1.1] satisfy assumption (A). Then for any a ∈ L2 ([0, 1]) and 0 ≤ t1 < t2 ≤ 1, the following inequality holds   t  2 2 1 t1  sup z(t, s) − a(s) ds ≥ z(t2 , s) − z(t1 , s) ds. [1.19] 4 0 t∈[0,1] 0 2) The equality in [1.19] implies that a(s) =

 1 z(t1 , s) + z(t2 , s) 2

a.e. on [0, t1 ),

[1.20]

and a(s) = z(t2 , s)

a.e. on [t1 , t2 ].

[1.21]

Projection of fBm on the Space of Martingales

13

Proof.– 1) The following inequalities are evident: 

t

sup 0

t∈[0,1]

z(t, s) − a(s)



≥ max  ≥ max ≥

1 2



0 t1  0

t1

t1 



0

2

ds

2 z(t1 , s) − a(s) ds, 2 z(t1 , s) − a(s) ds,

 

t2 

2 z(t2 , s) − a(s) ds

t1 

2 z(t2 , s) − a(s) ds

0

0

2  2  z(t1 , s) − a(s) + z(t2 , s) − a(s) ds.

 

[1.22]

Setting in the inequality [1.15] P = z(t1 , s), Q = z(t2 , s) and r = a(s), we obtain from [1.22] that 

t

sup t∈[0,1]

≥ ≥

1 2 1 4

0

 

2 z(t, s) − a(s) ds

t1 0

 2  2  z(t1 , s) − a(s) + z(t2 , s) − a(s) ds

t1  0

2 z(t2 , s) − z(t1 , s) ds.

[1.23]

Thus, inequality [1.19] is proved. 2) Now we show that equality in [1.19] implies [1.20] and [1.21]. Indeed, equality in [1.15] holds if and only if P + Q − 2r = 0. Equality in [1.23] has a form  2  2  1 t 1  z(t1 , s) − a(s) + z(t2 , s) − a(s) ds 2 0  2 1 t1  z(t1 , s) − z(t2 , s) ds, = 4 0 and it holds if and only if z(t1 , s) + z(t2 , s) − 2a(s) = 0

a.e. on [0, t1 ),

i.e. it holds if and only if condition [1.20] holds.

14

Fractional Brownian Motion

If [1.20] holds, then 

2 1 z(t1 , s) − a(s) ds = 4

t1 

0



t1 

z(t1 , s) − z(t2 , s)

0

2

ds,

and 

t2  0

2 1 z(t2 , s) − a(s) ds = 4 +

 

t1  0 t2 

2 z(t2 , s) − z(t1 , s) ds

2 z(t2 , s) − a(s) ds.

t1

It means that under condition [1.20], equality [1.19] holds if and only if 

t2

(z(t2 , s) − a(s))2 ds = 0,

t1

i.e. if and only if [1.21] holds.



Remark 1.2.– Let the kernel z of the functional f from [1.1] satisfy assumption (A). Then, for any a ∈ L2 ([0, 1]) and 0 ≤ t1 < t2 ≤ 1   t 2  2 1 t1  [1.24] z(t, s) − a(s) ds ≥ z(t2 , s) − z(t1 , s) ds. max 4 0 t∈{t1 ,t2 } 0 Equality in [1.24] holds if and only if [1.20] and [1.21] hold. Theorem 1.4.– (Example of functional f with infinite set Mf .) Take the kernel z(t, s) of the form z(t, s) = g(t)h(s), t, s ∈ [0, 1], where g(t) = (6t − 2)1 13 ≤t≤ 12 + (4 − 6t)1 12 ≤t≤ 56 + (6t − 6)1 56 ≤t≤1 and h(s) = 4s10≤s≤ 14 + (2 − 4s)1 14 ≤s≤ 12 (see Figure 1.1). Then,  t  2 1 max z(t, s) − a(s) ds = , min 6 a∈L2 ([0,1]) t∈[0,1] 0 and Mf consists of functions a(s) satisfying the conditions   a(s) = 0 a.e. on 0, 56

[1.25]

[1.26]

Projection of fBm on the Space of Martingales

and



t 5 6

a(s)2 ds ≤

− 6(1 − t)2 ,

1 6

5 6

≤ t ≤ 1.

15

[1.27]

Remark 1.3.– 1) Since z ∈ C([0, 1]2 ) and a ∈ L2 ([0, 1]), we have that 2  t z(t, s) − a(s) ds is continuous in t. Therefore, we can really replace 0 supt∈[0,1] with maxt∈[0,1] in equality [1.25]. 2) Some examples of functions satisfying√[1.26] and [1.27]: a(s) = 0, s ∈ [0, 1]; a(s) = (12(1 − s))1/2 15/6 0 and 0 < s ≤ t, z(ct, cs) = cα z(t, s) with α = H − 1/2.

18

Fractional Brownian Motion

Proof.– 1. Since z is the Molchan kernel of an fBm, we have that t

2H

 2 = E BtH = E



t 0

2 z(t, s) dWs

t Therefore, supt∈[0,1] 0 z(t, s)2 ds statements follow directly from [1.7].



t

= 0

=

z(t, s)2 ds.

1, and [1.2] is satisfied. Other 

Now we are in position to establish the uniqueness of minimizing function for the principal functional corresponding to the Molchan kernel of fBm. In order to do this, first, we prove an auxiliary statement concerning any minimizing function for this functional. For x ∈ L2 ([0, 1]), denote  gx (t) =

t 0

z(t, s) − x(s)

2

1/2 ds

.

Then, we have from the definition of the principal functional f that f (x) = supt∈[0,1] gx (t). It follows from Lemma 1.6 that gx ∈ C[0, 1] for any x ∈ L2 ([0, 1]). Using the self-similarity property 4) of the kernel z, it is easy to note that ga (t) = cα+1/2 gc−α a(c·) (t/c).

[1.31]

Lemma 1.7.– Let a ∈ Mf . Then, the maximal value of ga is reached at point 1, i.e. f (a) = ga (1). Proof.– Set a(t) = 0 for t > 1. Suppose that ga (1) < f (a). Since ga (t) is continuous in t, c > 1 exists such that ga (t) < fa for t ∈ [1, c]. This means that maxt∈[0,c] ga (t) = fa . Set b(t) = c−α a(tc). It follows from equation [1.31] that gb (t) = c−1/2−α ga (tc), t ∈ [0, 1]. We immediately obtain f (b) = c−α−1/2 f (a) < f (a), which leads to a contradiction.  Remark 1.4.– Similarly, if the function ga is differentiable at point 1, then ga (1) ≥ 2Hga (1). Theorem 1.5.– (Uniqueness of a minimizing function) For the principal functional f defined by [1.1] with the fractional Brownian kernel z from [1.7], there is a unique minimizing function. Proof.– Let us denote Mf as the minimal value of functional f . Recall that the set Mf is non-empty and convex. Let zˆ(s) = z(1, s), s ∈ [0, 1]. It follows from Lemma 1.7 that for any function x ∈ Mf the following equality holds:  f (x) =

1 0

x(s) − z(1, s)

2

1/2 ds

= x − zˆL2 ([0,1]) .

Projection of fBm on the Space of Martingales

19

For any x, y ∈ Mf , α ∈ (0, 1), we have that αx + (1 − α)y ∈ Mf . Hence,   Mf = f αx + (1 − α)y = αx + (1 − α)y − zˆL2 ([0,1]) ≤ α x − zˆL2 ([0,1]) + (1 − α) y − zˆL2 ([0,1]) = αf (x) + (1 − α)f (y) = Mf . Thus, αx + (1 − α)y − zˆL2 ([0,1]) = α x − zˆL2 ([0,1]) + (1 − α) y − zˆL2 ([0,1]) for all α ∈ (0, 1), in particular,  1  x + 1 y − zˆ = 2 2 L2 ([0,1])

1 2

x − zˆL2 ([0,1]) +

1 2

y − zˆL2 ([0,1]) ,

x − zˆ + y − zˆL2 ([0,1]) = x − zˆL2 ([0,1]) + y − zˆL2 ([0,1]) . For arbitrary vectors x and y in a Hilbert space, the equality x + y = x + y implies that x and y are parallel and in the same direction (or either one or both of them are zero vectors). Therefore, the functions x − zˆ and y − zˆ are equal up to a non-negative multiplier, x − zˆ = C(y − zˆ) or C(x − zˆ) = y − zˆ, with C ≥ 0, but since x − zˆL2 ([0,1]) = y − zˆL2 ([0,1]) , we have x − zˆ = y − zˆ.  Therefore, x = y (in L2 ([0, 1])), i.e. a.e. on [0, 1], as required. Remark 1.5.– In this section, we assume that 12 < H < 1. However, Theorem 1.5 is valid for all H, 0 < H < 1. Indeed, statements 1 and 4 of Lemma 1.6 hold true for all H ∈ (0, 1). In the proof, we refer to equation [1.5] instead of [1.7]. Statements 2) and 3) of Lemma 1.6 do not hold true for 0 < H < 12 . Lemma 1.7 also holds true for all H ∈ (0, 1), and Theorem 1.5 follows from Lemma 1.7. Theorem 1.6.– Let a be the function that minimizes f , i.e. a ∈ Mf . Then, a function φ : [0, 1] → R exists such that s ≤ φ(s) ≤ 1, s ∈ [0, 1], and a(s) = z(φ(s), s) a.e. Proof.– Since z(t, s) ≥ 0 for all t and s, we have that (z(t, s) − a(s))2 ≥ (z(t, s) − max(0, a(s)))2 , whence  0

t

2

(z(t, s) − a(s)) ds ≥

 0

t

t, s ∈ [0, 1],

(z(t, s) − max(0, a(s)))2 ds,

t ∈ [0, 1].

20

Fractional Brownian Motion

Finally, f (a) ≥ f (max(a, 0)). It means that function max(a, 0) ∈ Mf . Due to the uniqueness of the minimizing function (see Theorem 1.5), a(s) = max(a(s), 0) for almost all (a.a.) s, whence a(s) ≥ 0 a.e. on [0, 1]. Furthermore, since the kernel z(t, s) increases in t, we have z(t, s) ≤ z(1, s),

t, s ∈ [0, 1],

whence (z(t, s) − a(s))2 ≥ (z(t, s) − min(z(1, s), a(s)))2 ,

t, s ∈ [0, 1].

It means that  0

t

2

(z(t, s) − a(s)) ds ≥



t 0

(z(t, s) − min(z(1, s), a(s)))2 ds,

t ∈ [0, 1],

and finally, f (a) ≥ f (min(ˆ z , a)), where zˆ(s) = z(1, s). Thus, function min(ˆ z , a) ∈ Mf . Due to the uniqueness of the minimizing function, a(s) = min(z(1, s), a(s)) for a.a. s, whence a(s) ≤ z(1, s) a.e. on [0, 1]. We have just proved that 0 = z(s, s) ≤ a(s) ≤ z(1, s) a.e. in [0, 1]. Since the kernel z(t, s) is continuous in t, there exists a function φ(s), s ≤ φ(s) ≤ 1, such that a(s) = z(φ(s), s) for a.a. s ∈ [0, 1].  Corollary 1.3.– The minimizing function a ∈ Mf is non-negative. It is fully consistent with Corollary 1.1, where it is stated that we can restrict ourselves to non-negative functions a.

Projection of fBm on the Space of Martingales

21

1.7. Representation of the minimizing function The main problem of our minimization procedure is that the minimizing function has no explicit analytical representation. Therefore, we can only study its properties and give the approximation formulae. In this section we consider principal functional f corresponding to fBm and establish that the minimizing function has a special form. We start by proving several auxiliary results concerning the Molchan kernel and the minimizing function. 1.7.1. Auxiliary results Lemma 1.8.– For any 0 ≤ t ≤ 1 Molchan kernel satisfies the relation  t  1  2 z(1, s)2 ds = (1 − t)2H . z(1, s) − z(t, s) ds + 0

[1.32]

t

2  Proof.– By [1.4], the left-hand side of [1.32] is equal to E B1H − BtH = (1 − t)2H .  The next statement will be essentially generalized in what follows. However, we prove it because its proof clarifies the main ideas; moreover; it has interesting consequences concerning the properties of the minimizing function. In the remainder of this section, a = a(s), s ∈ [0, 1] denotes the minimizing function, i.e. the unique element of Mf . Lemma 1.9.– Let t∗ = sup{t ∈ (0, 1) : ga (t) = f (a)} (t∗ = 0 if this set is empty). If t∗ < 1, then a(t) = z(1, t) for a.e. t ∈ [t∗ , 1]. Proof.– Fix some t1 ∈ (t∗ , 1] and prove that for any h ∈ L2 ([0, 1]) the following equality holds: 

1

  h(s) a(s) − z(1, s) ds = 0.

t1

Evidently, the proof follows immediately from this statement. Assume the contrary. Then, without loss of generality, h ∈ L2 ([0, 1]) exists such that 

1

t1

  h(s) a(s) − z(1, s) ds =: κ > 0.

22

Fractional Brownian Motion

It follows from the continuity of the last integral w. r. t. upper bound that for some t2 ∈ (t1 , 1] we have 

t

  h(s) a(s) − z(t, s) ds ≥ κ/2

t1

for any t ∈ [t2 , 1]. Note also that our assumption implies that m := max ga (s) < f (a). s∈[t1 ,t2 ]

Consider now bδ (t) = a(t) − δh(t)1[t1 ,1] (t) for δ > 0. We have that gbδ (t) = ga (t) for t ∈ [0, t1 ], and 

gbδ (t)2 = ga (t)2 − 2δ

t



  h(s) a(s) − z(t, s) ds + δ 2

t1

t

h(s)2 ds

t1

for t > t1 . For t ∈ (t1 , t2 ] the following inequality holds: 2

2

gbδ (t) ≤ m − 2δ



t

  h(s) a(s) − z(t, s) ds + δ 2

t1



t

h(s)2 ds ≤ m2 + Cδ

t1

with the constant C that does not depend on t, δ. Then, for sufficiently small δ > 0, we have that gbδ (t) < f (a) for any t ∈ (t1 , t2 ]. Furthermore, if t ∈ (t2 , 1], then gbδ (t)2 ≤ f (a)2 − 2δ



t

  h(s) a(s) − z(t, s) ds + δ 2

t1

≤ f (a)2 − κδ + δ 2



t

h(s)2 ds

t1



1 0

h(s)2 ds.

Again, for sufficiently small δ > 0 and any t ∈ (t2 , 1], we have that gbδ (t) < f (a). Therefore, for sufficiently small δ > 0, we obtain f (bδ ) = f (a) and gbδ (1) < f (a) = f (bδ ). We obtain the contradiction with Lemma 1.7 whence the proof follows.  Corollary 1.4.– A point t ∈ (0, 1) exists such that ga (t) = f (a). Proof.– Assuming the contrary, we obtain from Lemma 1.9 that a(t) = z(1, t) for a.a. t ∈ [0, 1]. However, in this case, ga (1) = 0, which contradicts Lemma 1.7. 

Projection of fBm on the Space of Martingales

23

Denote Ga = {t ∈ [0, 1] : ga (t) = f (a)}, the set of the maximal points of the function ga . Lemma 1.10.– Let u ∈ [0, 1) be a point such that ga (u) < f (a). Then, there does not exist a function h ∈ L2 ([0, 1]) such that for any t ∈ Ga ∩ (u, 1] the   t inequality u h(s) a(s) − z(t, s) ds > 0 holds. Proof.– Assume the contrary, i.e. let for some function h ∈ L2 ([0, 1]) we have   t that u h(s) a(s) − z(t, s) ds > 0 for any t ∈ Ga ∩ (u, 1]. The set Ga ∩ (u, 1] is closed because ga (u) < f (a). Therefore,  κ :=

min

t∈Ga ∩(u,1]

t

  h(s) a(s) − z(t, s) ds > 0.

u

Denote Bε = {t ∈ (u, 1] : Ga ∩ (u, 1] ∩ (t − ε, t + ε) = ∅} the intersection of ε-neighborhood of the set Ga ∩ (u, 1] with interval (u, 1]. Continuity argument implies that for some ε > 0 it holds that 

t

  h(s) a(s) − z(t, s) ds > κ/2

u

for any t ∈ Bε . Similarly to the proof of Lemma 1.9, denote bδ (t) = a(t) − δh(t)1(u,1] (t) for any δ > 0. Then, we have that gbδ (t) = ga (t) for any t ∈ [0, u], and 2

2

gbδ (t) ≤ f (a) − 2δ



t

  h(s) a(s) − z(t, s) ds + δ 2

u

≤ f (a)2 − κδ + δ 2

max t∈[u,1]\Bε

ga (t) < f (a).

t u



1 0

h(s)2 ds

for any t ∈ Bε . It follows from the continuity of ga that m=



h(s)2 ds

24

Fractional Brownian Motion

Therefore, we have for t ∈ (u, 1] \ Bε that   gbδ (t)2 = ga (t)2 − 2δ a(s) − z(1, s) + δ 2 2



≤ m − 2δ

t



t

h(s)2 ds

t1

  h(s) a(s) − z(t, s) ds + δ 2

t1



t

h(s)2 ds

t1

2

≤ m + Cδ, with the constant C that does not depend on t and δ. It follows from the above bounds that for sufficiently small δ > 0 and for any t ∈ (u, 1] we have the inequality gbδ (t) < f (a). It means that for sufficiently small δ > 0, we obtain the equality f (bδ ) = f (a), and moreover, gbδ (1) < f (a) = f (bδ ), which contradicts Lemma 1.7.  Lemma 1.10 supplies the form of minimizing function on the part of the interval [0, 1]. All equalities below are considered almost surely (a.s). Lemma 1.11.– Let t1 = min{t ∈ (0, 1) : ga (t) = f (a)}. Then, t2 ∈ (t1 , 1] ∩ Ga and a random variable ξa exist with the values in [t1 , t2 ] ∩ Ga such that for t ∈ [0, t2 ), we have that P(ξa ≥ t) > 0, and the equality    a(t) = E z(ξa , t)  ξa ≥ t holds. Proof.– Consider the set of functions K = {kt (s) = z(t, s)1s≤t + a(s)1s>t , t ∈ Ga } and let  C=

1 0

 kt (s) F (dt), F is a distribution function on Ga

be the closure of the convex hull of K. According to Lemma 1.10, applied to u = 0, there does not exist h ∈ L2 ([0, 1]) such that (h, k) < (h, a) for any k ∈ K. Moreover, there is no h ∈ L2 ([0, 1]) such that (h, k) < (h, a) for any k ∈ C, i.e. the element a and the set K cannot be separated properly. Then, according to the proper separation theorem (see Corollary A1.2 in Appendix 1), a ∈ C, so there exists a distribution F on Ga such that  a(s) =

1 0

 kt (s) G(dt) =

 [s,1]

kt (s) F (dt) +

[0,s)

a(s) F (dt).

[1.33]

Projection of fBm on the Space of Martingales

25

Hence,  a(s)F ([s, 1]) =

[s,1]

kt (s) F (dt).

[1.34]

Note that the equality supp F = {t1 } is impossible because otherwise it follows from equation [1.34] that a(s) = z(t1 , s) for s ≤ t1 ; therefore, ga (t1 ) = 0 which contradicts the assumption ga (t) = f (a). Using the latter statement and [1.34], we obtain the statement of the theorem with t2 = max(supp F ) and a random variable ξa with the distribution F .  Conditions on minimizing function from Lemma 1.11 are sufficient in the following sense. Lemma 1.12.– Let y ∈ L2 ([0, 1]). Define the kernel zy (t, s) for s, t ∈ [0, 1] as  z(t, s), t ≥ s, zy (t, s) = y(s), t < s. Function y is the minimizing function of the principal functional f if and only if a random variable ξy exists, which takes values in [0, 1] such that the following conditions hold: y(s) = Ezy (ξy , s) gy (ξy ) = f (y)

for almost all s ∈ [0, 1],

[1.35]

a.s.

[1.36]

Proof.– The necessity was proved in Lemma 1.11. Indeed, if y = a, let us take ξy = ξa , where ξa was obtained in the course of the proof of Lemma 1.11. Then condition [1.35] follows from equality [1.33], while condition [1.36] follows from the fact that ξa ∈ Ga . The sufficiency is proved basically by reversing a proper separation argument from Lemma 1.11: if a function belongs to the convex set C, then it cannot be properly separated from this set, which means that it is a minimizer. To make this idea rigorous, assume the contrary: let a function y satisfy [1.35] and [1.36], but y ∈ / Mf . Then, a function a ∈ L2 ([0, 1]) exists such that f (y) > f (a) (e.g. we can take a as the minimizing function). Functional f 2 is convex, therefore    2 f y + δ(a − y) ≤ f (y)2 + δ f (a)2 − f (y)2 ,

0 ≤ δ ≤ 1.

26

Fractional Brownian Motion

It is easy to see that for any function b ∈ L2 ([0, 1]), 2

max zy (t, ·) − b

t∈[0,1]



t

= max t∈[0,1]

≤ max

t∈[0,1]

0



2 z(t, s) − b(s) ds +

t 0

2 z(t, s) − b(s) ds +



1

y(s) − b(s)

2

 ds

t



1 0

y(s) − b(s)

2

ds

2

= f (b)2 + y − b . Therefore, for 0 ≤ δ ≤ 1 we have that   2 2 max zy (t, ·) − y − δ(a − y) ≤ f (y)2 − δ f (y)2 − f (a)2 + δ 2 a − y .

t∈[0,1]

It means that for sufficiently small δ > 0 2

max zy (t, ·) − y − δ(a − y) < f (y)2 .

[1.37]

t∈[0,1]

On the one hand, choose arbitrary δ for which inequality [1.37] holds, and set b = y + δ(a − y). Then, 2

max zy (t, ·) − b < f (y)2 .

[1.38]

t∈[0,1]

On the other hand, 2

2

2

max zy (t, ·) − b ≥ E zy (ξy , ·) − b ≥ E zy (ξy , ·) − Ezy (ξy , ·)

t∈[0,1]

2

= E zy (ξy , ·) − y = Egy (ξy )2 = f (y)2 .

[1.39]

Inequalities [1.38] and [1.39] contradict each other. Thus, assuming that the function y is not minimizing for the principal functional f , we obtain a contradiction. Therefore, f (y) = min f .  Now we are in position to prove that ess sup ξa := min {t : P(ξa ≤ t) = 1} = max(supp ξa ) = 1, which will imply that t2 = 1 in Lemma 1.11.

Projection of fBm on the Space of Martingales

27

Lemma 1.13.– Let a be the minimizing function for principal functional f and let ξa be a random variable satisfying conditions [1.35] and [1.36] where y = a. Then, ess sup ξa = 1. Proof.– Denote t2 = ess sup ξa . Evidently, ξa takes values from [0, t2 ]. Consider a function b(s) = t−α 2 a(t2 s),

s ∈ [0, 1].

Then, in view of the self-similarity property (item 4 in Lemma 1.6), b(s) = Ezb (ξa /t2 , s), where zb (t, s) is defined in the formulation of Lemma 1.12. Using [1.31], we obtain gb (t) = t−H 2 ga (t2 t),

t ∈ [0, 1].

On the one hand, since a(s) satisfies [1.36], we have −H f (b) = max gb ≥ gb (ξa /t2 ) = t−H 2 ga (ξa ) = t2 f (a) [0,1]

a.s.; on the other hand, f (b) = max gb = t−H max ga ≤ t−H max ga = t−H 2 2 2 f (a). [0,1]

[0,t2 ]

[0,1]

This implies f (b) = gb (ξa /t2 ) = t−H 2 f (a)

a.e.

Therefore, the function b satisfies [1.35] and [1.36] and is therefore a minimizer of f . Hence, t−H 2 f (a) = f (b) = min f = f (a), L2 ([0,1])

so t2 = 1, as required.



28

Fractional Brownian Motion

1.7.2. Main properties of the minimizing function Now we can refine Lemma 1.11 in view of Lemma 1.13. Recall that a is the minimizing function for the principal functional f and Ga = {t ∈ [0, 1] : ga (t) = f (a)}. Theorem 1.7.– A random variable ξa exists, taking values in Ga , such that P(ξa ≥ s) > 0

for all s ∈ [0, 1),

a(s) = E[z(ξa , s) | ξa ≥ s]

a.e. in [0, 1].

[1.40]

Proof.– We can put ξ˜a = ξa , where ξa comes from Lemma 1.11. Then the present statement is a straightforward consequence of Lemma 1.13.  Without loss of generality we will assume in what follows that [1.40] holds for every s ∈ [0, 1]: a(s) = E[z(ξa , s) | ξa ≥ s] for any s ∈ [0, 1].

[1.41]

Corollary 1.5.– 1) The minimizing function a is left-continuous and has right limits. 2) For any s ∈ [0, 1), 0 < a(s) ≤ z(1, s),

[1.42]

moreover, a(s) < z(1, s) on a set of positive Lebesgue measure. Proof.– 1) Follows from [1.41], continuity of z and the dominated convergence. 2) Taking into account statement 2 of Lemma 1.6, for 0 < s < t ≤ 1 0 < z(t, s) ≤ z(1, s). Now [1.42] follows from [1.41] and the fact that P(ξa > s) > 0 for s < 1. Further, if a(s) = z(1, s) a.e., then ga (1) = 0, which contradicts Lemma 1.7.  Further, we investigate the distribution of ξ.

Projection of fBm on the Space of Martingales

29

Lemma 1.14.– There exists t∗ ∈ (0, 1) such that for all t ∈ (t∗ , 1), ga (t) < f (a). Proof.– By Corollary 1.5, u1 and u2 exist such that 0 < u1 < u2 < 1 and λ1 {s ∈ [u1 , u2 ] : a(s) < z(1, s)} > 0, where λ1 is the Lebesgue measure. Now estimate f (a)2 − ga (t)2 from below, asymptotically as t → 1−. For u2 ≤ t ≤ 1 make simple transformations f (a)2 − ga (t)2 = ga (1)2 − ga (t)2  1  t  2  2 = z(1, s) − a(s) ds − z(t, s) − a(s) ds 0

0



=−

t

z(1, s) − z(t, s)

0



u1

+2  +2

0 u2 

+

 t

2



1

ds +

2 z(1, s) − a(s) ds

t



  z(1, s) − z(t, s) z(1, s) − a(s) ds

u2

  z(1, s) − z(t, s) z(1, s) − a(s) ds.

u1

We estimate each of the terms separately. By Lemma 1.8, 

t 0

2 z(1, s) − z(t, s) ds < (1 − t)2H .

Furthermore, 

u1 0

+

 t

   z(1, s) − z(t, s) z(1, s) − a(s) ds ≥ 0,

u2

since the integrand is non-negative by Lemmata 1.6 and 1.5; 

1

z(1, s) − a(s)

2

≥ 0.

t

It is easy to see that z(1, s) − z(t, s) → zt (1, s) = s−α (1 − s)1−α 1−t

30

Fractional Brownian Motion

uniformly in [u1 , u2 ] as t → 1−. Therefore,  u2  u1

  z(1, s) − z(t, s) z(1, s) − a(s) ds 1−t

 →

u2

u1

  zt (1, s) z(1, s) − a(s) ds,

as t → 1−, and the limit is positive. Consequently, lim inf t→1−

f (a)2 − ga (t)2 > 0, 1−t 

and the statement follows. The lemma just proved means that 1 is an isolated point of Ga .

As an immediate corollary, we have the following theorem.   Theorem 1.8.– There exists t∗a < 1 such that P ξa ∈ (t∗a , 1) = 0, and the distribution of ξa has an atom at 1, i.e. P(ξa = 1) > 0. Consequently, a(s) = z(1, s) for all s ∈ [t∗a , 1]. Further, we prove that the distribution of ξa has no other atoms. Theorem 1.9.– For any t ∈ (0, 1), P(ξa = t) = 0. Consequently, a ∈ C[0, 1]. Proof.– We start by computing for t ∈ (0, 1) a(t+) − a(t) = E[z(ξa , t) | ξa > t] − E[z(ξa , t) | ξa ≥ t] =

E[z(ξa , t)1ξa >t ]P(ξa ≥ t) − E[z(ξa , t)1ξa ≥t ]P(ξa > t) P(ξa > t)P(ξa ≥ t)

=

E[z(ξa , t)1ξa >t ]P(ξa = t) − E[z(ξa , t)1ξa =t ]P(ξa > t) P(ξa > t)P(ξa ≥ t)

=

E[z(ξa , t)1ξa >t ]P(ξa = t) E[z(t, t)1ξa =t ] − P(ξa > t)P(ξa ≥ t) P(ξa ≥ t)

=

a(t+)P(ξa = t) . P(ξa ≥ t)

[1.43]

Further, it is easy to see that g = ga has left and right derivatives at t, and  t    (t) = a(t)2 + 2 z(t, s) − a(s) zt (t, s) ds, g− 0

 (t+) = a(t+)2 + 2 g+



t 0

 z(t, s) − a(s) zt (t, s) ds.

Projection of fBm on the Space of Martingales

31

  But for any t ∈ Ga , g− (t) ≥ 0, g+ (t+) ≤ 0, so a(t) ≥ a(t+), whence from [1.43] we have that a(t+) = a(t) and also P(ξa = t) = 0, as a(t+) > 0. For t∈ / Ga , P(ξa = t) = 0 (recall that ξa takes values in Ga ) and a(t+) = a(t). 

Remark 1.6.– Due to monotonicity of z in the first variable, the right-hand side of inequality [1.19] is maximal for t2 = 1, so we have that  t  2 1 z(1, s) − z(t, s) ds. [1.44] f (a) ≥ max 4 t∈[0,1] 0 Theorem 1.9 implies in particular that the inequality is strict, i.e. this lower bound is not reached. Indeed, if there were equality in [1.44], Lemma 1.5 would imply that the distribution of ξa is 12 (δt0 + δ1 ), where t0 is the point where the minimum of the right-hand side of [1.44] is reached, which would contradict Theorem 1.9. Remark 1.7.– From [1.41] it is easy to see that the minimizing function a = a(s) decreases on the intervals of the complement to Ga , particularly on (0, t1 ) and (t∗a , 1). The numerical experiments suggest that a decreases on the entire complement to Ga , except for H = 0.9, where the experiment gives a(t1 ) < a(t∗a ), but the difference a(t∗a )−a(t1 ) is small and can be explained by the effect of discretization of the time. 1.8. Approximation of a discrete-time fBm by martingales Obviously, in the previous sections we collected a lot of interesting properties of the minimizing function for the kernel z that corresponds to fBm. However, there is no clear analytical representation for this function. In this connection, let us give a numerical algorithm of its approximation. Of course, it is easier to consider the discrete-time approximations. So, in this section, we consider a problem of minimization of the principal functional, but in discrete time. This is an approximation to the original problem, so its solution can be considered as an approximate solution to the original problem. However, to be absolutely precise, we now simply solve a discrete analog of the minimization problem described above. 1.8.1. Description of the discrete-time model We consider the discretizing procedure on the interval [0, 1] for technical H simplicity. Let N be a natural number, and define bk = Bk/N , k = 0, . . . , N . The vector b = (b0 , b1 , . . . , bN ) will be called a discrete-time fBm. It generates

32

Fractional Brownian Motion

a discrete-time filtration Fk = σ(b0 , . . . , bk ), k = 0, . . . , N . For an arbitrary random vector ξ = (ξ0 , ξ1 , . . . , ξN ) with square integrable components denote G(ξ) =

max E(bk − ξk )2 .

k=0,...,N

We consider the problem of minimization of the functional G(ξ), where ξ is an Fk -martingale. Let us represent the vector b as a linear operator applied to a random vector with standard multivariate normal distribution. It follows that b0 = 0 a.s. The vector (b1 , . . . , bN ) has multivariate normal distribution with zero mean and some covariance matrix Σb . The matrix Σb is non-singular (see Theorem 1.1), and its ijth element is equal to (Σb )ij =

i2H + j 2H − |i − j|2H . 2N 2H

Since Σb is a positive definite, symmetric matrix, a decomposition exists: Σb = KK 

[1.45]

with lower-triangular K, and the matrix K is non-singular. The representation [1.45] is called the Cholesky decomposition. Cholesky decomposition is widely used for drawing a Gaussian random vector with specified covariance matrix. For the algorithm of computing the Cholesky decomposition and for its use in statistical simulations, see Section A3.1 in Appendix 3 and [CHA 15, Section 6.5.3]. We used chol function from R package. After applying Cholesky decomposition, we obtain the representation (b1 , . . . , bN ) = K(ζ1 , . . . , ζN ) , where ζ1 , . . . , ζN are independent random variables with standard normal distribution. Denote kts the elements of the matrix K, 1 ≤ t ≤ N , 1 ≤ s ≤ N . Since the matrix K is lower triangular, kts = 0 for t < s. Then, bt =

t

kts ζs .

[1.46]

s=1

Now compare the discrete-time representation [1.46] to the continuousH time Molchan representation [1.4]. We have bt = Bt/N . Both ζs in [1.46] and √ N (Ws/N − W(s−1)/N ) in [1.4] have a standard normal distribution. Thus, kts plays  the similar role in [1.46]  as the average value of the function s N −1/2 z Nt , u on interval s−1 , N N in [1.4]. Therefore, the matrix K can be regarded as the discrete-time counterpart of the Molchan kernel z(t, s).

Projection of fBm on the Space of Martingales

33

Further, we will show, as in the continuous case, that minimization of G over martingales is equivalent to minimization over Gaussian martingales. Indeed, let ξ = (ξ0 , ξ1 , . . . , ξN ), ξ0 = 0, be an arbitrary square integrable Fk -martingale. Owing to the fact that Fk = σ {ζ1 , . . . , ζk }, k = 1, . . . , N , we have the following martingale representation: ξn =

n

αk ζ k ,

n = 1, . . . , N,

k=1

where αk is a square integrable Fk−1 -measurable random variable, k = 1, . . . , N . Taking into account the martingale property of ξ and mutual independence of ζk , we obtain 2

G(ξ) = max E(bj − ξj ) = max j=0,...,N

= max

j=1,...,N

j=1,...,N

j=1,...,N

E(kjn − αn )2

n=1

j

  (kjn − Eαn )2 + Var(αn ) n=1 j

≥ max

j

(kjn − Eαn )2 .

n=1

Hence, we can assume that ξ has a form ξn = with some non-random a1 , . . . , an . Then, G(ξ) = max

t=1,...,N

t

n k=1

ak ζk , n = 1, . . . , N ,

(kts − as )2 =: F (a).

[1.47]

s=1

Thus, we are searching for the minimum of functional F (a) over a ∈ RN . 1.8.2. Iterative minimization of the squared distance using alternating minimization method Now reduce the problem of finding the minimum of the convex functional F (a) to the problem of finding the minimum of biconvex functional of two vectors. In order to introduce a functional of two vectors, denote F (a, b) = max

t=1,...,N

 t s=1

(as − kts )2 +

N

 (as − bs )2 ,

s=t+1

34

Fractional Brownian Motion

(Here, by convention,

N

s=N +1 (as

− bs )2 = 0.)

For fixed a, min F (a, b) is reached at point b = a, i.e. F (a) = F (a, a) = min F (a, b). b∈RN

[1.48]

Definition 1.2.– Let Z be a non-empty bounded set in RN . Chebyshev center of the set Z is the point a ∈ RN where the minimum of supz∈Z a − z is reached: a = arg min sup a − z a∈RN

[1.49]

z∈Z

The minimum in [1.49] is reached because the criterion function  Q(a) = sup a − z z∈Z

is continuous on RN and tends to +∞ as a → ∞. It is reached at a unique point, according to the next lemma. Lemma 1.15.– Square criterion function Q(a) = sup a − z2 z∈Z

is strongly convex. Proof.– To prove convexity, note that for every z ∈ RN , a1 ∈ RN , a2 ∈ RN , and t ∈ (0, 1) 2 2 ta1 + (1 − t)a2 − z2 = t2 a1 2 + 2t(1 − t) a 1 a2 + (1 − t) a2   2 − 2ta 1 z − 2(1 − t)a2 z + z .

This equality can be obtained easily by representing the square norm as the inner product of a vector by itself (z2 = z  z) and using linearity and symmetry of the inner product. Indeed, ta1 + (1 − t)a2 − z2 = (ta1 + (1 − t)a2 − z) (ta1 + (1 − t)a2 − z)   = t2 a 1 a1 + t(1 − t)a1 a2 − ta1 z 2   + (1 − t)ta 2 a1 + (1 − t) a2 a2 − (1 − t)a2 z

− tz  a1 − (1 − t)z  a2 + z  z

Projection of fBm on the Space of Martingales

35

  = (t − t(1 − t))a 1 a1 + t(1 − t)a1 a2 − ta1 z   + t(1 − t)a 2 a1 + ((1 − t) − t(1 − t))a2 a2 − (1 − t)a2 z

− tz  a1 − (1 − t)z  a2 + (t + (1 − t))z  z    = ta 1 a1 − t(1 − t)a1 a1 + t(1 − t)a1 a2 − ta1 z    + t(1 − t)a 2 a1 + (1 − t)a2 a2 − t(1 − t)a2 a2 − (1 − t)a2 z

− tz  a1 − (1 − t)z  a2 + tz  z + (1 − t)z  z    = ta 1 a1 − ta1 z − tz a1 + tz z    + (1 − t)a 2 a2 − (1 − t)a2 z − (1 − t)z a2 + (1 − t)z z    − t(1 − t)a 1 a1 + t(1 − t)a1 a2 + t(1 − t)a2 a1 − t(1 − t)a2 a2

= t(a1 − z) (a1 − z) + (1 − t)(a2 − z) (a2 − z) − t(1 − t)(a1 − a2 ) (a1 − a2 ) = t a1 − z2 + (1 − t) a2 − z2 − t(1 − t) a1 − a2 2 .



If, in addition to assumptions a1 ∈ RN , a2 ∈ RN and t ∈ (0, 1), the inequality a1 = a2 holds true, then Q(ta1 + (1 − t) a2 ) ≤ tQ(a1 ) + (1 − t)Q(a2 ) − t(1 − t)a1 − a2 2 < tQ(a1 ) + (1 − t)Q(a2 ). Thus, the square criterion function Q(a) is indeed strongly convex. Corollary 1.6.– As a consequence, square criterion function Q(a) cannot reach its minimum at more than one point. Remark 1.8.– Lemma 1.15 can be generalized for any Hilbert space. Namely, for H the Hilbert space, for any fixed element z ∈ H, the functional Q(a) = a − z2 is strictly convex. The proof is exactly the same if only the notation a b is used instead of the standard notation of the scalar product in H.

36

Fractional Brownian Motion

For fixed b, min F (a, b) is reached at point a being the Chebyshev center of the N -point set {k1 (b), k2 (b), . . . , kN (b)}, where ⎞ kt1 ⎜ .. ⎟ ⎜ . ⎟ ⎟ ⎜ ⎜ ktt ⎟ ⎟ kt (b) = ⎜ ⎜bt+1 ⎟ , ⎟ ⎜ ⎜ . ⎟ ⎝ .. ⎠ ⎛

⎛

t = 1, . . . , N − 1;

bN

kN 1 kN 2 kN 3 .. .

⎞

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ kN (b) = ⎜ ⎟ = kN • . [1.50] ⎟ ⎜ ⎟ ⎜ ⎝kN,N −1 ⎠ kN N

Due to [1.48], the problem of minimization of F (a) is equivalent to the problem of minimization of F (a, b): min F (a) = min min F (a, b).

a∈RN

a∈RN b∈RN

If the minimum of F (a) is reached at point a = a∗ , then the minimum of F (a, b) is reached at point a = b = a∗ . If the minimum of F (a, b) is reached at point a = a∗ , b = b∗ , then the minimum of F (a) is reached at point a = a∗ . In what follows, min F will mean a common minimum of functions F (a) and F (a, b). Summarize the properties of the functions F (a) and F (a, b) in the following proposition. Proposition 1.1.– 1) For a fixed a ∈ RN , the minimum minb∈RN F (a, b) = F (a) is reached at point b = a. 2) For a fixed b ∈ RN , the minimum mina∈RN F (a, b) is reached at point a being the Chebyshev center of the points k1 (b), k2 (b), . . . , kN (b). 3) The minimal values of the functions F (a, b) and F (a) coincide. We denote their common value by min F : min F := min F (a, b) = min F (a). a,b∈RN

a∈RN

Now, we are in position to find the minimum of F (a, b) by the procedure that is called an alternating minimization. Namely, let a(0) ∈ RN be the initial approximation. In our case we take kN • as a(0) . Then, the minimum of F (a(0) , b) is reached at point b = a(0) , and we minimize F (a, a(0) ) with respect to a: a(1) = arg min F (a, a(0) ). a∈RN

Projection of fBm on the Space of Martingales

37

Again, the minimum of F (a(1) , b) is reached at point b = a(1) . Define the sequence {a(n) , n ≥ 1} iteratively: a(n) = arg min F (a, a(n−1) ), a∈RN

n ≥ 1.

[1.51]

Then, we have that the criterion function is non-increasing: F (a(0) , a(0) ) ≥ F (a(1) , a(0) ) ≥ F (a(1) , a(1) ) ≥ F (a(2) , a(1) ) ≥ F (a(2) , a(2) ) ≥ . . . ,

[1.52]

and since F (a(k) , a(k) ) = F (a(k) ), F (a(0) ) ≥ F (a(1) ) ≥ F (a(2) ) ≥ . . . ,

[1.53]

According to the following theorem, the sequence {F (a(k) ), k ≥ 0} converges to min F . Theorem 1.10.– Let min kts ≤ a(0) s ≤ max kts

t=s,...,N

for all s, 1 ≤ s ≤ N .

t=s,...,N

[1.54]

Then, the sequence {a(n) , n ≥ 1} defined by [1.51] has the following properties: 1) lim F (a(n) ) = min F . n→∞

2) If the minimal value of F (a) is reached at the unique point a∗ (i.e. the equality F (a) = min F holds true if and only if a = a∗ ), then lim a(n) = a∗ .

n→∞

Proof.– 1) Introduce a following rectangle H ⊂ RN : H = {a ∈ RN | ∀s = 1, . . . , N : &

% =

min kt1 , max kt1 ×

t=1,...,N

t=1,...,N

min kts ≤ as ≤ max kts }

t=s,...,N

%

t=s,...,N

&

min kt2 , max kt2

t=2,...,N

t=2,...,N

× . . . × [min(kN −1,N −1 , kN,N −1 ), max(kN −1,N −1 , kN,N −1 )] × {kN N }.

[1.55]

38

Fractional Brownian Motion

Denote by d the length of the diagonal of the rectangle H, so that d2 =

N 

j=1

2 max kij − min kij

i=j,...,N

i=j,...,N

.

[1.56]

Condition [1.54] means that a(0) ∈ H. By induction, a(n) ∈ H for all n. Indeed, if a(n−1) ∈ H, then kt (a(n−1) ) ∈ H for all t = 1, . . . , N , where kt (b) is defined in [1.50]. The Chebyshev center of a set lies in the convex hull of the set. Thus, a(n) lies in the convex hull of ki (a(n−1) ), i = 1, . . . , N , whence a(n) ∈ H. The minimal value mina∈H F (a) is reached because F (a) is a continuous function and H is a non-empty closed bounded set. Denote by a∗ ∈ H the point where the minimal value mina∈H F (a) is reached. It holds that for all a ∈ RN F (a) ≥ F (nearest(H, a)) ≥ F (a∗ ),

[1.57]

whence F (a∗ ) = mina∈RN F (a). Here, nearest(H, a) is the point of H nearest to a; the coordinates of nearest(H, a) equal ⎧ ⎪ min kts , if as ≤ mint=s,...,N kts , ⎪ ⎪ t=s,...,N ⎪ ⎪ ⎨a , if mint=s,...,N kts ≤ as s (nearest(H, a))s = ⎪ ≤ maxt=s,...,N kts , ⎪ ⎪ ⎪ ⎪ ⎩ max kts , if as ≥ maxi=j,...,N kts . t=s,...,N

Due to inequality [1.57], the minimal value of F (a) on RN is reached at a∗ , namely min F = F (a∗ ) = min F (a) = min F (a, b). a∈RN

a,b∈RN

[1.58]

Since a(n) ∈ H and a∗ ∈ H, we have a(n) − a∗  ≤ d for all n ≥ 0. For all a ∈ H and b ∈ H, the inequality F (a) ≤ F (a, b) ≤ d2 holds true. The case d = 0 is trivial: in this case, the set H consists of only one point, a(n) = a(0) and F (a(n) ) = 0 for all n; therefore, the statement of this theorem holds true. Hence, in the rest of the proof assume that d > 0. Define the following pseudonorms on RN : 1/2 1/2   N t 2 2 xt = xs and x⊥t = xs . s=1

s=t+1

Projection of fBm on the Space of Martingales

39

For t = N we have that xN = x and x⊥N = 0. With this notation, x2t + x2⊥t = x2

for all t = 1, . . . , N and x ∈ RN ,

F (a) = max a − kt• 2t , t=1,...,N

F (a, b) = max

t=1,...,N

[1.59]

  a − ki• 2t + a − b2⊥t .

In what follows, we are going to use inequalities a − ki• 2i ≤ F (a) for all a ∈ RN , a − b2⊥i ≤ F (a, b) ≤ d2

for all a ∈ H and b ∈ H.

Now we are going to construct an upper bound for F (a(n) ), and it will be inequality [1.62]. Denote for fixed n

   F (a(n−1) ) − F (a∗ ) F (a(n−1) ) αn = .

 2  (n−1) 2 F (a ) − F (a∗ ) + d Then,

   d2 − F (a(n−1) ) − F (a∗ ) F (a∗ ) . 1 − αn =

  2  F (a(n−1) ) − F (a∗ ) + d2 Taking into account the relations 0 ≤ F (a∗ ) ≤ F (a(n−1) ) ≤ d2 , we obtain the inequality 0 ≤ αn < 1. The next auxiliary result also will be applied to obtain [1.62]. Namely, we construct the upper bound for F ((1 − αn )a(n−1) + αn a∗ , a(n−1) ). In order to do this, note that for every t = 1, . . . , N , (1 − αn )a(n−1) + αn a∗ − ki• t ≤ (1 − αn ) a(n−1) − ki• i + αn a∗ − ki• t +  ≤ (1 − αn ) F (a(n−1) ) + αn F (a∗ )

40

Fractional Brownian Motion

= 

d2



F (a(n−1) ) , 2  F (a(n−1) ) − F (a∗ ) + d2

and (1 − αn )a(n−1) + αn a∗ − a(n−1) ⊥t = αn a∗ − a(n−1) ⊥t ≤ αn d

   d F (a(n−1) ) − F (a∗ ) F (a(n−1) ) = .

 2  F (a(n−1) ) − F (a∗ ) + d2 Hence, (1 − αn )a(n−1) + αn a∗ − ki• 2t + (1 − αn )a(n−1) + αn a∗ − a(n−1) 2⊥t

 2  d4 F (a(n−1) ) + d2 F (a(n−1) ) − F (a∗ ) F (a(n−1) ) ≤ 

2 2   F (a(n−1) ) − F (a∗ ) + d2 = 

d2 F (a(n−1) ) . 2  F (a(n−1) ) − F (a∗ ) + d2

[1.60]

Take the maximum over t = 1, . . . , N in the left-hand side of [1.60], apply equation [1.59], and obtain d2 F (a(n−1) ) F ((1 − αn )a(n−1) + αn a∗ , a(n−1) ) ≤  . 2  (n−1) 2 F (a ) − F (a∗ ) + d Now we continue with the upper bound for F (a(n) ). Recall that F (a) = minb F (a, b) and F (a(n) , a(n−1) ) = mina F (a, a(n−1) ). Therefore, the following inequality holds true: F (a(n) ) ≤ F (a(n) , a(n−1) ) ≤ F ((1 − αn )a(n−1) + αn a∗ , a(n−1) ) d2 F (a(n−1) ) ≤ F (a(n−1) ). ≤  2  (n−1) 2 F (a ) − F (a∗ ) + d

[1.61]

Projection of fBm on the Space of Martingales

41

The sequence {F (a(n) ), n ≥ 0} is decreasing and bounded, more exactly, 0 ≤ F (a(n) ) ≤ F (a(0) ). Hence, it converges to a finite limit. From inequality [1.61], we finally obtain the desired upper bound: F (a(n) ) ≤ 

d2 F (a(n−1) ) . 2  F (a(n−1) ) − F (a∗ ) + d2

[1.62]

Take the limit in [1.62] as n → ∞: lim F (a(n) ) ≤  n→∞

d2 limn→∞ F (a(n) ) . 2  limn→∞ F (a(n) ) − F (a∗ ) + d2

Therefore,

+

lim F (a(n) ) −

n→∞

whence either



2  F (a∗ ) lim F (a(n) ) ≤ 0,

limn→∞ F (a(n) ) −

n→∞

 F (a∗ ) = 0, or limn→∞ F (a(n) ) ≤ 0.

  If limn→∞ F (a(n) ) − F (a∗ ) = 0, then limn→∞ F (a(n) ) = F (a∗ ). Since F (a∗ ) = mina F (a) ≥ 0, the inequality 0 ≤ F (a∗ ) ≤ limn→∞ F (a(n) ) holds true, and if limn→∞ F (a(n) ) ≤ 0, then F (a∗ ) = limn→∞ F (a(n) ) = 0. Thus, the equality limn→∞ F (a(n) ) = F (a∗ ) holds true in either case. The first statement of the theorem is proved. 2) Let us draw the reader’s attention to the fact that for the second statement of the theorem, we have an additional condition, namely it is assumed that the function F (a) reaches its minimum at the unique point a∗ . In the proof of the first part, we established that F (a) reaches its minimum in H. Thus, a∗ ∈ H. Fix  > 0 and prove that a(n) − a∗  <  for n large enough. Denote H = {a ∈ H : a(n) − a∗  ≥ }. The set H ⊂ H ⊂ RN is closed and bounded. Recall that a(n) ∈ H for all n. If H = ∅, then a(n) − a∗  <  for all n. If H = ∅, then the function F (a), as a continuous function on a non-empty compact set, reaches its minimum on H , and min F (a) > min F = F (a∗ ) = lim F (a(n) ),

a∈H

n→∞

42

Fractional Brownian Motion

since a∗ ∈ H and F (a) reaches its minimal value min F at the unique point, which necessarily must coincide with a∗ . Therefore, min F (a) > F (a(n) )

a∈H

for n large enough, whence a(n) ∈ H ,

a(n) ∈ H \ H ,

a(n) − a∗  < 

for n large enough. We proved that for any  > 0, n0 exists such that for n ≥ n0 the inequality a(n) − a∗  <  holds true. Then a(n) → a∗ . The second statement of this theorem is proved.  Remark 1.9.– Generally speaking, condition [1.54] in Theorem 1.10 can be omitted. In order to prove Theorem 1.10 without condition [1.54], we can dismiss definitions [1.55] and [1.56] and define H and d as follows. Let H be the axis-aligned minimum bounding box that contains the points a(0) , k1 (a(0) ), k2 (a(0) ), . . . , and kN (a(0) ). In other words, we extend H in such a way that a(0) ∈ H. Then, let d be the length of the diagonal of H. With these definitions, and without condition [1.54], the further proof remains correct. Remark 1.10.– 1) Theorem 1.10 is valid for arbitrary matrix K = (kts )N t,s=1 . The fact that K is a factor of the Cholesky decomposition of the covariance matrix Σb , is not used in the proof. 2) In Theorem 1.5, we proved that continuous-time functional f (a), a ∈ L2 ([0, 1]), has a unique minimum. We have not proved a similar result for discrete-time functional F (a), a ∈ RN . Remark 1.11.– From inequality [1.62], we obtain the following bound: 2 +  d2 (F (a(n−1) ) − F (a(n) )) . F (a(n−1) ) − F (a∗ ) ≤ F (a(n) )

Projection of fBm on the Space of Martingales

43

Hence, 0 ≤ F (a(n−1) ) − F (a∗ ) ⎧ , ⎪ F (a(n−1) )(F (a(n−1) ) − F (a(n) )) d2 (F (a(n−1) ) − F (a(n) )) ⎪ ⎪ 2d − , ⎪ ⎪ ⎪ F (a(n) ) F (a(n) ) ⎪ ⎪ ⎪ ⎪ d2 (F (a(n−1) ) − F (a(n) )) ⎪ ⎨ if F (a(n−1) ) ≥ , F (a(n) ) ≤ 2 (n−1) ⎪ ) − F (a(n) )) ⎪ ⎪ d (F (a ⎪ , ⎪ ⎪ F (a(n) ) ⎪ ⎪ ⎪ ⎪ d2 (F (a(n−1) ) − F (a(n) )) ⎪ ⎩ if F (a(n−1) ) ≤ . F (a(n) ) [1.63] Finally, it follows from [1.63] that 0 ≤ F (a(n−1) ) − F (a∗ ) , F (a∗ )(F (a(n−1) ) − F (a(n) )) d2 (F (a(n−1) ) − F (a(n) )) + ≤ 2d F (a(n) ) F (a(n) ) + d2 (F (a(n−1) ) − F (a(n) )) . ≤ 2d F (a(n−1) ) − F (a(n) ) + F (a(n) ) Using inequalities F (a(0) ) ≤ d2 and [1.62], we can construct the following bounds. If F (a∗ ) = 0, then 0 ≤ F (a(n) ) ≤ d2 /(n + 1). Otherwise, if F (a∗ ) > 0, then F (a∗ ) ≤ F (a(n) ) ≤ F (a∗ ) +

d4 . 4nF (a∗ )

1.8.3. Implementation of the alternating minimization algorithm We approximate the vector that minimizes the functional F (a) as follows. As the initial approximation, we take the bottom row of the matrix K, i.e. a(0) = kN • . Then we iteratively perform minimization in [1.51] using the Chebyshev center algorithm, which is presented in Appendix 2. We stop iterations [1.51] when both conditions become true: the upper bound [1.63] is lower than the threshold and the distance a(n−1) − a(n)  is less than the threshold. Finally, we take a(n) as an approximation of the point of minimum. The convergence of the algorithm is demonstrated in the following experiment. We took H = 0.6 and N = 1000. We performed 10 iterations of

44

Fractional Brownian Motion

the algorithm and obtained 10 approximations a(n) , n = 1, . . . , 10 of the point a∗ in which the function F (a) reaches minimum, taking a(0) = kN • as the initial approximation. In Table 1.1 we present the following items: values of criterion function at a(n) , n = 1, . . . , 10, the right-hand side of inequality [1.63], which serves as the guaranteed upper bound for F (a(n) ) − min F , and distances between some a(n) ’s. n

F (a(n) )

0 1 2 3 4 5 6 7 8 9 10

0.015646356 0.006397266 0.005156821 0.005117567 0.005117560 0.005117560 0.005117560 0.005117560 0.005117560 0.005117560 0.005117560

Upper bound a(n) − a(n−1)  a(n) − a(10)  for F (a(n) ) − min F 378.0 62.89 2.006 2.426 × 10−3 1.956 × 10−6 — 7.377 × 10−8 — 1.594 × 10−7 1.413 × 10−7 —

— 7.495 × 10−2 1.915 × 10−2 3.961 × 10−4 1.284 × 10−6 1.477 × 10−9 2.019 × 10−12 5.969 × 10−14 6.227 × 10−14 6.261 × 10−14 6.735 × 10−14

7.154 × 10−2 1.943 × 10−2 3.958 × 10−4 1.286 × 10−6 1.479 × 10−9 2.022 × 10−12 6.353 × 10−14 5.938 × 10−14 6.527 × 10−14 6.735 × 10−14 0

Table 1.1. Iterative minimization of the function F (a): the value of criterion function F (a(n) ), the upper bound for F (a(n) ) − min F obtained from inequality [1.63], the distances between consecutive approximation to the point of minimum and between each approximation and the last approximation

We can observe that the inequality F (a(6) ) < F (a(5) ), which should hold true theoretically, actually does not hold true due to rounding errors. The upper bound for F (a(n) ) − min F is very loose for n ≤ 4, and does not make sense for those values n that do not meet the inequality F (a(n+1) ) < F (a(n) ). The approximations a(n) of the minimum point a∗ seem to converge up to machine precision. 1.8.4. Computation of the minimizing function Table 1.2 gives the minimum values of the functional F (a) for various H and N . We tried N = 100, N = 200 and N = 1000 to see if the difference of N can explain the discrepancy between our results and those of [SHK 14]. The results in [SHK 14] for N = 200 are close to our results for N = 100.

Projection of fBm on the Space of Martingales

N

H

.51

.55

.6

.65

.7

.75

.8

.85

.9

.95

45

.99

100 min F (a) .00005 .0013 .0051 .0112 .0200 .0320 .0482 .0704 .1022 .1511 .2190 200 min F (a) .00005 .0013 .0051 .0113 .0201 .0321 .0483 .0706 .1025 .1515 .2193 1000 min F (a) .00006 .0013 .0051 .0113 .0202 .0322 .0485 .0708 .1027 .1518 .2196

Table 1.2. Values of min F for various H from 0.51 to 0.99 and for N = 100, N = 200, and N = 1000

0.00

0.05

0.10

minF

0.15

0.20

0.25

Figure 1.2 shows the values of min F for H from 0.51 to 0.99 with a step 0.010, and also for H = 0.501, 0.995 and 0.999, for N = 1000.

0.5

0.6

0.7

0.8

0.9

1.0

H

Figure 1.2. Values of min F for H from 0.501 to 0.999 with step 0.010 and for H = 0.501, 0.995 and 0.999

As a by-product, the Chebyshev center algorithm computes weights wt in the linear combination a=

N

wt kt• ,

i=1

which can be used to obtain the distribution of the random variable ξa introduced in Lemma 1.12. Our numerical experiment shows that the random variable ξa is concentrated in the interval (t1 , t∗a ) and at point 1; thus

46

Fractional Brownian Motion

Ga = [t1 , t∗a ] ∪ {1}. The support of ξa and P[ξa = 1] for H = 0.51, 0.52, . . . , 0.99 are shown on Figure 1.3. For instance, for H = 0.6 the random variable ξa is concentrated in the interval (0.8, 0.868) and at point 1, with P[ξa ∈ (0.8, 0.868)] = 0.693,

[1.64]

and P[ξa = 1] = 0.307. For H = 0.75, the random variable ξa is concentrated in (0.080, 0.868) and at 1, with P[ξa ∈ (0.080, 0.868)] = 0.583,

[1.65]

and P[ξa = 1] = 0.417. For H = 0.9, ξa is concentrated in (0.035, 0.208) and at 1, with P[ξa ∈ (0.035, 0.208)] = 0.52,

[1.66]

and P[ξa = 1] = 0.48. The distribution function of the continuous part of the distribution of ξa is shown on Figures 1.6, 1.7 and 1.8 for H = 0.6, H = 0.75 and H = 0.9, respectively. We numerically evaluate the minimizing function for some H < 0.5, namely for H = 0.1 and H = 0.4, see Figures 1.4 and 1.5. The plots suggest that for 0 < H < 1/2 the minimizing function a(s) is equal to E[z(ξa , s) | ξa ≥ s], (i.e. Theorem 1.7 holds true), where the random variable ξa satisfies the following properties: 1) the support of ξa is an interval [t1 , 1] for some t1 ∈ (0, 1); 0;

2) the distribution of ξa is absolutely continuous. Particularly, P(ξa = 1) =

3) the probability density function (pdf) pξa = pξa (t), t ∈ [0, 1] of ξa satisfies the relation limt→1− pξa (t) = +∞. However, this is an observation rather than a theoretically proven statement. Figures 1.4–1.8 contain graphs of the minimizing vector (thick solid line) t and the scaled squared distance R(t) = s=1 (kts − as )2 (thin solid line) for H = 0.1, 0.4, 0.6, 0.75, 0.9 and N = 1000. The key point of the whole theory we have constructed is that we consider the minimizing function a = a(t) as a result of successive constructions of the vector that minimizes the functional F = F (a), defined by equality [1.47]. As a result of the plotting, the minimizing function a(t) varies slightly on the interval where R(t) reaches its maximum, and decreases outside the interval.

0.0

0.1

0.2

P[xi=1]

0.3

0.4

0.5

Projection of fBm on the Space of Martingales

0.5

0.6

0.7

0.8

0.9

1.0

0.8

0.9

1.0

0.0

0.2

0.4

T

0.6

0.8

1.0

H

0.5

0.6

0.7 H

Figure 1.3. Graph of P[ξa = 1] and the support of the random variable ξa for H from 0.51 to 0.99 with step 0.010. (H, T ) is the point from the shaded area that is the support of ξa

47

a(t), g_a(t), and pdf_xi(t)

Fractional Brownian Motion

0.0

0.2

0.4

0.6

0.8

1.0

t

Figure 1.4. The minimizing function a(t) (thick solid line), the scaled square distance ga (t) (thin solid line), and the pdf of ξa (dashed line) for H = 0.1

a(t), g_a(t), and pdf_xi(t)

48

0.0

0.2

0.4

0.6

0.8

t

Figure 1.5. The minimizing function a(t), the scaled square distance ga (t), and the pdf of ξa for H = 0.4

1.0

a(t), g_a(t), and pdf_xi(t)

Projection of fBm on the Space of Martingales

69.3%

0.0

0.2

0.4

0.6

0.8

1.0

t

a(t), g_a(t), and pdf_xi(t)

Figure 1.6. The minimizing function a(t), the scaled square distance ga (t), and the pdf of ξa for H = 0.6. Percentage corresponds to equality [1.64]

58.3%

0.0

0.2

0.4

0.6

0.8

1.0

t

Figure 1.7. The minimizing function a(t), the scaled square distance, and the pdf of ξa for H = 0.75. Percentage corresponds to equality [1.65]

49

Fractional Brownian Motion

a(t), g_a(t), and pdf_xi(t)

50

52%

0.0

0.2

0.4

0.6

0.8

1.0

t

Figure 1.8. The minimizing function a(t), the scaled distance ga (t), and the pdf of ξa for H = 0.9. Percentage corresponds to equality [1.66]

1.9. Exercises Exercise 1.1.– Let z : [0, 1]2 → R be a function such that  t z(t, s)2 ds < ∞. sup t∈[0,1]

0

(Here we do not assume that z is the Molchan kernel of fBm.) For a ∈ L2 ([0, 1]) define functions  t 2 2 ga (t) = (a(s) − z(t, s)) ds , t ∈ [0, 1]; [1.67] 0

f (a) = sup ga (t). t∈[0,1]

Prove the following statements: 1) The functional f reaches its minimal value on L2 ([0, 1]).

[1.68]

Projection of fBm on the Space of Martingales

51

2) Suppose that for every function a ∈ L2 ([0, 1]) where f (a) reaches minimum, ga (1) = f (a). Then the functional f reaches minimum at “essentially” unique point, i.e. if f (a1 ) = f (a2 ) = min f , then the functions a1 and a2 are equal a.e. on [0, 1]. Exercise 1.2.– Consider the Riemann–Liouville process  t 1 β R (t) = (t − s)β−1 dWs Γ(β) 0 for β > 12 . In this representation, which is similar to [1.4], the kernel is equal to  1 (t − s)β−1 , if t < s, z(t, s) = Γ(β) 0, if t ≥ s. Prove the following statements: 1) The Riemann–Liouville process, considered for β > the mean-square sense, i.e.

1 2,

is continuous in

lim E(Rβ (t2 ) − Rβ (t1 ))2 = 0.

t2 →t1

2) The functional f (a) that is constructed in [1.67]–[1.68] for this z(t, s), reaches its minimal value at “essentially” unique point. Exercise 1.3.– Give an example of such a lower-triangular N × N matrix K (with entries kts , 1 ≤ t ≤ N , 1 ≤ s ≤ N ) that the functional F (a) = max

t=1,...,N

t

(as − kts )2

s=1

reaches its minimum at multiple points.

2 Distance Between fBm and Subclasses of Gaussian Martingales

As we know from the previous chapter, the minimization problem in the whole class L2 ([0, T ]) is a problem without an explicit analytical solution. Therefore, let us consider the following simplification: introducing several subclasses of L2 ([0, T ]), for which it is possible to explicitly calculate the distance to fractional Brownian motion (fBm). Naturally, since the Molchan and Mandelbrot–Van Ness kernels contain power functions, subclasses under consideration contain various combinations of power functions. Most of the results are proved for the case H ∈ (1/2, 1), unless otherwise stated. Chapter 2 is organized as follows. Section 2.1 describes the procedure of evaluation of the minimizing function and the distance between fBm and the respective class of Gaussian martingales in the following cases: integrand is a constant function; it is a power function with a fixed exponent; it is a power function with arbitrary non-negative exponent; integrand a(s) is such that a(s)s1/2−H is non-decreasing; integrand is a power function with a negative exponent; it contains a linear combination of two power functions with different exponents. Section 2.2 compares the distances received in section 2.1 and some inequalities involving normalizing coefficient cH for an fBm. The new upper bound for sinx x is provided. Comparison is illustrated by the plots. Section 2.3 is devoted to distance between fBm and class of “similar” functions. In section 2.4, the distance between fBm and Gaussian martingales in the integral norm is evaluated, while section 2.5 describes the distance between fBm with Mandelbrot–Van Ness kernel and Gaussian martingales. Section 2.6 presents the calculations of the distance for an fBm with H ∈ (0, 1/2). Finally, in section 2.7, the opposite problem is solved, i.e. the distance from a Wiener process to the integrals with respect to an fBm is evaluated. The methods and techniques presented in this chapter can be applied for the approximation of the more involved processes (see, for example, [CHE 12, SHE 16, YAN 15]). Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

54

Fractional Brownian Motion

2.1. fBm and Wiener integrals with power functions Let T > 0 be fixed, B H = {BtH , t ∈ [0, T ]} be an fBm with the Hurst index 12 < H < 1, M = {Mt , t ∈ [0, T ]} be a Gaussian martingale of the t form Mt = 0 a(s)dWs with a ∈ L2 ([0, T ]), where W is the underlying Wiener process, so that W is the Wiener process from the Molchan representation [1.4] of B H . Denote  M(K) = {M = {Mt =

t 0

a(s)dWs , t ∈ [0, T ]}, a ∈ K}.

As before, consider the difference E(BtH − Mt )2 , 0 ≤ t ≤ T . That is, the problem is to find ρ2 (B H , M(K)) = min max E(BtH − Mt )2 a∈K 0≤t≤T

[2.1]

for some subclasses K ⊂ L2 ([0, T ]) such that the problem has an analytical or simple numerical solution. Recall that it was established in Chapter 1 that the solution of the minimization problem can be restricted to non-negative functions, so we assume that any of classes K under consideration consists of non-negative functions. Since a great deal of calculations involve special functions, we recall the following notations: Γ(x) stands for the gamma function; B(x, y) = Γ(x)Γ(y) Γ(x+y) stands for the beta function. Some properties of these functions are defined in section A1.1. 2.1.1. fBm and Wiener integrals with constant integrands Consider the simplest case. Namely, let the function a = a(s) be a constant, i.e. K1 = {a(s) = a ∈ (0, +∞)}. It means that we simply consider a Wiener process aW , i.e. the non-normalized Wiener process. Introduce the notation α = H − 1/2 > 0, and recall that a constant cH was introduced by equality [1.6]. Then, we can prove the following result. Theorem 2.1.– The following equality holds: ρ2 (B H , M(K1 )) := min max E(BtH − Mt )2 = T 2H (1 − c21 (H)), a∈K1 0≤t≤T

Distance Between fBm and Subclasses of Gaussian Martingales

55

where α πα B(1 − α, α)cH = cH . α+1 (α + 1) sin(πα)

c1 (H) =

[2.2]

This minimum is reached on the function amin = c1 (H) · T α . Proof.– Since a(s) is a constant, we have the obvious equalities  Mt =



t 0

a(s)dWs =

t 0

adWs = a(Wt − W0 ) = aWt .

Then, it follows that E(BtH − Mt )2 = E(BtH − aWt )2 = E(BtH )2 − 2aE(BtH Wt ) + a2 EWt2 = t2H − 2aE(BtH Wt ) + a2 t. Let us calculate E(BtH Wt ):  E(BtH Wt )



t

=E 0

z(t, s)dWs · Wt = 

= α cH

t

s

−α

0



t

z(t, s)ds 0

t

·

u (u − s) α

α−1

 du ds.

[2.3]

s

Let us change the order of integration in [2.3]. This leads to the equalities  EBtH Wt = α cH



t

uα 0

0

u

s−α (u − s)α−1 ds

 du.

Replacing the variables s = ux, we reduce the latter integral to  1   t H α −α α−1 u x (1 − x) dx du E(Bt Wt ) = α cH 0

= α cH ·

0

α+1

t · B(1 − α, α) = c1 (H)tα+1 . α+1

Hence, E(BtH − aWt )2 = t2H − 2a c1 (H)tα+1 + a2 t =: f (t).

56

Fractional Brownian Motion

Differentiate function f in t and equate the derivative to zero: 2H · t2α − 2a · c1 (H) · tα · (α + 1) + a2 = 0. Changing the variable tα =: x we obtain the following quadratic equation: 2Hx2 − 2ac1 (H)(α + 1)x + a2 = 0.

[2.4]

The discriminant of quadratic equation [2.4] equals   D = 4(α + 1)2 c21 (H)a2 − 4 · 2H · a2 = 4a2 · (α + 1)2 c21 (H) − 2H . [2.5] Let us establish that D < 0 for any 1/2 < H < 1. Indeed,  2 B(1 − α, α) (α + 1)2 α − 2H cH α+1 2    12 2HΓ(1 − α) B(1 − α, α) 2 · − 2H = (α + 1) α · Γ(α + 1) · Γ(1 − 2α) α+1  =

 α·

2HΓ(1 − α) Γ(α + 1) · Γ(1 − 2α)

 12

2 · Γ(1 − α)Γ(α)

 (Γ(1 − α))3 (Γ(α))2 = 2H · α2 · −1 Γ(α + 1) · Γ(1 − 2α) 

(Γ(1 − α))3 −1 = 2H · (Γ(α + 1))2 · Γ(α + 1) · Γ(1 − 2α)  3 Γ(α + 1) (Γ(1 − α)) −1 . = 2H · Γ(1 − 2α)

− 2H

[2.6]

It follows from Lemma A1.1 that the right-hand side and consequently, [2.6], the discriminant of equation [2.5] is negative. Therefore, equation [2.4] has no real roots. Since the coefficient at x2 is greater than zero, we obtain df dt > 0. In other words, f (t) is increasing in t, whence max[0,T ] f (t) = f (T ). Furthermore, we have f (T ) = E(BTH − aWT )2 = a2 T − 2c1 (H)T α+1 a + T 2H ,

[2.7]

Distance Between fBm and Subclasses of Gaussian Martingales

57

and now we look for the minimal value of the quadratic function [2.7] in a: amin =

2c1 (H) · T α+1 = c1 (H) · T α . 2T

Therefore, 2

min max E(BtH − aWt )2 = (c1 (H)T α ) T − 2c21 (H)T α+1 T α + T 2H a

t∈[0,T ]

= c21 (H)T 2α+1 − 2c21 (H)T 2α+1 + T 2H = T 2H (1 − c21 (H)).

[2.8] 

Corollary 2.1.– For any 1/2 < H < 1, the following inequality holds √ 2H c1 (H) < , α+1 which is equivalent to 1

cH
0 for any H ∈ R. Consequently, [2.8]. Indeed, α+1 √ 1 2 2H < H + H + 4 , or that is the same for H > 0, 2H < H + 1/2 = α + 1. √ At the same time, inequality cH < 2H will be useful in what follows. 2.1.2. fBm and Wiener integrals involving power integrands with a positive exponent 2.1.2.1. Exponent is fixed To start with the power functions, first consider the power function with a fixed exponent α = H − 1/2 and varying coefficient k, i.e. K2 = {a(s) = k · sα , k > 0}.

58

Fractional Brownian Motion

Theorem 2.2.– The following equality holds: ρ2 (B H , M(K2 )) = min max E(BtH − Mt )2 = T 2H



a∈K2 0≤t≤T

1−

c2H 2H

 ,

[2.11]

and the value of kmin when the minimum is reached in [2.11] equals kmin = cH . Proof.– Since a(s) = k · sα , we can write the equality  Mt =



t 0

a(s)dWs = k

0

t

sα dWs .

In turn, this formula implies the following equalities:  t 2

H sα dWs E Bt − k 0

 2   t  t = (EBtH )2 − 2kE BtH sα dWs + k 2 E sα dWs 

= t2H − 2kE BtH

0





t

sα dWs

0

+ k2

0



t

s2α ds =: g(t, k).

[2.12]

0



t Let us calculate E BtH 0 sγ dWs for arbitrary γ > 0:    t   t  t  t E BtH z(t, s) sγ ds sγ dWs = E z(t, s)dWs sγ dWs = 0

0

 = αcH

sγ−α 

= αcH



t 0



t

uα 0

0

0

0

t

 uα (u − s)α−1 du ds

s

u

 sγ−α (u − s)α−1 ds du

= αcH B(γ − α + 1, α) = αcH B(γ − α + 1, α)



t

uα+γ du 0

tα+γ+1 . α+γ+1



t 2α+1 2α+1 In particular, E BtH 0 sα dWs = αcH B(1, α) t2α+1 = cH t2α+1 .

[2.13]

Distance Between fBm and Subclasses of Gaussian Martingales

59

t 2H Further, 0 s2α ds = t2H . Therefore, the function g(t, k) from [2.12] can be transformed to the following form: g(t, k) = t

2H

cH t2H t2H − 2k · + k2 · = t2H 2H 2H



cH 1 1−k· + k2 · H 2H

 .

Evidently, g(t, k) is non-negative and non-decreasing in t ∈ [0, T ]. It means that for any k > 0 we have the following equality: max g(t, k) = g(T, k) = T

2H



t∈[0,T ]

cH 1 1−k· + k2 · H 2H

 .

[2.14]

Let us calculate the point at which the minimum of the triple term 1−k·

cH 1 + k2 · H 2H

is reached. This point is kmin = cH . At the same time, first, g(T, kmin ) = 1 − kmin ·

cH 1 c2 2 · + kmin = 1 − H > 0, H 2H 2H

and second,  min max E(BtH − k k

0≤t≤T

0

t

1

 cH 1 + c2H · H 2H   2 c  = T 2H 1 − H . 2H

sH− 2 dWs )2 = T 2H



1 − cH ·

Remark 2.2.– As a by-product, again, we obtain the inequality c2H < 2H (note that another independent proof is given in Lemma A1.6). 2.1.2.2. Arbitrary non-negative exponent Consider a wider class of power functions with an arbitrary non-negative exponent. Thus, we now introduce the class K3 = {a(s) = k · sγ , k > 0, γ ≥ 0}. Theorem 2.3.– The following equality holds: ρ2 (B H , M(K3 )) = min max E(BtH − Mt )2 = T 2H (1 − c21 (H)). a∈K3 0≤t≤T

This minimum is reached at γmin = 0 and kmin = c1 (H)T α .

60

Fractional Brownian Motion

Proof.– According to our assumption, a = a(s) is a power function of the form a(s) = k · sγ ; therefore, we can write that  Mt =



t 0

a(s)dWs = k

0

t

sγ dWs .

Hence, we can use [2.13] and equality 

2

t

E 0

sγ dWs



t

=

s2γ ds =

0

t2γ+1 , 2γ + 1 > 0, 2γ + 1

to obtain the following relations:  t 2

H sγ dWs E Bt − k 0

 2  t   t  2 = EBtH − 2kE BtH sγ dWs + k 2 E sγ dWs 0

= t2H − 2k · c2 (H, γ) · tα+γ+1 + k 2 · where c2 (H, γ) =

0

2γ+1

t =: h(t), 2γ + 1

[2.15]

α cH B(1−α+γ, α) . α+γ+1

Differentiate function h in t: dh = 2H · t2H−1 − 2k c2 (H, γ) · (α + 1 + γ)tα+γ dt 1 + k2 · · (2γ + 1)t2γ 2γ + 1   = t2H−1 2H − 2k c2 (H, γ) · (α + 1 + γ) t−α+γ + k 2 · t−2α+2γ . Let us verify whether there is t ∈ [0, T ] such that

dh dt

= 0, i.e.

k 2 · t−2α+2γ − 2k c2 (H, γ) · (α + 1 + γ) t−α+γ + 2H = 0. Changing the variable k · t−α+γ =: x, we obtain the following quadratic equation: x2 − 2 c2 (H, γ) · (α + 1 + γ) · x + 2H = 0.

[2.16]

The discriminant D1 of quadratic equation [2.16] equals D1 = 4c22 (H, γ)(α + 1 + γ)2 − 8H.

[2.17]

Distance Between fBm and Subclasses of Gaussian Martingales

61

Let us prove that D1 < 0 for any 1/2 < H < 1 and γ > 0. Indeed,  D1 = 4 ·

αcH B(1 − α + γ, α) α+γ+1

2

· (α + γ + 1)2 − 8H

2  2H · Γ(1 − α) Γ(1 − α + γ) · Γ(α) =4·α · − 8H · Γ(α + 1) · Γ(1 − 2α) Γ(1 + γ)  2  Γ(α + 1) · Γ(1 − α) Γ(1 − α + γ) −1 . [2.18] = 8H · · Γ(1 − 2α) Γ(1 + γ) 2

If we substitute γ = 0 in equality [2.18], we obtain the right-hand side of equation [2.6]. According to Lemma A1.1, this value is negative. Now, our goal is to establish that D1 < 0 for any γ > 0. To this end, we establish that the function Γ(1−α+γ) Γ(1+γ) =: z(γ) from [2.18] is decreasing in γ > 0. Since z(γ) > 0 for γ > 0, this is equivalent to proving the fact that log z(γ) is decreasing in γ > 0. Consider   Γ(1 − α + γ) log (Γ(1 + γ)) − log (Γ(1 − α + γ)) log = (−α) · Γ(1 + γ) (1 + γ) − (1 − α + γ) and denote log (Γ(1 + γ)) − log (Γ(1 − α + γ)) =: ω(γ). (1 + γ) − (1 − α + γ) Note that log Γ(x) is a convex function as its derivative is increasing according to the Gauss formula, see [A1.4]. Then, it follows from Corollary A1.2 that the slope function ω(γ) is increasing in γ. Furthermore, −α = 1/2 − H < 0 and does not depend on γ. Therefore, log z(γ) decreases in γ, whence z(γ) for γ > 0 is less than Γ(1−α) Γ(1) = Γ(1 − α). Γ(1−α) It means that for α > 0, γ > 0, we have the relations Γ(1−α+γ) Γ(1+γ) < Γ(1) . Therefore, D1 < D < 0, where the value of D is determined by formula [2.6]. Coefficient at x2 in [2.16] is strongly positive, which means that this function is convex, whence dh dt > 0, and consequently function h increases in t.

2

T Then, max[0,T ] h(t) = h(T ) = E BTH − k 0 sγ dWs .

62

Fractional Brownian Motion

Hence,  ψ(γ, k) := max E

 BtH

0≤t≤T

−k

2

t γ

0

s dWs

= T 2H − 2k · c2 (H, γ) · T α+γ+1 + k 2 ·

T 2γ+1 . 2γ + 1

It is obvious that the function ψ(γ, k) is continuous in γ > 0 and k > 0, and it is bounded from below. In addition, ψ(γ, k) → +∞ as k → +∞, and lim ψ(γ, k) =

γ→∞

T 2H , if T ≤ 1 +∞, if T > 1.

Therefore, inf γ,k>0 ψ(γ, k) is reached at some point (γ0 , k0 ) with finite coordinates. At the same time, it is obvious that k0 = k0 (γ0 ) =

2c2 (H, γ0 ) · T α+γ0 +1 2·

T 2γ0 +1 2γ0 +1

= (2γ0 + 1) · c2 (H, γ0 ) · T α−γ0 ,

since ψ(γ0 , k) ≥ ψ(γ0 , k0 (γ0 )) for any k > 0. Thus, instead of inf γ≥0,k>0 ψ(γ, k), it is possible to calculate inf γ≥0 ψ(γ, k0 (γ)). However, ψ(γ, k0 (γ)) = T 2H − 2(2γ + 1) · c22 (H, γ) · T α−γ · T α+1+γ + (2γ + 1)2 · c22 (H, γ) · T 2α−2γ ·

T 2γ+1 2γ + 1

= T 2H − 2(2γ + 1) · c22 (H, γ) · T 2H + (2γ + 1) · c22 (H, γ) · T 2H

 = T 2H 1 − (2γ + 1) · c22 (H, γ) . [2.19] Now, let us minimize the right-hand side of [2.19] with respect to γ ≥ 0. Thus, we need to find out the point at which the maximum max(2γ + 1)c22 (H, γ) γ≥0

is reached, or the point at which √

max γ≥0

2γ + 1 Γ(1 − α + γ) α + γ + 1 Γ(1 + γ)

Distance Between fBm and Subclasses of Gaussian Martingales

63

is reached, depending on α ∈ (0, 12 ). Consider the function √

g(γ) :=

2γ + 1 Γ(1 − α + γ) , α + γ + 1 Γ(1 + γ)

γ ≥ 0.

Let us calculate the derivative of this function: g  (γ) =

1 (α + γ + · (Γ(1 + γ))2

Γ(1 − α + γ)  √ + 2γ + 1 · Γ (1 − α + γ) × 2γ + 1 1)2

· (α + γ + 1) · Γ(1 + γ)



− 2γ + 1 · Γ(1 − α + γ) · Γ(1 + γ) + (α + γ + 1) · Γ (1 + γ) . It follows from the calculations above that the derivative of the function √ g(γ), up to a positive factor 2γ + 1(α + γ + 1)−1 (Γ(1 + γ))−1 Γ(1 − α + γ), coincides with Γ (1 − α + γ) Γ (1 + γ) 1 1 − + − . Γ(1 − α + γ) Γ(1 + γ) 2γ + 1 α + γ + 1 Let us determine the sign of the function p(γ, α) :=

Γ (1 − α + γ) Γ (1 + γ) 1 1 − + − , Γ(1 − α + γ) Γ(1 + γ) 2γ + 1 α + γ + 1

It follows from Gauss’s formula [A1.4] that 

1 γ

p(γ, α) = 0

t − tγ−α 1 1 dt + − , 1−t 2γ + 1 α + γ + 1

If α = 0, then p(γ, 0) < 0. In addition, ∂p(γ, α) = ∂α



1 γ−α

t

0

log t 1 . dt + 1−t (α + γ + 1)2

γ ≥ 0.

γ ≥ 0.

64

Fractional Brownian Motion

Note that in the interval (0, 1] ∂p(γ, α) 0 0≤t≤T

0

0≤t≤T

= T 2H (1 − c22 (H, 0)) = T 2H (1 − c21 (H)), and the best approximation in the class K3 is given by the constant function.  Remark 2.3.– Since K3 ⊃ K2 , we can conclude that ρ2 (B H , M(K3 )) ≤ ρ2 (B H , M(K2 )). We have plotted the graph that not only confirms this observation but also shows that the latter inequality is strict. Indeed, Figure 2.1 plots the graphs of ρ2 (B H , M(K2 )) and ρ2 (B H , M(K3 )) as the functions of H ∈ 12 , 1 using the Mathematica software. Therefore, we conclude that c21 (H) > 1 c1 (H) = α+1 · sinπαπα cH , we obtain the inequality sin πα < πα

√

2α + 1 α2 √ , =1− α+1 (α + 1)(α + 1 + 2α + 1)

1 , α ∈ 0, 2

c2H 2H .

Since

[2.20]

with the equality at zero, in order to insert the respective limit into the lefthand side. Let us construct the graph of the difference and right of  the left√ 2α+1 1 hand sides of inequality [2.20] as a function of α ∈ 0, 2 , i.e. α+1 − sinπαπα (Figure 2.2). Thus, comparing distances in various metrics, as a by-product, we obtain an interesting and non-trivial inequality which compares elementary functions.

Distance Between fBm and Subclasses of Gaussian Martingales

65

Figure 2.1. The graphs of ρ2 (B H , M(K2 ))  and ρ2 (B H , M(K3 )), as the functions of H ∈ 12 , 1

Figure 2.2. The graph of the difference − sinπαπα as a function of α ∈ (0, 1/2)

√ 2α+1 α+1

2.1.3. fBm and integrands a(s) with a(s)s−α non-decreasing Consider another class of functions for which we can calculate the distance to fBm in our metric and find out at which function this distance is reached. Let K4 = {a ∈ C[0, T ], a(s)s−α is nondecreasing}.

66

Fractional Brownian Motion

Theorem 2.4.– Within the class K4 , the minimum in [2.1] is reached at the  c2H α 2H 1 − 2H . function a(s) = cH s and equals T Proof.– We rewrite the right-hand side of [2.1] as follows: ϕ(t) := t2H − 2cH α





t

u

uα 0

0

s−α (u − s)α−1 a(s)dsdu +



t

a2 (s)ds.

[2.21]

0

Differentiating the right-hand side of [2.21] with respect to t and taking into account continuity of a = a(s), we obtain ϕ (t) = 2H t2α − 2cH α tα



t 0

s−α (t − s)α−1 a(s)ds + a2 (t).

Changing the variable s = t · u in the latter integral, we obtain ϕ (t) = 2H t2α − 2cH α tα



1 0

s−α (1 − s)α−1 a(ts)ds + a2 (t).

Let s−α · a(s) =: b(s). Then, a(s) = b(s) · sα and ϕ (t) takes the form ϕ (t) = ψ(t) · t2H−1 , where ψ(t) = 2H − 2cH α

1 0

3

(1 − s)H− 2 b(ts)ds + b2 (t).

If the function b is non-decreasing with respect to t, then b(ts) ≤ b(t) for any t > 0 and s ≤ 1. Thus,  ψ(t) ≥ 2H − 2cH α

1 0

3

(1 − s)H− 2 ds · b(t) + b2 (t)

or ψ(t) ≥ 2H − 2cH · b(t) + b2 (t). On the one hand, the minimal value of the right-hand side is reached at b(t) = cH , and in this case ψ(t) = cH . On the other hand, the bigger the derivative ϕ (t) for all t, the bigger is ϕ(t). Thus, among all a(s) such that b(s) = a(s)s−α is non-decreasing, the minimum is reached at b(s) = cH . Thus, the lemma is proved.  Remark 2.4.– Obviously, Theorem 2.4 generalizes Theorem 2.2.

Distance Between fBm and Subclasses of Gaussian Martingales

67

2.1.4. fBm and Gaussian martingales involving power integrands with a negative exponent Consider now the classes of power functions with a negative power. Introduce the class K5 = {a(s) = k · s−α , k > 0}. First, we prove some auxiliary results. Let  2  t H −α f (t, k) := E Bt − k s dWs . 0

Lemma 2.1.– The following assertions hold: i) The function f (t, k) admits the representation f (t, k) = t2H − 4tk

t2−2H H + k2 . cH 2 − 2H

[2.22]

ii) For all k > 0 the function f (t, k), t ≥ 0 has a point of local maximum 1

1

t1 = t1 (k) = k 2α (p1 (H))− 2α ,  

where p1 (H) =

2H cH

+

2

2H cH

1

[2.23]

− 2H, and the point of local minimum 1

t2 = t2 (k) = k 2α (p2 (H))− 2α ,  

where p2 (H) =

2H cH

−

2

2H cH

[2.24]

− 2H.

iii) If we denote, klim := T 2α p1 (H), we can then claim that t1 (k) < T for k < klim , t1 (k) = T for k = klim , and t1 (k) > T for k > klim . Remark 2.5.– Equalities [2.23] and [2.24] obviously imply that t2 (k) > t1 (k). Proof.– Assertion i) follows from straightforward calculations. Indeed, we have  E(BtH

−k

t

s 0

−α

2

dWs ) = t

2H

 −

2kEBtH

t 0

s−α dWs + k 2

t2−2H . 2 − 2H

68

Fractional Brownian Motion

t According to representation [1.4], BtH = 0 z(t, s)dWs . Then, it is easy to verify that  t  t  t  t EBtH s−α dWs = z(t, s)s−α ds = αcH s−2α uα (u − s)α−1 duds 0

0



= αcH = αcH

0



t

u

uα 0

0

s

s−2α (u − s)α−1 dsdu = αcH B(1 − 2α, α)t

Γ(1 − 2α)Γ(α) Γ(1 − 2α)Γ(α + 1) t = cH t Γ(1 − α) Γ(1 − α)

= 2HcH c−2 H t=

2H t. cH

To prove assertion ii), we differentiate f (t, k) with respect to t:   ∂f (t, k) 4H + k 2 t−4α = t2α 2H − t−2α k · ∂t cH and change the variables as follows: x = kt−2α . Then, we obtain the quadratic equation x2 −

4H x + 2H = 0, cH

[2.25]

with two real-valued roots x1 = p1 (H) and x2 = p2 (H). This implies equalities [2.23] and [2.24]. Since the derivative ∂f ∂t is positive for t < t1 (k) and negative for t1 (k) < t < t2 (k), we prove that t1 (k) and t2 (k) are the points of local maximum and minimum, respectively. To prove assertion iii), we first calculate klim , i.e. a point such that t1 (k) = T . It is obvious that klim = p1 (H)T 2α . Second, noting that t1 (k) increases with respect to k, we therefore establish that t1 (k) < T for k < klim , while t1 (k) > T for k > klim . Thus, the lemma is proved.  Corollary 2.2.– The following equality holds: ⎧ if t1 (k) ≥ T, ⎨ f (T, k), max f (t, k) =   ⎩ t∈[0,T ] max f (t1 (k), k) , f (T, k) , if t1 (k) < T. Lemma 2.2.– inequality:

For any 1/2 < H < 1 constant c2H satisfies the following

c2H > 8H 2 (1 − H).

[2.26]

Distance Between fBm and Subclasses of Gaussian Martingales

69

Proof.– Let us substitute t = 1 in equation [2.22]. This gives us a strictly k2 positive quadratic function 1 − 4k cHH + 2−2H . Now, let us calculate its discriminant, which will obviously be negative: D=

16H 2 2 − . c2H 1−H

This is the formula from which the required inequality follows.



Remark 2.6.– Inequality [2.26] can be rewritten in the form Γ( 32 − H) > 4H(1 − H), Γ(H + 12 )Γ(2 − 2H)

[2.27]

or 2H
0 t∈[0,T ]

inf

0klim

Now, let us compare the functions ϕ1 (k) := f (t1 (k), k) and ϕ2 (k) := f (T, k) for 0 < k < klim . First, we establish the coincidence of these functions and their derivatives at the point k = klim . We have ϕ1 (klim ) = ϕ2 (klim ) by the definition of klim . It turns out that the derivatives of these functions are the same at the point klim . Lemma 2.4.– We have ϕ1 (klim ) = ϕ2 (klim ). Proof.– It is clear that ϕ1 (k) = (t1 (k))2H −

(t1 (k))2−2H 4kt1 (k)H + k2 cH 2 − 2H 2−2H

1

= (kp1 (H)

−1

)

2H 2H−1

(kp1 (H)−1 ) 2H−1 4k · (kp1 (H)−1 ) 2H−1 · H − + k2 cH 2 − 2H

2H

= k 2H−1 · d(H),

2H where d(H) = (p1 (H))− 2H−1 1 − 2H

= (p1 (H))− 2H−1 ·

[2.29] 4Hp1 (H) cH

+

p21 (H) 2−2H

 2H − 1 2H p1 (H) − 1 . · 1−H cH



72

Fractional Brownian Motion

Furthermore, ϕ2 (k) = T 2H −

T 2−2H 4kH T + k2 . cH 2 − 2H

[2.30]

Hence, ϕ1 (k) =

1 2H k 2H−1 d(H) 2H − 1

ϕ2 (k) =

kT 2−2H 4H T. − 1−H cH

and

Substituting klim in place of k, we obtain   1 2H 4Hp1 (H) 2H p2 (H) + 1 T p1 (H) 2H−1 · p1 (H)− 2H−1 1 − 2H − 1 cH 2 − 2H   4H p1 (H) 2H . + = T p1 (H)−1 − 2H − 1 cH 2 − 2H

ϕ1 (klim ) =

Similarly, ϕ2 (klim ) = T



p1 (H) 4H − 1−H cH

 .

It remains to note that p1 (H) is a root of equation [1.4], i.e. p21 (H) = − 2H. Then, we can rewrite ϕ1 (klim ) as follows:

4H cH p1 (H)

ϕ1 (klim ) =

2H 1−H



 2H − p1 (H)−1 T. cH

Finally, p1 (H) 4H 2H 4H 2 − + p1 (H)−1 − 1−H cH cH (1 − H) 1 − H   p1 (H)−1 4H p21 (H) − = p1 (H) + 2H = 0, 1−H cH which means that the derivatives are equal. Thus, the lemma is proved.



Distance Between fBm and Subclasses of Gaussian Martingales

73

Now, we are in a position to establish that klim is the largest abscissa of the two points of intersection of the functions ϕ1 (k) and ϕ2 (k). This follows from Lemma 2.5, applied together with the established equalities ϕ1 (klim ) = ϕ2 (klim )

and

ϕ1 (klim ) = ϕ2 (klim ).

Lemma 2.5.– For all k ≥ klim we have the following relation: ϕ1 (k) > ϕ2 (k). Proof.– Relations [2.29] and [2.30] imply that ϕ1 (k) =

2−2H 2H k 2H−1 d(H), 2 (2H − 1)

and that ϕ1 (k) increases with respect to k, while ϕ2 (k) = k = klim , we have

T 2−2H 1−H .

At the point

2−2H 2H T 2−2H (p1 (H)) 2H−1 d(H) 2 (2H − 1)   2H 2H 2−2H −2 = (p1 (H)) p1 (H) − 1 . T (1 − H)(2H − 1) cH

ϕ1 (klim ) =

It is sufficient to prove that 2H 2H − 1



 2H p1 (H) − 1 > p21 (H) cH

which is equivalent to 2H 2H − 1



 2H 4H p1 (H) − 1 > p1 (H) − 2H. cH cH

In turn, inequality [2.31] is equivalent to the following: p1 (H) > cH or 2H + cH



2H cH

2 − 2H > cH ,

[2.31]

74

Fractional Brownian Motion

or finally, 

2H cH

2 − 2H >

c2H − 2H . cH

This is true, since in Lemma A1.6 we showed that c2H < 2H.



Now, we show that there exists a unique point 0 < k ∗ < klim of intersection of the functions ϕ1 (k) and ϕ2 (k). This means that ϕ1 (k) < ϕ2 (k) for 0 < k < k ∗ and ϕ1 (k) > ϕ2 (k) for k ∗ < k < klim and for k > klim . Lemma 2.6.– A unique point k ∗ exists such that 0 < k ∗ < p2 (H)T 2α and ϕ1 (k ∗ ) = ϕ2 (k ∗ ). Proof.– The point k ∗ necessarily satisfies the relations ⎧ ∗ 2H ∗ 2 T 2−2H ∗ H 2H ⎪ = 0, ⎨ (k ) 2H−1 d(H) − (k ) 2−2H + 4k T cH − T ⎪ ⎩

1 2H ∗ 2H−1 d(H) 2H−1 (k )

2−2H

− k ∗ T1−H + 4T cHH > 0,

whence, in particular, we obtain that 2H 2H − 1

  2−2H T 2−2H T 2H H H ∗ T k − k∗ + ∗ > 0. − 4T + 4T 2 − 2H cH k 1−H cH

The latter inequality can be rewritten as follows: k ∗ T 1−2H − 4

H T 2H−1 + 2H > 0. cH k∗

Denoting y := k ∗ T 1−2H , we obtain the inequality y2 −

4H y + 2H > 0, cH

which is true for y > p1 (H) and for y < p2 (H). Note that if y > p1 (H), then k ∗ > T 2α p1 (H) = klim and the preceding lemmas imply that there are no points of intersection greater than klim . It remains to consider the behavior of the function 2H

θ(x) := x 2H−1 d(H) − x2

T 2−2H H − T 2H + 4xT 2 − 2H cH

Distance Between fBm and Subclasses of Gaussian Martingales

75

in the interval (0, p2 (H)T 2α ). It is obvious that θ(0) < 0. Thus, it is sufficient to show that θ(p2 (H)T 2α ) > 0, whence the existence of the point k ∗ follows (its uniqueness is obvious). Rewrite θ(p2 (H)T 2α ) as follows: θ(p2 (H)T

2α

  2H − 1 2H )=T p1 (H) − 1 1−H cH  (p2 (H))2 p2 (H)H −1 . − +4 2 − 2H cH 2H



p2 (H) p1 (H)

2H  2H−1

For simplicity, let us substitute u := p1 (H) > z := p2 (H). We need to prove the inequality   2H

z  2H−1 2H − 1 2H z2 zH u−1 − − 1 > 0, +4 u 1−H cH 2 − 2H cH where z 2 = z

2H 2H−1

4H cH z



[2.32]

− 2H. Inequality [2.32] is equivalent to the following:

2H u−1 cH

 >u

2H 2H−1



 2H z−1 , cH

or 1 2H 1 2H 4H 2 2H−1 4H 2 2H−1 z − z 2H−1 > u − u 2H−1 , cH cH

since zu = 2H. Consider the function β(s) := equals   2−2H 4H 2 1 β  (s) = − 2Hs , s 2H−1 2H − 1 cH

1 4H 2 2H−1 cH s

[2.33] 2H

− s 2H−1 . Its derivative

and becomes zero at the point s0 = 2H cH . In turn, β(u) reaches its maximum at this point. The points u and z are symmetric with respect to s0 . The absolute value of the derivative β  (s) at the point s0 + α is equal to 2−2H 1 (s0 + α) 2H−1 |2Hα| 2H − 1

if α > 0, or to |2Hα|

2−2H 2−2H 1 1 (s0 − α) 2H−1 < |2Hα| (s0 + α) 2H−1 2H − 1 2H − 1

76

Fractional Brownian Motion

if α < 0. This implies that β(z) > β(u), which is equivalent to inequality [2.33]. The lemma is proved.  Remark 2.7.– We can show that kmin < p2 (H)T 2α . Indeed, this bound is equivalent to the inequality  2 4H(1 − H) 2H 2H < − − 2H, cH cH cH which can be transformed to the equivalent ones:  2 2H 4H 2 − 2H − 2H < , cH cH or −2H
8H 2 (1 − H), which was proved in Lemma 2.2. Note that the inequality kmin < p2 (H)T 2α improves Lemma 2.3 essentially. Remark 2.8.– The following inequality holds: kmin > k ∗ ,

[2.34]

which, in fact, is equivalent to ϕ1 (kmin ) > ϕ2 (kmin )

[2.35]

or 2H

(kmin ) 2H−1 d(H) +

8H 2 (1 − H) 2H T − T 2H > 0. c2H

Let us demonstrate that inequalities [2.34] and [2.35] are equivalent. Indeed, at the point k = 0 function ϕ1 = 0, while ϕ2 > 0. If ϕ1 (kmin ) > ϕ2 (kmin ), then the function ϕ1 (k) − ϕ2 (k) has different signs at points 0 and kmin ; therefore, it equals zero at some point (0, kmin ). The smallest root of the equation ϕ1 (k) = ϕ2 (k) will be equal to k ∗ . Therefore, if ϕ1 (kmin ) > ϕ2 (kmin ), then k ∗ is situated between 0 and kmin , i.e. k ∗ < kmin . Figure 2.5 contains plots of the functions ϕ1 (kmin ) and ϕ2 (kmin ) as the functions of H ∈ (1/2, 1).

Distance Between fBm and Subclasses of Gaussian Martingales

77

Figure 2.5. Plots of the functions ϕ1 (kmin ) and ϕ2 (kmin ) as the functions of H ∈ (1/2, 1)

Figure 2.6. Difference ϕ1 (kmin ) − ϕ2 (kmin ) as a function of H ∈ (1/2, 1)

Figure 2.6 demonstrates the graph of the difference of the functions, ϕ1 (kmin ) − ϕ2 (kmin ). The graph clearly shows that ϕ1 (kmin ) > ϕ2 (kmin ) for 1/2 < H < 1. Note that the latter inequality essentially improves inequality [2.32]. The plots of the functions ϕ1 (k) and ϕ2 (k) are as shown in

78

Fractional Brownian Motion

Figure 2.7 for H = 23 . For other H, the plots are similar. All the plots are constructed for T = 1.

Figure 2.7. Plots of the functions ϕ1 (k) and ϕ2 (k) as the functions of k for H = 2/3

We also give a plot of k ∗ as a function of H (Figure 2.8).

Figure 2.8. Plot of k∗ as a function of H

Distance Between fBm and Subclasses of Gaussian Martingales

79

Now, we prove the main result of this section. Theorem 2.5.– Let the subspace K5 = {a(s) = ks−α , k > 0}. Then, ρ2 (B H , M(K5 )) = min max E(BtH − Mt )2 = ϕ1 (k ∗ ) a∈K5 0≤t≤T

and the minimum is reached at the function a(s) = k ∗ s−α , where k ∗ is the smallest abscissa of the two points of intersection of the functions 2H

ϕ1 (k) = k 2H−1 d(H), where d(H) = (p1 (H))

2H − 2H−1

2H − 1 1−H



 2H p1 (H) − 1 , cH

and ϕ2 (k) = T 2H −

T 2−2H 4kH T + k2 . cH 2 − 2H

Proof.– We apply Corollary 2.2 and Remark 2.8. If k ≤ klim , then max{f (t1 (k), k), f (T, k)} = max{ϕ1 (k), ϕ2 (k)} = ϕ2 (k)1kk∗ . If k < k ∗ , then Lemma 2.6 implies that ϕ2 (k) > ϕ2 (k ∗ ); otherwise, if k > k ∗ , then ϕ1 (k) > ϕ1 (k ∗ ) (and ϕ1 (k ∗ ) = ϕ2 (k ∗ )). Finally if k > klim , then ϕ2 (k) = f (T, k) > ϕ2 (klim ) = ϕ1 (klim ) > ϕ1 (k ∗ ). Therefore, mink maxt∈[0,T ] f (t, k) = ϕ1 (k ∗ ) (= ϕ2 (k ∗ )). Thus, the theorem is proved. 

80

Fractional Brownian Motion

2.1.5. fBm and the integrands a(s) = a0 sα + a1 sα+1 Let us expand the class K2 of power functions, i.e. estimate the distance between fBm and the class of functions  6 = {a(s) = a0 sα + a1 sα+1 , a0 , a1 ∈ R}. K First, we present some preliminary arguments. Let, as before,  Mt =



t

a(s)dWs , BtH =

0

the kernel z(t, s) = cH α s−α which it is proved that 

t 0

α 2

t s

s 0

z(t, s)dWs ,

uα (u − s)α−1 du. We apply Theorem 2.2, in

2H



(z(t, s) − cH s ) ds = t

c2 1− H 2H

 .

This equality implies the following:  t E(BtH − Mt )2 = (z(t, s) − a(s))2 ds 0



t

= 0



((z(t, s) − cH sα ) + (cH sα − a(s)))2 ds

t

= 0



α 2

(z(t, s) − cH s ) ds + 2



t

2

0



− 2αcH 

t 0

0 2H

(cH s − a(s)) ds = t

+

α

+

t



t

u 0

u

α 0

(z(t, s) − cH sα ) · (cH sα − a(s)))ds 

c2 1− H 2H



(u − s)α−1 · (a(s)s−α − a(u)u−α )dsdu

(cH sα − a(s))2 ds.

[2.36]

Let us introduce the functions b(u) = a(u)u−α and c(u) = b(u) − cH . Then, we can transform the distance as follows:    t  u c2 E(BtH − Mt )2 = t2H 1 − H − 2αcH uα (u − s)α−1 · (b(s) 2H 0 0  t s2α (b(s) − cH )2 ds [2.37] − b(u))dsdu + 0

Distance Between fBm and Subclasses of Gaussian Martingales

=t

2H



c2 1− H 2H



 − 2αcH

 − c(u))dsdu +

t



t

u

u

α

0

81

0

(u − s)α−1 · (c(s)

s2α c2 (s)ds.

0

Let us try to pick a function c(u) so as to minimize the components. To do this, on the right-hand side of equation [2.37], we set c(u) = a0 + a1 u, which is equivalent to a(u) = (a0 + cH + a1 u)uα in [2.36]. Since simultaneous minimization in a0 and a1 leads to overly cumbersome calculations, thereby requiring some additional restriction on these coefficients, we shall minimize the distance, applying Lemma 1.7 and Remark 1.4. Consider the following subclass of functions:  6 : fa (T ) = 2Hfa (T )}, K6 = {a ∈ K

[2.38]

2

t where fa (t) = E BtH − 0 a(s)dWs , and the derivatives are taken in t. Obviously, the distance between fBm and subclass K6 has the form  t 2

(a0 sα + a1 sα+1 )dWs . [2.39] ρ2 (B H , M(K6 )) = min max E BtH − a∈K6 0≤t≤T

0

Theorem 2.6.– A minimal value in the distance [2.39] is reached at the point a0 = a∗0 , a1 = a∗1 , where α(2α + 1)(2α + 3) cH , (α + 1)   2α + 3 αcH , a∗1 = − · a0 + T (2α + 2) α+1

a∗0 =

and this minimal value equals  c2H α2 (2α + 3)c2H  T 2α+1 1 − . − 2α + 1 (α + 1)2   6 )) ≤ T 2α+1 1 − Therefore, ρ2 (B H , M(K

c2H 2α+1

−

α2 (2α+3)c2H (α+1)2

 .

82

Fractional Brownian Motion

Proof.– Let us substitute c(u) = a0 + a1 u in formula [2.37] and obtain   c2 fa (t) = E(BtH − Mt )2 = t2H 1 − H 2H  t  u  t + 2αcH uα (u − s)α · a1 dsdu + s2α (a20 + 2a0 a1 s + a21 s2 )ds = t2H

0

 1−

c2H



0

0

+ 2αcH a1

2H

2α+2

1 t α + 1 2α + 2

t t2α+2 t2α+3 + 2a1 a0 + a21 2α + 1 2α + 2 2α + 3   c2H a20 = t2α+1 1 − + 2α + 1 2α + 1   αcH a21 2α+2 a1 a0 + + t2α+3 · +t . α+1 α+1 2α + 3 + a20

2α+1

Let us take in [2.40] the derivative in t and obtain the following equality: 

fa (t) = t2α (κ0 + κ1 t + κ2 t2 ),

[2.40]

where κ0 = 2α + 1 −

c2H

+

a20 ,

 κ1 = 2a1

αcH a0 + α+1

 ,

κ2 = a21 .

[2.41]

Taking into account the fact that we consider functions from the class K6 , we obtain from [2.38] that at point T , T

2α

2α + 1 2α+1 (κ0 + κ1 T + κ2 T ) = T T 2



κ2 κ0 κ1 +T + T2 2α + 1 2α + 2 2α + 3



whence κ2 = −κ1 ·

2α + 3 . 2T · (2α + 2)

[2.42]

The resulting equality is equivalent to the following: a1 = −

  2α + 3 αcH , · a0 + T · (2α + 2) α+1

[2.43]

,

Distance Between fBm and Subclasses of Gaussian Martingales

83

whence fa (T ) = T 2α+1 [1 − − a0 ·

c2H 1 1 (a2 · + 2α + 1 (2α + 2)2 0 2α + 1

2(2α + 3)αcH (2α + 3)α2 · c2H )]. − α+1 (α + 1)2

[2.44]

A minimal value of [2.44] in a0 is reached at the point a 0 = cH

α(2α + 1)(2α + 3) , (α + 1)

and equals  min fa (T ) = T 2α+1 1 −

c2H 2α + 1

α2 (2α + 3)2 (2α + 1)c2 (2α + 3)α2 c2H  1 H − − + (2α + 2)2 (α + 1)2 (α + 1)2  c2H α2 (2α + 3)c2H  . [2.45] = T 2α+1 1 − − 2α + 1 (α + 1)2

a0

It remains to prove that at the point a0 = a∗0 , a1 = a∗1 function fa (t) is increasing in t. Then, it will be true that mina0 ,a1 max0≤t≤T fa (T ) ≤ fa∗ (T ), where a∗ = (a∗0 , a∗1 ). In turn, taking into account equation [2.40], we see that it is enough to prove that the smallest root of the equation κ0 + κ1 t + κ2 t2 = 0 exceeds T . As it is sufficient to consider T = 1, the root mentioned above equals t1 =

−κ1 −

 κ21 − 4κ0 κ2 1 =α+1− c2H α2 (α + 1)2 − κ0 . 2κ2 αcH

Inequality t1 > 1 is equivalent to the following: α2 > (α + 1)2 − or 2α + 1


(2α + 1)2 (2α + 3)2 > (2α + 1)2 > 2α + 1, (α + 1)2

therefore, it is really true that t1 > 1.



Corollary 2.3.– It follows from [2.45] that 1−

c2H α2 (2α + 3)c2H > 0, − 2α + 1 (α + 1)2

hence, we can obtain the following upper bound for cH :  (2α + 1)(α + 1)2 cH < . (α + 1)2 + α2 (2α + 1)(2α + 3)

[2.46]

This upper bound for any H ∈ (1/2, 1) clearly improves the upper bound c2H ≤ 2H, obtained in Lemma A1.6, both are true for any H ∈ (1/2, 1) 2.1.6. fBm and Wiener integrals involving integrands k1 + k2 sα Introduce the class of functions K7 = {k1 + k2 · sα , k1 , k2 ∈ R}, where generally α = H − 12 . As before, our goal is to calculate  t 2

ρ2 (B H , M(K7 )) = min max E BtH − a(s)dWs . a∈K7 0≤t≤T

0

It turns out that the situation depends on the value of H and is different for H < H0 and H ≥ H0 , where H0 ≈ 0.94 (this value will be defined later). The case H ≥ H0 , which is simpler and easier, is considered in section 2.1.6.1. The values H ≥ H0 will be called “close to one”. The case H < H0 is considered in section 2.1.6.2.

Distance Between fBm and Subclasses of Gaussian Martingales

85

2.1.6.1. fBm with Hurst index close to one and Wiener integrals involving integrands k1 + k2 sα Let H ≥ H0 . Denote f (t, k1 , k2 ) := E(BtH − Mt )2  2  tα+1 k2 cH 2H =t − k2 · + 1 − 2k1 (c1 (H)(α + 1) − k2 ) + tk12 2H H α+1 =: t2H · a(k2 ) − 2k1

tα+1 · b(k2 ) + tk12 , α+1

where a(k2 ) :=

k22 cH − k2 · + 1 > 0, 2H H

b(k2 ) := c1 (H)(α + 1) − k2 , α c1 (H) = cH · · B(1 − α, α). α+1 The inequality a(k2 ) > 0 has been proved in Lemma A1.7. It will follow from further calculations that for some values of the coefficients k1 and k2 it holds that maxt∈[0,T ] f (t, k1 , k2 ) = f (T, k1 , k2 ). Therefore, we prove the following result. Lemma 2.7.– Local and global minima in k1 , k2 of the function f (T, k1 , k2 ) = T 2H · a(k2 ) − 2k1 ·

T α+1 · b(k2 ) + k12 T, α+1

are reached at the point (k1∗ , k2∗ ), where k1∗ = k2∗ =

(α+1)(c1 (H)(α+1)−cH )T α , α2

(α + 1) (cH (α + 1) − c1 (H)(2α + 1)) . α2

Proof.– Let us calculate partial derivatives of the function f (T, k1 , k2 ). We have ∂f (T, k1 , k2 ) T α+1 = −2T α+1 c1 (H) + 2k1 T + 2k2 , ∂k1 α+1 T α+1 T 2α+1 cH ∂f (T, k1 , k2 ) = −2T 2α+1 + 2k1 + 2k2 . ∂k2 2α + 1 α+1 2α + 1 According to the necessary condition of a local extremum, which we know has the form ∂f (T, k1 , k2 ) ∂f (T, k1 , k2 ) = 0, = 0, ∂k1 ∂k2

86

Fractional Brownian Motion

we obtain a system of linear equations: ⎧ T α+1 ⎪ ⎨ 2k1 T + 2k2 α+1 − 2T α+1 c1 (H) = 0, ⎪ ⎩ 2k

T α+1 1 α+1

2α+1

cH + 2k2 T2α+1 − 2T 2α+1 2α+1 = 0.

Solving this system of equations, we obtain k1 =

(α+1)(c1 (H)(α+1)−cH )T α , α2

k2 =

(α+1)(cH (α+1)−c1 (H)(2α+1)) . α2

[2.47]

Denote the values obtained in [2.47] by k1 =: k1∗ and k2 =: k2∗ . Now, let us verify that [2.47] is the point of local minimum. That is, we need to verify ⎛ ∂ 2 f (T,k1 ,k2 ) ∂ 2 f (T,k1 ,k2 ) ⎞ ⎜ whether the matrix ⎝

∂k12

∂k1 ∂k2

2

∂ f (T,k1 ,k2 ) ∂k1 ∂k2

2

∂ f (T,k1 ,k2 ) ∂k22

⎟ ⎠ is positive definite. In turn,

matrix 2 × 2 is positive definite by the Silvester criterion if and only if ∂ 2 f (T,k1 ,k2 ) > 0, and the determinant of this matrix is positive. ∂k2 1

∂ 2 f (T,k1 ,k2 ) ∂k12

We have    Δ =   =

1 Tα 1 T α (α+1)

= 2T > 0 and the determinant is equal to 1 α+1

1 2α+1

    1 1 = 1  T α 2α + 1 − (α + 1)2 

α2 1 · 2 > 0. α T (2α + 1) (α + 1)

Therefore, (k1∗ , k2∗ ) (with k1∗ and k2∗ defined in [2.47]) is the point of a local minimum. Now, we are in a position to establish that [2.47] is, moreover, the point of the global minimum. That is, we want to prove that for any k1 , k2 ∈ R f (T, k1 , k2 ) > f (T, k1∗ , k2∗ ).

Distance Between fBm and Subclasses of Gaussian Martingales

87

In order to do this, we apply the Taylor formula for polynomials of the second degree and obtain f (T, k1 , k2 ) = f (T, k1∗ , k2∗ ) ⎛ ⎜ + (k1 − k1∗ , k2 − k2∗ ) · ⎝

T T α+1 α+1

T α+1 α+1 T 2α+1 2α+1

⎞ ⎛ ⎞ k1 − k1∗ ⎟ ⎝ ⎠. ⎠· ∗ k2 − k2

Since the last matrix is positive definite, the following value is non-negative: ⎛ ⎜ (k1 − k1∗ , k2 − k2∗ ) · ⎝

T T α+1 α+1

T α+1 α+1 T 2α+1 2α+1

⎞ ⎛ ⎞ k1 − k1∗ ⎟ ⎝ ⎠ ≥ 0. ⎠· k2 − k2∗

Thus, we have f (T, k1 , k2 ) ≥ f (T, k1∗ , k2∗ ); equality is achieved for k1 = k1∗ , k2 = k2∗ .  Let us verify that k1∗ > 0. Indeed, using [A1.24], we can conclude that k1∗ =

(α + 1)(c3 (H) − cH )T α > 0, α2

[2.48]

where c3 (H) := c1 (H)(α + 1) = αB(1 − α, α)cH .

[2.49]

We also verify equality k1∗ =

b(k2∗ )T α , α+1

[2.50]

which we will use in section 2.1.6.2. Indeed, Tα b(k2∗ )T α = (c1 (H)(α + 1) − k2∗ ) α+1 α+1   Tα (α + 1)(cH (α + 1) − c1 (H)(2α + 1)) = c1 (H)(α + 1) − α2 α+1   cH (α + 1) − c1 (H)(2α + 1) Tα = c1 (H) − α2 =

c1 (H)(α2 + 2α + 1) − cH (α + 1) α T = k1∗ . α2

88

Fractional Brownian Motion

A more elegant proof is to note that k1∗ is the point at which the quadratic 2T α+1 b(k2∗ ) polynomial f (T, k1 , k2∗ ) = T k12 − k1 + T 2H a(k2∗ ) reaches a minimum. α+1 Due to [2.48] and [2.50], we have b(k2∗ ) > 0.

[2.51]

Inequality [2.51] means that k2∗ < c1 (H)(α + 1) = c3 (H). For some reason, which will be clear in the proof of Theorem 2.7, denote k20 =

c23 (H) − 2H . 2(c3 (H) − cH )

[2.52]

Figure 2.9. Graphs of k2∗ (line b) and k20 (line a) as the functions of H ∈ (1/2, 1)

We present the graphs of k20 (line a) and k2∗ (line b) as the functions of H ∈ (1/2, 1), constructed, as before, using the Mathematica software (Figure 2.9). From these graphs, it is clear that k20 = k2∗ , i.e.   (H + 12 ) (H + 12 )cH − 2Hc1 (H) c23 (H) − 2H = , 2(c3 (H) − cH ) (H − 12 )2

[2.53]

Distance Between fBm and Subclasses of Gaussian Martingales

89

has a unique solution H in the interval ( 12 , 1). Denote H0 as the solution to [2.53]. Then, k2∗ < k20 if 12 < H < H0 , and k2∗ > k20 if H0 < H < 1. The value of H0 computed numerically is H0 ≈ 0.94. In Theorem 2.7 we consider only the values of H for which k2∗ ≥ k20 , i.e. H ≥ H0 . Theorem 2.7.– Let a = a(s) be a function of the form a(s) = k1 + k2 sα , k1 , k2 ∈ R, s ∈ [0, T ]. If H ≥ H0 so that k2∗ ≥ k20 , then the minimal value of the function f (k1 , k2 ) := max f (t, k1 , k2 ) t∈[0,T ]

is reached at the point k1 = k1∗ , k2 = k2∗ taken from equalities [2.47], and this minimal value equals   2B(1 − α, α) (α + 1)2 2α+1 2α+1 2 2 T . [2.54] −T cH B (1 − α, α) − + 2 α α (2α + 1) Proof.– Since a = a(s) is a function of the form a(s) = k1 + k2 sα , k1 , k2 ∈ R, we can rewrite  Mt =



t

(k1 + k2 · s ) dWs = k1 Wt + k2

t

α

0

0

sα dWs .

To continue,  E(BtH − k1 Wt − k2 =t

2H

t 0

sα dWs )2

− 2k1 · c1 (H) · tα+1 − k2 ·

cH 2H · t + k12 t H

t2H tα+1 + 2k1 k2 · 2H α+1  2  k2 cH tα+1 2H =t − k2 · + 1 + 2k1 · (k2 − c3 (H)) + k12 t 2H H α+1 + k22

=: t2H · a(k2 ) − 2k1

tα+1 · b(k2 ) + tk12 = f (t, k1 , k2 ). α+1

Let us differentiate the function f (t, k1 , k2 ) in t:   ∂f = t2α · k22 − 2k2 · cH + 2H + 2tα k1 (k2 − c3 (H)) + k12 . ∂t

[2.55]

90

Fractional Brownian Motion

Note that in [2.55], the constant term k12 ≥ 0. Additionally, it follows from Lemma A1.1 that k22 − 2k2 · cH + 2H > 0. Consider ∂f ∂t , i.e. the right-hand side of [2.55], as a quadratic polynomial in t . Its discriminant D is equal to α

    D = k12 k22 − 2k2 c3 (H) + c23 (H) − k12 2H − 2k2 · cH + k22 4   = k12 −2k2 c3 (H) + c23 (H) − 2H + 2k2 · cH   = k12 c23 (H) − 2H − 2k2 (c3 (H) − cH ) = 2k12 (k20 − k2 )(c3 (H) − cH )

[2.56]

with k20 defined in [2.52]. According to Lemma A1.8 we have k20 < 0; and according to [A1.24] c3 (H)− cH > 0. Since the discriminant D in [2.56] decreases in k2 , we have: i) D < 0 if k2 > k20 ; ii) D = 0 if k2 = k20 ; iii) D > 0 if k2 < k20 . Let us study f (t, k1 , k2 ) as a function of t, depending on k1 , k2 as the parameters. 1) Let k1 ≤ 0. In this case, f (t, k1 , k2 ) strictly increases in t, since ∂f ∂t > 0 for all t > 0 and k2 ∈ R. Indeed, k12 ≥ 0, and we have already proved that in [2.55] the coefficient at t2α is positive for all k2 ∈ R. Consider the coefficient 2k1 (k2 − c3 (H)) at tα : a) If k2 < c3 (H), then it is obvious that

∂f ∂t

> 0, t > 0.

b) If k2 ≥ c3 (H), then it is necessary to analyze the value in the brackets in the right-hand side of [2.56]. Using [A1.24] and Lemma A1.6, we have: c23 (H) − 2H − 2k2 (c3 (H) − cH ) ≤ c23 (H) − 2H − 2c3 (H) (c3 (H) − cH ) 2

= 2cH c3 (H) − c23 (H) − 2H = c2H − 2H − (c3 (H) − cH ) < 0. That is, the discriminant D < 0 and leading coefficient k22 −2k2 ·cH +2H > 0 in [2.55]. Hence, ∂f ∂t > 0, t > 0 and maxt∈[0,T ] f (t, k1 , k2 ) = f (T, k1 , k2 ). 2) Let k1 > 0. Then, consider the following cases:

Distance Between fBm and Subclasses of Gaussian Martingales

91

a) Let D < 0, i.e. k2 > k20 . We apply Lemma A1.7. It follows from this lemma that the values of the quadratic polynomial [2.55] are positive, i.e. ∂f ∂t > 0, and f (t, k1 , k2 ) strictly increases; therefore, max f (t, k1 , k2 ) = f (T, k1 , k2 ).

t∈[0,T ]

b) Let D = 0, i.e. k2 = k20 . In this case, we have k2 < c3 (H), because k2 = k20 , and k20 < c3 (H) (it is obvious that c3 (H) > 0 and k20 < 0 according to Lemma A1.8). From here it follows that the coefficient at tα in [2.55] is negative: 2k1 (k2 − c3 (H)) < 0. Thus, we have the following situation: D = 0, coefficient at k22 − 2k2 · cH + 2H > 0, coefficient at tα is negative, and a constant term k12 ≥ 0. Therefore, quadratic function [2.55] has the unique root −2c t= = b



k1 c3 (H) − k2

 α1

 =

−k1 b(k2 )

 α1 ,

and this root is positive. It means that ∂f ∂t ≥ 0, whence f (t, k1 , k2 ) strictly increases in t > 0; as before, maxt∈[0,T ] f (t, k1 , k2 ) = f (T, k1 , k2 ). c) Let D > 0. In this case, we have k2 < c3 (H). Hence, it follows that in [2.55] the coefficient at tα 2k1 (k2 − c3 (H)) < 0. Thus, we have the following situation: D > 0, coefficient at t2α is positive, k22 −2k2 ·cH +2H > 0, coefficient at tα is negative, and a constant term k12 ≥ 0. Hence, the roots of the quadratic function [2.55] have the form t1 = t1 (k1 , k2 ) α1 

k1 (c3 (H) − k2 ) − |k1 | c23 (H) − 2H − 2k2 (c3 (H) − cH ) , = 2H − 2k2 cH + k22 [2.57] t2 = t2 (k1 , k2 ) α1 

k1 (c3 (H) − k2 ) + |k1 | c23 (H) − 2H − 2k2 (c3 (H) − cH ) . = 2H − 2k2 cH + k22 The roots t1 and t2 are positive. Denote xi = tα i , i = 1, 2. Then: 

1 1 ∂f α α > 0 if 0 < t < x , i.e. f (t, k , k ) increases in the interval 0, x 1 2 1 1 ; ∂t 

1 1 1 1 ∂f α α α α ∂t < 0 if x1 < t < x2 , i.e. f (t, k1 , k2 ) decreases in the interval x1 , x2 ; ∂f ∂t

1  1 > 0 if t > x2α , i.e. f (t, k1 , k2 ) increases in the interval x2α , +∞ .

92

Fractional Brownian Motion

Thus, we have the following cases: i) Let k1 ≤ 0 and k2 ∈ R. In this case, we have min

max f (t, k1 , k2 ) =

k1 ≤0,k2 ∈R t∈[0,T ]

min

k1 ≤0,k2 ∈R

f (T, k1 , k2 ) = T 2H

 1−

c2H 2H

 .

Indeed, the function (k1 , k2 ) → f (T, k1 , k2 ) has no local extrema on the set {(k1 , k2 ) : k1 ≤ 0} (this function has one local extremum (k1∗ , k2∗ ), but k1∗ > 0). Therefore, in our case, the minimum is reached on the line k1 = 0 and this minimum is calculated in Theorem 2.2. More precisely, this minimum is reached at point k1 = 0, k2 = cH and equals   c2 [2.58] T 2H 1 − H . 2H ii) Let k1 > 0 and k2 ≥ k20 . ∗ 0 a) Additionally, let H > H 0 . In this case we have k2 > k2 . Hence,  0 ∈ (k1 , k2 )| k1 > 0, k2 ≥ k2 . Since the function f (T, k1 , k2 ) reaches a minimum at the point (k1∗ , k2∗ ), then

(k1∗ , k2∗ )

min

k1 >0,k2 >k20

=T

f (T, k1 , k2 ) =

2α+1

−

T 2α+1 c2H

min

k1 ∈R,k2 ∈R



f (T, k1 , k2 ) = f (T, k1∗ , k2∗ )

2B(1 − α, α) (α + 1)2 B (1 − α, α) − + 2 α α (2α + 1) 2

 .

b) Let H < H0 . In this case we have k2∗ < k20 . Then, according to cases 2a) and 2b) (see page 91), function f (t, k1 , k2 ) strictly increases in t. Therefore, min

max f (t, k1 , k2 ) =

k1 ≥0,k2 ≥k20 t∈[0,T ]

 = f (T )

min

k1 ≥0,k2 ≥k20

f (T, k1 , k2 )

  k20 Tα k1 = c1 (H) − α+1 0 k2 = k2 2  k20 = − c1 (H) − · T 2α+1 + T 2α+1 α+1 − 2k20

2 T 2α+1 cH · T 2α+1 + k20 · . 2α + 1 2α + 1

[2.59]

Distance Between fBm and Subclasses of Gaussian Martingales

93

Indeed, the function (k1 , k2 ) → f (T, k1 , k2 ) does  not have local extremes (minimums) on the set (k1 , k2 ) : k1 ≥ 0, k2 ≥ k20 (this function has one local extremum (k1∗ , k2∗ ), but we consider k2∗ < k20 ). Hence, in our case, the minimum is reached on the line k2 = k20 . That is, min

k1 ≥0,k2 ≥k20

f (T, k1 , k2 ) =

= min T 2H



k1 ≥0

− 2k1

min

k1 ≥0,k2 =k20

(k20 )2 cH − k20 · +1 2H H

f (T, k1 , k2 ) = min f (T, k1 , k20 ) k1 ≥0



  T α+1  c1 (H)(α + 1) − k20 + T k12 , α+1

[2.60]

and this minimum of the quadratic polynomial in k1 in [2.60] is reached at the point k1 =

2·

T α+1 α+1

  · c1 (H)(α + 1) − k20 2T

=

 c1 (H) −

k20 α+1

 T α > 0.

iii) Let k1 > 0, k2 < k20 . Since the function f (t, k1 , k2 ) is increasing on [0, t1 ], [t2 , +∞) and decreasing on [t1 , t2 ], we have: a) if t1 > T (where t1 is the root of the form [2.57]), then f (t, k1 , k2 ) is increasing on [0, T ], and maxt∈[0,T ] f (t, k1 , k2 ) = f (T, k1 , k2 ); b) if t1 < T , then max f (t, k1 , k2 ) = max{f (T, k1 , k2 ), f (t1 , k1 , k2 )};

t∈[0,T ]

c) if t1 = T , then maxt∈[0,T ] f (t, k1 , k2 ) = f (T, k1 , k2 ) = f (t1 , k1 , k2 ). Hence, min

max f (t, k1 , k2 )

k1 ≥0,k2 0, k2 < k20 we have to compare f (T, k1 , k2 ) and f (t1 (k1 , k2 ), k1 , k2 ). Thus, let us first establish some auxiliary results concerning the points ti (k1∗ , k2∗ ), i = 1, 2. Lemma 2.8.– If H < H0 , then the following inequality holds: t1 (k1∗ , k2∗ ) < T.

Distance Between fBm and Subclasses of Gaussian Martingales

95

Proof.– The inequality H < H0 is equivalent to the inequality k2∗ < k20 . Note that k1∗ > 0. Furthermore, for k1 > 0, we have  t1 (k1 , k2 ) =

 α1  k1 c3 (H) − k2 − c23 (H) − 2H − 2k2 (c3 (H) − cH ) . 2H − 2k2 cH + k22

Multiplying and dividing by expression, conjugate to the numerator, i.e. by c3 (H) − k2 +

 c23 (H) − 2H − 2k2 (c3 (H) − cH ),

we obtain t−α 1 (k1 ,

k2 ) =

c3 (H) − k2 +

c23 (H) − 2H − 2k2 (c3 (H) − cH ) . k1

c3 (H) − k2 −

c23 (H) − 2H − 2k2 (c3 (H) − cH ) , k1

Similarly, t−α 2 (k1 , k2 ) = and −α t−α 1 (k1 , k2 ) > t2 (k1 , k2 ).

[2.63]

We can add t−α 1 (k1 , k2 ) to the left- and the right-hand sides of [2.63] in order to obtain 2 1 1 2(c3 (H) − k2 ) . > α + α = tα (k , k ) t (k , k ) t (k , k ) k1 1 2 1 2 1 2 1 1 2 Substitute k1 = k1∗ > 0, k2 = k2∗ in the above relations to derive the following: 2 ∗ tα 1 (k1 ,

k2∗ )

>

2(c3 (H) − k2∗ ) k1∗

 2 3 (H) 2 c3 (H) − (α+1) cH −(2α+1)c α2

 = H) T α (α+1)(cα3 (H)−c 2 =

2 (α + 1)2 c3 (H) − (α + 1)2 cH 2 2 = α (α + 1) > α . Tα (α + 1)(c3 (H) − cH ) T T

[2.64]

96

Fractional Brownian Motion ∗ ∗ 0 ∗ α ∗ ∗ It follows that tα 1 (k1 , k2 ) < T , i.e. t1 (k1 , k2 ) < T for k2 < k2 .



As a result, our goal is to find min

k1 ≥0,k2 f (T, k1∗ , k2∗ ), which would be undesirable to some extent, but would bring some certainty. However, this inequality is wrong, which follows from the next enhancement of the above result. Lemma 2.9.– For any H < H0 , the following inequality holds: t2 (k1∗ , k2∗ ) < T. Proof.– Since f (T, k1 , k2 ) = T 2α+1 a(k2 ) − 2k1

T α+1 b(k2 ) + T k12 , α+1

and (k1∗ , k2∗ ) is the point of minimum of f (T, k1 , k2 ), we state that k1∗ =

Tα b(k ∗ ), α+1 2

see equation [2.50]. Note that f (T, k1 , k2 ) > 0 for all (k1 , k2 ). The function f (T, k1 , k2 ) is the quadratic polynomial in k1 . Since f (T, k1 , k2 ) > 0 for all k2 ∈ R, the discriminant of the quadratic polynomial is negative, i.e. T 2(α+1)



b2 (k2 ) − a(k2 ) (α + 1)2

 < 0.

Let us calculate the derivative of f (t, k1 , k2 ) with respect to t: ∂f (t, k1 , k2 ) = (2α + 1)t2α a(k2 ) − 2k1 tα b(k2 ) + k12 . ∂t

[2.65]

Distance Between fBm and Subclasses of Gaussian Martingales

97

1 ,k2 ) Since t1 (k1 , k2 ) and t2 (k1 , k2 ) are the roots of the equation ∂f (t,k = 0, ∂t α ∗ ∗ we conclude that t1,2 (k1 , k2 ) =: x1,2 are the roots of the quadratic function

x2 (2α + 1)a(k2∗ ) − 2k1∗ xb(k2∗ ) + (k1∗ )2 .

[2.66]

Substitute T α instead of x and let us establish that the quadratic function is positive at T α . Indeed, T 2α (2α + 1)a(k2∗ ) − 2k1∗ T α b(k2∗ ) + (k1∗ )

2

T α b(k2∗ ) α ∗ T b(k2 ) + α+1   2α + 1 2 ∗ 2α ∗ (2α + 1)a(k2 ) − =T b (k2 ) (α + 1)2   b2 (k2∗ ) 2α ∗ > 0, = T (2α + 1) a(k2 ) − (α + 1)2 = T 2α (2α + 1)a(k2∗ ) − 2



T α b(k2∗ ) α+1

2

as the expression in the brackets is positive according to [2.65]. Since the quadratic polynomial with positive highest coefficient [2.66] has a positive value at T α , point T α is not between its roots, i.e. ∗ ∗ α ∗ ∗ Tα ∈ / [tα 1 (k1 , k2 ), t2 (k1 , k2 )].

However, it follows from Lemma 2.8 that T > t1 (k1∗ , k2∗ ). Therefore, T > t2 (k1∗ , k2∗ ).  Lemmas 2.8 and 2.9 demonstrate that it is difficult to compare the values at the points t1 (k1 , k2 ) and T because the point t2 (k1 , k2 ) of local minimum is situated between these two points. It has been shown that we should consider min

k1 ≥0,k2 0, 1 1 ∂k1 ∂ (f (t1 (k1 , k2 ), k1 , k2 ) − f (T, k1 , k2 )) |k1 =k¯1 = 0. ∂k1 If k1 < k1 then f (t1 (k1 , k2 ), k1 , k2 ) < f (T, k1 , k2 ) and vice versa. Proof.– Introduce the notation  c(k2 ) :=

b(k2 ) −

α1

b2 (k2 ) − 2Ha(k2 ) , 2Ha(k2 )

and ϕ(k2 ) = (c(k2 ))2H a(k2 ) − 2

(c(k2 ))α+1 b(k2 ) + c(k2 ). α+1

k2

Here, as before, a(k2 ) = 2H2 − k2 · cHH + 1 > 0, b(k2 ) = c3 (H) − k2 . Consider the value k2 < k20 < 0. Note that the values t1 (k1 , k2 ) and f (t1 (k1 , k2 ), k1 , k2 ) can be rewritten in the form 1

2H

t1 (k1 , k2 ) = k1α · c(k2 ) , f (t1 (k1 , k2 ), k1 , k2 ) = k1α ϕ(k2 ). First, let us find the value k1 = k1 (k2 ), such that for any k2 < k20 the following equalities will be performed: f (t1 (k1 , k2 ), k1 , k2 ) = f (T, k1 , k2 ) and fk 1 (t1 (k1 , k2 ), k1 , k2 ) = fk 1 (T, k1 , k2 ). For this purpose let us equate f (t1 (k1 , k2 ), k1 , k2 ) to f (T, k1 , k2 ), in order to obtain 2H

k1 α ϕ(k2 ) = T 2H a(k2 ) − 2k1

T α+1 b(k2 ) + k12 T. α+1

By multiplying the left- and right-hand sides of [2.67] by

[2.67] 2H α k1 ,

−1 2H 2H 2H 2H T 2H a(k2 ) 4H · T α+1 k1 α · b(k2 ) + k1 T. − ϕ(k2 ) = α α k1 α(α + 1) α

we obtain [2.68]

Distance Between fBm and Subclasses of Gaussian Martingales

99

Now, differentiate the left- and right-hand sides of [2.67] with respect to k1 : −1 2H 2H 2T α+1 ϕ(k2 ) = − k1 α b(k2 ) + 2k1 T. α α+1

[2.69]

Since in [2.68] and [2.69] left-hand sides are equal, we can equate their right-hand sides to obtain 2H T 2H a(k2 ) 4H · T α+1 2H 2T α+1 − · b(k2 ) + k1 T + b(k2 ) − 2k1 T = 0. α k1 α(α + 1) α α+1 Let us collect the coefficients at b(k2 ) and at k1 : 2H T 2H a(k2 ) 2 1 − T α+1 b(k2 ) + k1 T = 0. · α k1 α α

[2.70]

Multiply [2.70] by αk1 : 2H · T 2H a(k2 ) − 2k1 T α+1 b(k2 ) + k12 T = 0.

[2.71]

We obtain a quadratic equation with respect to k1 . Its discriminant equals T 2α+2 (b2 (k2 ) − 2Ha(k2 )), and it is positive if and only if k2 < k20 . Therefore, with such values k2 , equation [2.71] has two roots of the form (1,2)

k1



= T α b(k2 ) ± b2 (k2 ) − 2Ha(k2 ) . (1)

Choose the root that matches +, and denote it by k1 :



(1) k1 = T α b(k2 ) + b2 (k2 ) − 2Ha(k2 ) =

T α · 2Ha(k2 ) Tα

. = (c(k2 ))α b(k2 ) − b2 (k2 ) − 2Ha(k2 ) (2)

Compare another root, k1 , to c(k2 ): (2)

k1 = 2Ha(k2 )(c(k2 ))α T α .

[2.72]

It follows from [2.71] and [2.72] that 2HT 2H a(k2 ) − 4Ha(k2 )(c(k2 ))α b(k2 )T 2H + 4H 2 a2 (k2 )(c(k2 ))2α T 2H = 0,

100

Fractional Brownian Motion

which is equivalent to 1 − 2b(k2 )(c(k2 ))α + 2Ha(k2 )(c(k2 ))2α = 0.

[2.73] 1

(1)

Let us return to the root k1 . Recall that t1 (k1 , k2 ) = k1α c(k2 ). Therefore, (1) (1) (1) 1 at point k1 , we have t1 (k1 , k2 ) = (k1 ) α c(k2 ) = T . So the following equality is indeed true: (1)

(1)

(1)

f (t1 (k1 , k2 ), k1 , k2 ) = f (T, k1 , k2 ). Let us verify that (1)

(1)

(1)

∂f (t1 (k1 , k2 ), k1 , k2 ) ∂f (T, k1 , k2 ) = . ∂k1 ∂k1 Indeed, (1)

(1)

∂f (t1 (k1 , k2 ), k1 , k2 ) 2H (1) 2H −1 2H T α+1 = ϕ(k2 ), (k1 ) α ϕ(k2 ) = ∂k1 α α (c(k2 ))α+1 while (1)

∂f (T, k1 , k2 ) T α+1 T α+1 = −2 . b(k2 ) + 2 ∂k1 α+1 (c(k2 ))α Thus, it remains to verify that H ϕ(k2 ) b(k2 ) 1 =− , + α (c(k2 ))α+1 α + 1 (c(k2 ))α which is equivalent to H(α + 1)ϕ(k2 ) = α(α + 1)c(k2 ) − αb(k2 )(c(k2 ))α+1 . Let us substitute the value of ϕ(k2 ) into [2.74]: H(α + 1)ϕ(k2 ) = H(α + 1)c(k2 )2H a(k2 ) − 2H(c(k2 ))α+1 b(k2 ) + H(α + 1)c(k2 ).

[2.74]

Distance Between fBm and Subclasses of Gaussian Martingales

101

Thus, it is necessary to verify the equality H(α + 1)c(k2 )α a(k2 ) − 2H(c(k2 ))α b(k2 ) + H(α + 1) = α(α + 1) − αb(k2 )(c(k2 ))α , or H(α+1)c(k2 )α a(k2 )−(α+1)(c(k2 ))α b(k2 )+ 12 (α+1) = 0, which is equivalent to Hc(k2 )2α a(k2 ) − (c(k2 ))α b(k2 ) +

1 = 0. 2

[2.75]

However, if we consider [2.75] as a quadratic equation 2Hx2 a(k2 ) − 2xb(k2 ) + 1 = 0, then we see that its roots equal x1,2 =

(c(k2 ))α is its root. Thus, we need to verify following equalities hold: (1)

(1)

√

b2 (k2 )−2Ha(k2 ) . It 2Ha(k2 ) (1) that at point k1

b(k2 )±

means that both of the

(1)

f (t1 (k1 , k2 ), k1 , k2 ) = f (T, k1 , k2 ), and (1)

(1)

(1)

∂f (t1 (k1 , k2 ), k1 , k2 ) ∂f (T, k1 , k2 ) = . ∂k1 ∂k1 Now, let us establish that a unique point 0 <  k1 < k1

(1)

exists, at which

k1 , k2 ),  k1 , k2 ) = f (T,  k1 , k2 ), f (t1 ( and that at this point we have inequality k1 , k2 ),  k 1 , k2 ) ∂f (t1 ( ∂f (T,  k1 , k2 ) > . ∂k1 ∂k1 To this end, consider the difference g(k1 , k2 ) := f (t1 (k1 , k2 ), k1 , k2 ) − f (T, k1 , k2 ) 2H

= k1α ϕ(k2 ) − T 2H a(k2 ) + 2k1

T α+1 b(k2 ) − k12 T. α+1

102

Fractional Brownian Motion

Let us calculate the second partial derivative of function g with respect to k1 :  1 ∂ 2 g(k1 , k2 ) 2H 2H = − 1 k1α ϕ(k2 ) − 2T. ∂k12 α α α

2 This derivative is negative for k1 < " k1 = H(H+T α1 )ϕ(k ) but positive for 2 2 ∂g(k1 ,k2 ) " k1 > k1 . It means that the first partial derivative ∂k1 is decreasing at the " " set k1 < k1 and is increasing at the set k1 > k1 . Now, let us establish that (1) " k1 < k . Indeed, let us write the inequality 1

T α α2α Tα , < H α (α + 1)α ϕα (k2 ) (c(k2 ))α which is the same as c(k2 )
0. Lemma 2.10.– There exists 0 < t0 < T such that for t ≤ t0 the inequality f (T, k1 , k2∗ ) ≥ f (t, k1 , k2∗ ) holds for any k1 > 0, while for t0 < t ≤ T , two numbers k11 and k12 exist such that f (T, k1 , k2∗ ) < f (t, k1 , k2∗ ) for k11 < k1 < k12 , and f (T, k1 , k2∗ ) > f (t, k1 , k2∗ ) for k1 ∈ / (k11 , k12 ). Proof.– Consider quadratic polynomials in k1 > 0 for any fixed 0 < t < T : 2k1 b(k ∗ )T α+1 + T 2H a(k2∗ ) = f (T, k1 , k2∗ ), α+1 2 2k1 k12 t − b(k ∗ )tα+1 + t2H a(k2∗ ) = f (t, k1 , k2∗ ). α+1 2 k12 T −

Here a(k2 ) =

k22 2H

− k2 ·

cH H

+ 1 > 0, b(k2 ) = c3 (H) − k2 , k2 < k20 < 0.

Consider the difference Δ = Δf (T, t, k1 , k2∗ ) := f (T, k1 , k2∗ ) − f (t, k1 , k2∗ ) = k12 (T − t) −

2k1 b(k ∗ )(T α+1 − tα+1 ) + a(k2∗ )(T 2H − t2H ). α+1 2

104

Fractional Brownian Motion

By dividing Δ by (T − t), we obtain a quadratic polynomial: Δ1 = k12 −

2k1 T α+1 − tα+1 T 2H − t2H b(k2∗ ) + a(k2∗ ) . α+1 T −t T −t

[2.78]

Its discriminant is equal to D=

b2 (k2∗ ) (α + 1)2



T α+1 − tα+1 T −t

2

− a(k2∗ )

T 2H − t2H . T −t

(This D is not related to D defined in the proof of Theorem 2.7.) Consider three possible cases: 1) D < 0. Then, f (T, k1 , k2∗ ) > f (t, k1 , k2∗ ) for all k1 > 0, in particular, f (T, k1∗ , k2∗ ) > f (t, k1∗ , k2∗ ). 2) D > 0. Then, two roots exist: k11,2 =

b(k2∗ ) T α+1 − tα+1 √ ± D; α+1 T −t

[2.79]

moreover, f (T, k1 , k2∗ ) < f (t, k1 , k2∗ ) for k11 < k1 < k12 and an opposite inequality holds for k1 < k11 , k1 > k12 . 3) D = 0. Then, f (T, k1 , k2∗ ) > f (t, k1 , k2∗ ) for all k1 except k1 =

b(k2∗ ) T α+1 − tα+1 . α+1 T −t

For this value of k1 , we have an equality. Let us prove that for H < H0 , a unique t0 exists such that for 0 < t < t0 , it holds that D < 0, and for t0 < t ≤ T, it holds that D > 0. To this end, transform the discriminant D to a form in which it is easier to examine its sign. Instead of D, first consider D1 =

b2 (k2∗ ) ϕ(T, t) − , ∗ 2 a(k2 )(α + 1) ψ(T, t)

where ϕ(T, t) = ratio

ϕ(T,t) ψ(T,t)

Therefore,

T 2H −t2H , T −t

ψ(T, t) =



T α+1 −tα+1 T −t

2 . It is easy to see that the

coincides with the function ζ(t) introduced by equality [A1.25]. ϕ(T,t) ψ(T,t)

= ζ(t). The behavior of ζ(t) is investigated in Lemma A1.2.

Distance Between fBm and Subclasses of Gaussian Martingales

If t = 0, then ζ(t) = 1, and since the inequality a(k2∗ ) > have D1 < 0.

b2 (k2∗ ) (α+1)2

105

holds, we

If t → T , then lim ζ(t) = lim t↑T

t↑T

(T 2H − t2H )(T − t) (T α+1 − tα+1 )2

= lim

−2Ht2H−1 T − T 2H + (2H + 1)t2H −2(α + 1)tα T α+1 + 2(α + 1)t2α+1

= lim

−2H(2H − 1)t2H−2 T + (2H + 1)2Ht2H−1 −2(α + 1)α tα−1 T α+1 + 2(α + 1)(2α + 1)t2α

t↑T

t↑T

=

−4H 2 + 2H + 4H 2 + 2H     −2(H + 12 ) H − 12 + 2 H + 12 2H

=

H+

1 2



2H 4H = 2 , (4H − 2H + 1) H + 12

and b2 (k2∗ ) − 2Ha(k2∗ ) > 0. It means that D1 > 0. In general, D1 is strictly monotonically increasing with respect to t, as by Lemma A1.2, ζ(t) is strictly monotonically decreasing with respect to t. Hence, a unique t0 exists such that for 0 < t < t0 D < 0, and for t0 < t ≤ T D > 0, whence the proof follows.  In particular, if 0 < t < t0 , then we have at point k1∗ that Δ = Δf (T, t, k1∗ , k2∗ ) := f (T, k1∗ , k2∗ ) − f (t, k1∗ , k2∗ ) 2k1∗ b(k ∗ )T α+1 + T 2α+1 a(k2∗ ) α+1 2 2k1∗ 2 > (k1∗ ) t − b(k ∗ )tα+1 + t2α+1 a(k2∗ ). α+1 2 2

= (k1∗ ) T −

Note that we do not know whether the point t = t1 (k1∗ , k2∗ ) satisfies inequality t ≤ t0 or vice versa, t > t0 . If we could establish one of the following facts, D3 (k1∗ , k2∗ ) < 0, i.e. 0 < t1 (k1∗ , k2∗ ) ≤ t0 , or vice versa, t1 (k1∗ , k2∗ ) ≥ t0 , but k1∗ ≤ k11 , then we would obtain inequality f (T, k1∗ , k2∗ ) ≥ f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ). Since testing these facts is just as difficult as comparing the values f (T, k1∗ , k2∗ ) and f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ), we perform the comparison numerically.

106

Fractional Brownian Motion

Consider the discriminant D, substituting t1 (k1∗ , k2∗ ) instead of t. We obtain the value 2  α+1 − (t1 (k1∗ , k2∗ ))α+1 T b2 (k2∗ ) ∗ ∗ D = D(k1 , k2 ) = [2.80] (α + 1)2 T − t1 (k1∗ , k2∗ ) − a(k2∗ )

T 2α+1 − (t1 (k1∗ , k2∗ ))2α+1 . T − t1 (k1∗ , k2∗ )

Again, due to the complexity of the expressions on the right-hand side of equality [2.80], we examine the sign of D(k1∗ , k2∗ ) numerically. The discriminant D(k1∗ , k2∗ ) is plotted in Figure 2.10 as a function of H, 12 < H ≤ H0 (line a). On Figure 2.10, T = 1. For other T > 0, the value shown on Figure 2.10 should be multiplied by T 2α . From Figure 2.10, we can see that H2 exists such that D(k1∗ , k2∗ ) > 0 if 12 < H < H2 , and D(k1∗ , k2∗ ) < 0 if H2 < H < H0 . The expressions t1 (k1∗ , k2∗ ) and D(k1∗ , k2∗ ) are well-defined for 12 < H ≤ H0 . Here H0 ≈ 0.94 is the solution to [2.53].

2 2 (b(k∗ ))2 (T α −tα 1 ) t1 (T −t1 )2

2 Figure 2.10. The graph of (a) D(k1∗ , k2∗ ) and (b) (α+1) 2 1 as the functions of H, H ∈ ( 2 , H0 ), for T = 1.

Now, find whether the expression k1∗ − k11 is less than zero, equal to zero or greater than zero. We apply formula [2.50] for k1∗ . The inequality k1∗ ≤ k11 is equivalent to each of the following: b(k2∗ ) α b(k2∗ )(T α+1 − (t1 (k1∗ , k2∗ ))α+1 ) √ − D, T ≤ α+1 (α + 1)(T − t1 (k1∗ , k2∗ )) √

D≤

) b(k2∗ )(T α t1 − tα+1 1 , (α + 1)(T − t1 )

Distance Between fBm and Subclasses of Gaussian Martingales

D−

2 2 (b(k2∗ ))2 (T α − tα 1 ) · t1 ≤ 0, 2 2 (α + 1) (T − t1 )

107

[2.81]

where, in order to reduce the notation, we have denoted t1 = t1 (k1∗ , k2∗ ). In 2 2 (b(k2∗ ))2 (T α −tα 1 ) ·t1 Figure 2.10, graph (b) (line b) plots (α+1) against H. From 2 (T −t1 )2 Figure 2.10, we can see that for H ∈ [H1 , H0 ] inequality [2.81] holds true. Hence, k1∗ > k11 if H ∈ ( 12 , H1 ), k1∗ = k11 if H = H1 , and k1∗ < k11 if H ∈ (H1 , H2 ]. The approximate values H1 and H2 computed numerically are H1 ≈ 0.911328, H2 ≈ 0.930755. Note that the inequality k1∗ < k12 (where k12 is the greater root defined in [2.79]) holds true for all H ∈ ( 12 , H0 ] because k12 − k1∗ = =

) √ b(k2∗ )(T α+1 − tα+1 b(k2∗ ) α 1 + D− T (α + 1)(T − t1 ) α+1 ) √ b(k2∗ )(T α t1 − tα+1 1 + D > 0, (α + 1)(T − t1 )

where t1 is a short form of t1 (k1∗ , k2∗ ). Here we used inequality [2.51] and Lemma 2.8. Now, compare f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) and f (T, k1∗ , k2∗ ). 1) If H ∈ ( 12 , H1 ), then k1∗ ∈ (k11 , k12 ), i.e. k1 lies between roots of quadratic polynomial [2.78]. In this case Δ1 < 0 and f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) > f (T, k1∗ , k2∗ ). 2) If H = H1 , then k1∗ = k11 , Δ1 = 0, and f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) = f (T, k1∗ , k2∗ ). 3) If H ∈ (H1 , H2 ), then k1∗ < k11 , k1∗ ∈ [k11 , k12 ], Δ1 > 0, and f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) < f (T, k1∗ , k2∗ ). 4) If H ∈ (H2 , H0 ], then the quadratic polynomial [2.78] has no real roots. In this case Δ1 > 0 and again f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) < f (T, k1∗ , k2∗ ). The results of a comparison between f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) and f (T, k1∗ , k2∗ ) are shown in Figure 2.11. Plot (b) (solid line) corresponds to f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ), while plot (a) (dashed line) corresponds to f (T, k1∗ , k2∗ ). From Figure 2.11 you can see that for H ∈ (H1 , H0 ) the following inequality holds true: f (T, k1∗ , k2∗ ) ≥ f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ).

108

Fractional Brownian Motion

Figure 2.11. Comparison results for f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) (solid line) and f (T, k1∗ , k2∗ ) (dashed line)

The maximum of the function f (t, k1∗ , k2∗ ) can be reached either at the point of local maximum t1 (k1∗ , k2∗ ) or at the edges of the interval [0, T ], i.e. max f (t, k1∗ , k2∗ ) = max (f (0, k1∗ , k2∗ ), f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ), f (T, k1∗ , k2∗ ))

t∈[0,T ]

= max (f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ), f (0, k1∗ , k2∗ )) . The maximum cannot be reached at t = 0 because f (0, k1∗ , k2∗ ) = 0 and f (t, k1∗ , k2∗ ) > 0 for all t > 0. If H ∈ [H1 , H0 ], then f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) ≤ f (T, k1∗ , k2∗ ) and maxt∈[0,T ] f (t, k1∗ , k2∗ ) = f (T, k1∗ , k2∗ ). Therefore, for H ∈ [H1 , H0 ] it holds that  t H (k1 + k2 sα )dWs )2 min max E(Bt − k1 ,k2 0≤t≤T

= =

0

min

max{f (T, k1 , k2 ), f (t1 , k1 , k2 )}

min

f (T, k1 , k2 )

k1 ≥0,k2 f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ), holds true for all H that belongs to some left-sided neighborhood (H0 − δ, H0 ] of H0 . T Proof.– For H = H0 , we have t1 (k1∗ , k2∗ ) = (α+1) 1/α < T . Furthermore, ∗ ∗ f (t, k1 , k2 ) for H = H0 is strictly increasing in t. This follows from the positivity of the derivative taken in t, which, in turn, was justified in the proof of Theorem 2.7, item 2b). Therefore, for H = H0

f (T, k1∗ , k2∗ ) > f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ).

[2.83]

Since all the above functions are continuous in H, inequality [2.83] keeps a sign on some interval (H0 − δ, H0 ].  2.2. The comparison of distances between fBm and subspaces of Gaussian martingales 2.2.1. Summary of the results concerning the values of the distances In section 2.1, we established the distances between fBm and the Gaussian t martingales of the form 0 a(s)dWs , where function a can be taken from various

110

Fractional Brownian Motion

subclasses of L2 ([0, T ]). Now, for the reader’s convenience, we summarize the results and present several comparisons. Recall that, as always,  α = H − 1/2, cH =

2HΓ( 32 − H) Γ(H + 12 )Γ(2 − 2H)

1/2 ,

α B(1 − α, α)cH α+1 πα = cH . (α + 1) sin(πα)

c1 (H) =

Hence, the following classes were considered: 1) K1 = {a(s) = a ∈ (0, +∞)}; 2) K2 = {a(s) = k · sα , k > 0}; 3) K3 = {a(s) = k · sγ , k > 0, γ ≥ 0}; 4) K4 = {a ∈ C[0, T ], a(s)s−α is non-decreasing}; 5) K5 = {a(s) = ks−α , k > 0};  6 = {a(s) = a0 sα + a1 sα+1 , a0 , a1 ∈ R}; 6) K # $

  6 : fa (T ) = 2Hfa (T ) , where fa (t) = E BtH − t a(s) 7) K6 = a ∈ K 0 2

dWs ) ; 8) K7 = {a(s) = k1 + k2 sα , k1 , k2 ∈ R}. For technical simplicity, we denote mi = ρ2 (B H , M(Ki )). For these classes we have established the following facts: 1) It holds that m1 := min max E(BtH − Mt )2 = T 2H (1 − c21 (H)), a∈K1 0≤t≤T

α B(1−α, α)cH , and the minimum is reached at the function where c1 (H) = α+1 α amin = c1 (H) · T . It was proved in Theorem 2.1.

2) It holds that m2 := min max

a∈K2 0≤t≤T

E(BtH

2

− Mt ) = T

2H



c2 1− H 2H

 ,

and the minimum is reached at the function amin (s) = cH sα , i.e. kmin = cH . It was proved in Theorem 2.2.

Distance Between fBm and Subclasses of Gaussian Martingales

111

3) It holds that m3 := min max E(BtH − Mt )2 = T 2H (1 − c21 (H)), a∈K3 0≤t≤T

and the minimum is reached at the function amin = c1 (H) · T α . It was proved in Theorem 2.3. 4) It holds that m4 := min max E(BtH − Mt )2 = T 2H



a∈K4 0≤t≤T

1−

c2H 2H

 ,

and the minimum is reached at the function amin (s) = cH sα , i.e. kmin = cH . It was proved in Theorem 2.4. 5) It holds that m5 := min max E(BtH − Mt )2 = ϕ1 (k ∗ ), a∈K5 0≤t≤T

and the minimum is reached at the function a(s) = k ∗ s−α , where k ∗ is the smallest point in terms of its abscissa from two points where the following 2H equality holds: ϕ1 (k) = k 2H−1 = ϕ2 (k). Here d(H) = (p1 (H)) and ϕ2 (k) = T 2H −

2H − 2H−1

4kH cH T

2H − 1 1−H



 2H p1 (H) − 1 , cH

2−2H

+ k 2 T2−2H . It was proved in Theorem 2.5.

6) It holds that m  6 := min max E(BtH − Mt )2 6 0≤t≤T a∈K  c2H α2 (2α + 3)c2H  ≤ T 2α+1 1 − − 2α + 1 (α + 1)2 = min max E(BtH − Mt )2 =: m6 . a∈K6 0≤t≤T

It was proved in Theorem 2.6. 7) i) Let H ∈ (H1 , 1), where H1 ≈ 0.911328. Then, m7 := min max E(BtH − Mt )2 = T 2α+1 a∈K7 0≤t≤T

(α + 1)2  2B(1 − α, α) + 2 , − T 2α+1 c2H B 2 (1 − α, α) − α α (2α + 1)

112

Fractional Brownian Motion

and the minimum is reached at (k1∗ , k2∗ ) (i.e. if k1 = k1∗ , k2 = k2∗ ), where (α + 1)(c1 (H)(α + 1) − cH )T α , α2 (α + 1) (cH (α + 1) − c1 (H)(2α + 1)) k2∗ = . α2 k1∗ =

It was proved in Theorem 2.7 (for H ∈ [H0 , 1)) and in section 2.1.6.2 (for H ∈ [H1 , H0 ]). ii) Let H ∈ (0.5, H1 ). Then, m7 ≤ max{f (T, k1∗ , k2∗ ), f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ )} =: m7 , where  t1 (k1 , k2 ) =

 α1  k1 c3 (H) − k2 − c23 (H) − 2H − 2k2 (c3 (H) − cH ) , 2H − 2k2 cH + k22

and c3 (H) = c1 (H)(α + 1). It was established in section 2.1.6.2. 2.2.2. The comparison of distances Let us compare some of the distances mentioned in section 2.2.1. First, it was established in Theorem 2.3 that from all power functions from classes K1 , K2 and K3 , the best approximation to fBm is provided by the constant function from K1 , i.e. by amin = c1 (H) · T α . Second, compare the distances in the classes  6 and K7 . Obviously, K1 ⊂ K7 and K2 ⊂ K7 . Therefore, m1 ≥ m7 K1 , K 5 , K6 , K and m2 ≥ m7 . Lemma 2.12.– For any H ≥ H1 the strict inequality m1 > m7 holds. Proof.– There are at least two possibilities to establish this statement. As the first possibility, we have m1 = f (T, c1 (H)T α , 0), and m7 = f (T, k1∗ , k2∗ ), where (k1∗ , k2∗ ) is the unique point of minimum of f (T, k1 , k2 ), and what is more, k2∗ = 0. Therefore, m1 > m7 . Another possibility is that the difference m1 − m7 can be bounded as follows:

2B(1 − α, α) m1 − m7 = T 2α B 2 (1 − α, α) − α

2  2 (α + 1) α + 2 − · B(1 − α, α) α (2α + 1) α+1

Distance Between fBm and Subclasses of Gaussian Martingales

113

2α + 1 2B(1 − α, α) (α + 1)2  B 2 (1 − α, α) − + 2 2 (α + 1) α α (2α + 1) [2.84]

B(1 − α, α) α + 1 2  = T 2α (2α + 1) > 0.  − α+1 α(2α + 1) = T 2α

Now, compare graphically the minimal values, obtained for classes K5 , K6 and K7 . In Figure 2.12, plot (m7 ) (solid line) corresponds to the values of the function max(f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ), f (T, k1∗ , k2∗ )). In turn, plots (m5 ) and (m6 ) correspond to the values of the functions m5 and m6 respectively, considered as the functions of H. From the picture you can see the following: for H ∈ (0.5; 0.728148) m6 is better, while for H ∈ (0.728148; 1) m5 is better.

Figure 2.12. Comparison results for the distances m5 , m6 and m7

2.2.3. The comparison of upper and lower bounds for the constant cH In the previous sections, various upper estimates for the coefficient cH were obtained, namely: cH
H0 ,  cH


(5) 8H 2 (1 − H) =: CH . (2)

[2.89] (1)

Obviously, bound CH is better than CH . Let us graphically compare the other bound using the Mathematica software (Figure 2.13). (2)

(3)

We can see that CH is the better upper bound for cH than CH , on the whole interval (0.5, 1) (Figure 2.13). This fact is confirmed by the graph of the (3) (2) difference CH − CH in Figure 2.14. (2)

Finally, in the interval H ∈ (0.5, 0.605702) we see that CH is the better (4) upper bound for cH than CH . This is numerically established using the (4) Mathematica software. Conversely, in the interval H ∈ (0.605702, 1), CH is (2) the better upper bound for cH than CH . This fact is confirmed by the graph (2) (4) of the difference CH − CH in Figure 2.15. (2)

Hence, CH is the best upper bound for cH in the interval (4) H ∈ (0.5, 0.605702), and CH is the best upper bound for cH in the interval H ∈ (0.605702, 1). Of course, we have in mind the best upper bound from the bounds mentioned above.

Distance Between fBm and Subclasses of Gaussian Martingales

115

Figure 2.13. Comparison of the upper and the lower bounds for cH .

(3)

(2)

Figure 2.14. Graph of the value of CH − CH

Corollary 2.5.– Inequalities [2.86] and [2.87] imply the new upper bound for sinπαπα . We compared the right-hand sides of these inequalities and made

116

Fractional Brownian Motion (2)

sure that CH is the better bound on the whole interval H ∈ (0.5, 1). This is (3) (2) confirmed by the graph of the difference CH − CH in Figure 2.14. Therefore, sin πα α+1 . < 2 πα (α + 1) + α2 (2α + 1)(2α + 3)

[2.90]

(2)

(4)

Figure 2.15. Graph of the value of CH − CH

We constructed the plot of the difference

α+1 (α +

1)2

+

α2 (2α

+ 1)(2α + 3)

−

sin πα πα

of

the right- and the left-hand sides of [2.90], considered as a function of α ∈ 0, 12 (Figure 2.16). Let us compare graphically the bounds for sinπαπα , contained in inequalities [2.20] and [2.90]. Figure 2.17 presents sinπαπα (line a), the right-hand side of inequality [2.20] (line b) and the right-hand side of inequality [2.90] (line c). We can see that the bound obtained in inequality [2.90] is better. Figure 2.18 demonstrates the comparison of the plots that correspond to differences of the left-hand and right-hand sides of inequalities [2.20] and √  1 2α+1 [2.90] as the functions of α ∈ 0, 2 , i.e. α+1 − sinπαπα (line a) and α+1 √ − sinπαπα (line b), respectively. 2 2 (α+1) +α (2α+1)(2α+3)

Distance Between fBm and Subclasses of Gaussian Martingales

Figure 2.16. Plot of the difference √

α+1 (α+1)2 +α2 (2α+1)(2α+3)

− sinπαπα as a function of α ∈ (0, 1/2)

Figure 2.17. Comparison of

sin πα πα

with its lower and upper bound

117

118

Fractional Brownian Motion

Figure 2.18. The values of α+1 and √ 2 2

√

2α+1 α+1

(α+1) +α (2α+1)(2α+3)

−

− sinπαπα (line sin πα (line b) πα

a)

as the functions of α ∈ (0, 1/2)

2.3. Distance between fBm and class of “similar” functions Consider an fBm with the Hurst index H ∈ ( 12 , 1) and the class K8 of functions from L2 ([0, 1]) having the following form: a(s) = at0 (s) = z(t0 , s)10 t0 , we have

z(t0 , s)dWs = BtH0 ,

whence gt0 (t) = E(BtH − Mt )2 = E(BtH − BtH0 )2 = (t − t0 )2H .



Thus, supt0 ≤t≤1 gt0 (t) = (1 − t0 )2H . Let Dα =

Cα2



p

sup

s

0≤p≤1

2α



0

2

1/s

x (x − 1) α

α−1

dx

ds.

[2.95]

p/s

Lemma 2.15.– The following equality holds: Dα m8 = min f (at ) =

2H . t∈[0,1] 1/2H 1 + Dα Proof.– Consider two functions: u(t) = Dα t2H , and v(t) = (1−t)2H , t ∈ [0, 1]. Then, for any t ∈ [0, 1], we have f (at ) = max{u(t), v(t)}. The function u increases while the function v decreases in the interval [0, 1]. Both functions u and v are continuous. Moreover, 0 = u(0) < v(0) = 1 and Dα = u(1) > v(1) = 0. Thus, a unique point t∗ ∈ (0, 1) exists such that u(t∗ ) = v(t∗ ) and u(t∗ ) = v(t∗ ) = min max{u(t), v(t)}. t∈[0,1]

Now, we evaluate the point t∗ . It immediately follows from the equality Dα t∗ 2H = (1 − t∗ )2H that t∗ =

1 1/2H

1 + Dα

.

Distance Between fBm and Subclasses of Gaussian Martingales

121

Therefore, Dα m8 := min f (at ) = u(t∗ ) =

2H . t∈[0,1] 1/2H 1 + Dα



2.3.1. Lower bounds for the distance A lower bound for the distance m8 was obtained in section 1.3. Indeed, according to Lemma 1.5, for arbitrary function a ∈ L2 ([0, 1]) and all 0 ≤ t1 < t2 ≤ 1, we have 

t

sup t∈[0,1]

0

(z(t, s) − a(s))2 ds ≥

1 4



t1 0

(z(t2 , s) − z(t1 , s))2 ds.

[2.96]

It means that 1 m8 ≥ sup 4 0≤t≤1



t 0

(z(1, s) − z(t, s))2 ds.

Substitute m8 =

1 sup 4 0≤t≤1



t 0

(z(1, s) − z(t, s))2 ds.

Obviously, m8 is a lower bound for m8 . Let us study in more detail the behavior of the function 

t

g(t) = 0

(z(1, s) − z(t, s))2 ds.

[2.97]

Lemma 2.16.– 1) The function g defined by [2.97] has a unique point of maximum in the interval [0, 1]. 2) Lower bound m8 satisfies the equality m8 =

Dα . 4

122

Fractional Brownian Motion

Proof.– 1) It is obvious that g(0) = g(1) = 0. Thus, the points of maximum of the function g belong to the open interval (0, 1). The derivative of g is given by g  (t) = z 2 (1, t) − 2



t 0

(z(1, s) − z(t, s))zt (t, s)ds.

Consider the integral It =

t 0

(z(1, s) − z(t, s))zt (t, s)ds, where

zt (t, s) = Cα s−α tα (t − s)α−1 . Obviously, integral It can be transformed as follows:  t  1 2 α α−1 −2α (t − s) s uα (u − s)α−1 duds I(t) = Cα t = Cα2 tα



0

t

uα t

=

t



1

Cα2 tα−1

0



1

u

s−2α (t − s)α−1 (u − s)α−1 dsdu



1

α

t

0

s−2α (1 − s)α−1

u t

−s

α−1 dsdu.

[2.98]

It is proved in the paper [NOR 99b] that  0

1

sμ−1 (1 − s)ν−1 (c − s)−μ−ν ds = c−ν (c − 1)−μ B(μ, ν)

for all c > 1, μ > 0, and ν > 0. We apply the latter equality to the right-hand side of [2.98] and obtain I(t) = Cα2 B(1 − 2α, α)t2α−1 = Cα2 B(1 − 2α, α)





1

u

t 1

t

−1

2α−1 du

(u − t)2α−1 du

t

= Cα2

B(1 − 2α, α) (1 − t)2α , 2α

[2.99]

where Cα = αcH . Using equality [2.99], we transform the equation g  (t) = 0 to the following form: z 2 (1, t) = Cα2

B(1 − 2α, α) (1 − t)2α , α

Distance Between fBm and Subclasses of Gaussian Martingales

123

where Cα2

B(1 − 2α, α) α(2α + 1)Γ(1 − α) Γ(1 − 2α)Γ(α) = = 2H. α Γ(α + 1)Γ(1 − 2α) Γ(1 − α)

Similarly, we obtain the equation z(1, t) =

√

2H(1 − t)α ,

which is the same as  1 √ Cα t−α uα (u − t)α−1 du = 2H(1 − t)α .

[2.100]

t

Since z(1, 1) = 0, t = 1 is a root of equation [2.100]. For t > 0, let us rewrite √  1/t equation [2.100] as follows: Cα tα 1 uα (u − 1)α−1 du = 2H(1 − t)α , or that is the same for non-zero t,  1/t α √ 1 Cα uα (u − 1)α−1 du = 2H [2.101] −1 . t 1 α √

 1/t Consider the function G(t) = Cα 1 uα (u−1)α−1 du− 2H 1t −1 in the √ 1 −2α interval (0, 1). It can be easily seen that G(t) ∼ Cα 2α t − 2Ht−α → +∞ as t → 0, while G(t) → 0 as t → 1. The derivative of the function G(t) is given by the formula G (t) =

1 t

α−1  √  t−2 α 2H − Cα t−α . −1

Obviously, G (t) equals zero at a unique point which is located in the open interval (0, 1), because the value in the bracket tends to −∞ as t → 0, and for t = 1, the value in the brackets equals √ √ α 2H − Cα = α( 2H − cH ) > 0, see Lemma A1.6. Therefore, the function G(t) decreases in some subinterval (0, t0 ) of (0, 1) and then increases to zero on (t0 , 1). Thus, the function has a unique point of minimum in the interval (0, 1) and its minimal value is negative. In turn, it implies that the function equals zero at a unique point between the origin and t0 . Thus, statement 1) of the lemma is proved.

124

Fractional Brownian Motion

2) Note that g(t) = g1 (t), t ∈ [0, 1], where the function gt0 is defined by formula [2.92]. According to Lemma 2.13 and the definition of Dα , we obtain sup g1 (t) =

0≤t≤1

Cα2



p

sup

0≤p≤1

y

2α



2

1/y

x (x − 1) α

0

α−1

dx

dy = Dα .



p/y

Corollary 2.6.– For all H ∈ ( 21 , 1) m8 =

Dα Dα = m8 . ≤ m8 ≤ 1/2H 4 (1 + Dα )2H

[2.102]

2.3.2. Evaluation First we evaluate Dα , then we find m8 and m8 in inequality [2.102]. Numerical procedure for the evaluation of Dα is reduced to the evaluation of the integral 

B

s

2α



2

v(s)

x (x − 1) α

A

α−1

dx

ds,

[2.103]

u(s)

where A, B ∈ [0, 1], A ≤ B, and u and v are some functions on [A, B] such that 1 ≤ u(s) ≤ v(s), s ∈ [A, B]. We show how such integrals can be evaluated numerically. The function Q is defined by  z Q(z) = xα (x − 1)α−1 dx, z ∈ [1, ∞). 1

B Then, we can rewrite the integral from [2.103] as A s2α (Q(v(s)) −Q(u(s)))2 ds. The problem in the procedure of the evaluation of Q is that the integrand has a singularity at the point 1. To remove the singularity, we integrate by parts and obtain  z  z z α (z − 1)α Q(z) = xα−1 (x − 1)α dx. xα (x − 1)α−1 dx = − α 1 1 Put q(x) = xα−1 (x − 1)α , z ∈ [1, ∞). i) Algorithm for the evaluation of Q(z), z ∈ [1, ∞). Let N = 100000 and Δ = 0.1. Consider a partition πN of the interval [1, 1 + N Δ] = [1, 10001] such that N N N πN = {1 = tN 0 < t1 = 1 + Δ < . . . < tk = 1 + kΔ < . . . < tN = 1 + N Δ}.

Distance Between fBm and Subclasses of Gaussian Martingales

125

The function H(z), z ∈ [1, ∞), is evaluated numerically as follows. Step 1. If k exists such that z = tN k , then we use the Simpson formula for the numerical integration (see, for example, [COU 88], V. 1, Chapter VII, section 1.3), and obtain N α % (tN N N N k (tk − 1)) (q(tN − i−1 ) + 4q((ti−1 + ti )/2) + q(ti ))/6. α i=1 k

Q(tN k )≈

N Step 2. If tN k < z < tk+1 , 1 ≤ k ≤ N − 1, then we use Step 1 and consider the linear interpolation

Q(z) ≈

(tk+1 − z) (z − tk ) Q(tN Q(tN k )+ k+1 ). Δ Δ

Step 3. If z > tN N , then Q(z) ∼ thus we can substitute Q(z) ≈

z 2α 2α

as z → ∞ by the L’Hôspital’s rule and

z 2α . 2α

ii) Algorithm for the evaluation of the value 

B

I(A, B, u, v) =

s2α (Q(v(s)) − Q(u(s)))2 ds.

A

Let N = 1000 and Δ = (B − A)/N . Consider the following partition π of the interval [A, B]: N N N π = {A = sN 0 < A + Δ = s1 < . . . A + kΔ = sk < . . . B = sN }. ∗ N = (sN Substitute sN i i−1 + si )/2.

Then, 

B

2α

2

s (Q(v(s)) − Q(u(s))) ds = A

≈

i=1 N %

∗ (Q(v(sN i ))

i=1

−

∗ 2 Q(u(sN i )))



sN i sN i−1

sN i

s2α (Q(v(s)) − Q(u(s)))2 ds

s2α ds

sN i−1

1 % ∗ N∗ 2 N 2α+1 2α+1 (Q(v(sN − (sN ). i )) − Q(u(si ))) ((si ) i−1 ) 2α + 1 i=1 N

=

N  %

126

Fractional Brownian Motion

From the definition of Dα we obtain that Dα = max g(t) = max I(0, t, ut , v), 0≤t≤1

[2.104]

0≤t≤1

where ut (y) = t/y, v = 1/y, y ∈ (0, 1]. iii) Algorithm for the evaluation of Dα . We find the maximum of the function g(t), t ∈ [0, 1], by using standard methods of optimization of unimodal functions (note that g is unimodal according to Lemma 2.16). We use formula [2.104] to evaluate g. Table 2.1 presents the results of evaluation for some H. H

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Dα 0.0260 0.0289 0.0451 0.0742 0.1171 0.1767 0.2537 0.3368 0.3586 t∗0 0.9650 0.9504 0.9156 0.8651 0.8069 0.7471 0.6914 0.6467 0.6317 m8 0.0065 0.0072 0.0113 0.0185 0.0293 0.0442 0.0634 0.0842 0.0897 m8 0.0250 0.0272 0.0402 0.0606 0.0849 0.1108 0.1355 0.1537 0.1499 Table 2.1. Results of evaluation for some H

The values of m8 and m8 are shown in Figure 2.19 for H, ranging from 0.51 to 0.99 with the step 0.01.


























































Figure 2.19. m8 (dashed line), m8 (solid line). They are evaluated for H, ranging from 0.51 to 0.99 with the step 0.01



Distance Between fBm and Subclasses of Gaussian Martingales

127

In the next sections we continue to consider the distance between an fBm and Gaussian martingales. In particular, we consider the distance between fBm and Gaussian martingales in the integral norm for all H ∈ (0, 1); an fBm represented via the Mandelbrot–Van Ness kernel, for  all  H ∈ (0, 1); and an fBm represented via the Molchan kernel, with H ∈ 0, 12 . 2.4. Distance between fBm and Gaussian martingales in the integral norm Let us relate an fBm to Gaussian martingales in the quadratic integral norm, for any H ∈ (0, 1). For technical simplicity, we consider T = 1. Theorem 2.9.– Consider the value  1

 t 2 H inf E Bt − a(s)dWs dt mint = a∈L2 ([0,1])

=



0

1

0



t

inf

a∈L2 ([0,1])

0

0

2

(z(t, s) − a(s)) dsdt.

[2.105]

This infimum is reached at function 1 a0 (s) =

s

z(t, s)dt , 1−s

[2.106]

and function a0 ∈ L2 ([0, 1]). The value of the infimum equals: i) For H > 1/2 mint =

1 − 2H + 1

1 = − 2H + 1 = ≥



1

0



(1 − s)a20 (s)ds

1

0

1 − Cα2 2H + 1

−1

2

1

(1 − s)

z(t, s)dt

ds

[2.107]

s



1 0

(1 − s)−1 s−2α

c2 H · 2πα 1 − H , 2H + 1 sin(2πα)

where Cα = αcH .





2

1

uα (u − s)α−1 (1 − u)du s

ds

128

Fractional Brownian Motion

ii) For H < 1/2 1 − 2H + 1

mint =

 0

1

(1 − s)−1



1

2 z(t, s)dt ds,

[2.108]

s

where the kernel z(t, s) is given by formula [1.5]. Proof.– Let us transform the distance as follows:  1 t 2 (z(t, s) − a(s)) dsdt 0

0



1



t

z (t, s)dsdt − 2

= 

0 1



0 t

= 0

0



2

z (t, s)dsdt − 2





t

1



t

z(t, s)a(s)dsdt + 

2

1 0

0 1

0





1

1

z(t, s)dtds +

a(s) 0

0

s

a2 (s)dsdt

0

a2 (s)(1 − s)ds.

Since we have a quadratic function under the sign of integral, its minimal value is reached at the function 1 a0 (s) = 1 s

whenever

s

z(t, s)dt , 1−s

z(t,s)dt 1−s

∈ L2 [0, 1]. Thus, we need to verify that a0 ∈ L2 [0, 1].

i) For H > 12 , we have 0 < α = H − 12 < 12 . Taking into account that in the following calculations s ≤ u ≤ 1 and uα ≤ 1, we obtain 1 t Cα s−α s ( s uα (u − s)α−1 du)dt a0 (s) = 1−s   1 −α 1 α Cα s u (u − s)α−1 (1 − u)du s = uα (u − s)α−1 du ≤ Cα s−α 1−s s ≤

Cα s−α (1 − s)α = cH s−α (1 − s)α ∈ L2 [0, 1], α

and [2.107] follows. ii) For H < 12 , we have − 12 < α = H − a0 (s) =

cH (1 − s)



1 s

1 2

< 0 and

 t

 tα s−α (t − s)α − αs−α uα−1 (u − s)α du dt. s

Distance Between fBm and Subclasses of Gaussian Martingales

129

Taking into account that tα ≤ sα , we obtain  1   Cα s−α Cα s−α 1 t α−1 tα (t − s)α dt + u (u − s)α dudt |a0 (s)| ≤ α(1 − s) s 1−s s s  1  Cα s−α 1 α−1 Cα (t − s)α dt + u (u − s)α (1 − u)du ≤ α(1 − s) s 1−s s  Cα (1 − s)α+1 Cα s−α 2α 1/s α−1 ≤ x (x − 1)α dx · (1 − s) + s α(1 + α) 1−s 1  1/s α+1 cH (1 − s) = xα−1 (x − 1)α dx, + C α sα (1 + α) 1 ∞  and integral 1 xα−1 (x − 1)α dx converges. 2.5. Distance between fBm with Mandelbrot–Van Ness kernel and Gaussian martingales Consider E(BtH − Mt )2 , 0 ≤ t ≤ 1, where BtH is fBm having representation [1.8] with Hurst index 0 < H < 1, M is the square-integrable martingale of t &s , and W & is the same Wiener process that the form Mt = 0 a(s)dW participates in the Mandelbrot–Van Ness representation [1.8] of fBm. 1 Suppose, as usual, that the condition 0 a2 (s)ds < ∞ holds, i.e. a ∈ L2 ([0, 1]). Let us find mina∈L2 ([0,1]) max0≤t≤1 E(BtH − Mt )2 in some partial cases. 2.5.1. Constant function as an integrand Consider the simplest case. Let the function a(s) = a be a constant. Theorem 2.10.– The following equality holds:  t 2

c2H &s = 1 − a(s)dW min max E BtH − a 0≤t≤1 (α + 1)2 0 and the value of amin that supplies the minimal value of [2.109] equals cH amin = . α+1

[2.109]

130

Fractional Brownian Motion

Proof.– Consider  t  t 2 2

 t

H H 2 H & & &s a(s)dWs = E(Bt ) + E a(s)dWs − 2EBt a(s)dW E Bt − 0

= t2H +



t 0

a2 (s)ds − 2

0



0

t

z1 (t, s)a(s)ds.

0

Assuming that a(s) = a, we obtain  t  t 2

H 2H 2 α & a(s)dWs = t + a t − cH 2a ((t − u)α E Bt − + − (−u+ ))du. 0

0

Evidently,  0

t α ((t − u)α + − (−u+ ))du =

tα+1 , α+1

whence  t 2

α+1 H &s = t2H + a2 t − 2acH t a(s)dW =: f (t). E Bt − α+1 0 Differentiating function f with respect to t, we obtain (2α + 1)t2α − 2acH tα + a2 = 0. Changing the variable t2α =: y, we obtain a quadratic equation (2α + 1)y 2 − 2acH y + a2 = 0.

[2.110]

The discriminant of the quadratic equation [2.110] equals D = 4a2 c2H − 4a2 (2α + 1) = (2a)2 (c2H − (2α + 1)). According to Lemma A1.6, this discriminant is negative. Therefore, max f (t) = f (1) = 1 + a2 −

0≤t≤1

2acH . α+1

[2.111]

The right-hand side of [2.111] achieves its minimal value at the point amin =

cH . α+1

Distance Between fBm and Subclasses of Gaussian Martingales

131

Therefore,  t 2

&s = a(s)dW min max E BtH − a

0≤t≤1

0

c2H 2c2H − +1 2 (α + 1) (α + 1)2

=1−

c2H . (α + 1)2



2.5.2. Power function as an integrand Consider a more general case that corresponds to power functions with a constant exponent β, i.e. a(t) = atβ , a > 0. Theorem 2.11.– Let H > 1/2, a(s) = asβ , where β > 0, a > 0 and s ∈ [0, 1]. Then,  t 2

&s a(s)dW m := min max E BtH − a

0≤t≤1

 = min max a

0≤t≤1

0

t2H + a2

 t2β+1 − 2acH B(α + 1, β + 1)tα+β+1 , 2β + 1 [2.112]

and the value of amin that supplies the minimal value of [2.112] equals amin = cH (2β + 1)B(α + 1, β + 1), where the minimal value itself equals m = 1 − c2H (2β + 1)B 2 (α + 1, β + 1). Proof.– Recall that H > 1/2. Equality  t 2

2β+1 H &s = t2H + a2 t asγ dW − 2acH B(α + 1, β + 1)tα+β+1 E Bt − 2β + 1 0 can be established by direct calculations. Let us differentiate the function g(t) = t2H + a2

t2β+1 − 2acH B(α + 1, β + 1)tα+β+1 2β + 1

in t: 2acH Γ(α + 1)Γ(β + 1) α+β t Γ(α + β + 1)   = t2β 2Hz 2 + a2 − 2a rH z ,

g  (t) = 2Ht2α + a2 t2β −

132

Fractional Brownian Motion

where rH = cH Γ(α+1)Γ(β+1) , z = tα−β . Let us calculate the discriminant D of Γ(α+β+1) the quadratic function given above: D=



 2 − 2Ha2 = a r 2 − 2H. a2 rH H

Similarly to the calculations provided in the proof of Theorem 2.3, we can Γ(β+1) decreases in β > 0. Therefore, establish that the function z(β) = Γ(α+β+1) 2 2 rH ≤ cH < 2H, according to Lemma A1.6. It means that D < 0, and max g(t) = g(1) = 1 +

0≤t≤1

a2 − 2acH B(α + 1, β + 1). 2β + 1

In turn, a minimal value of this quadratic function is reached at the point a = cH (2β + 1)B(α + 1, β + 1) and equals 1 − c2H (2β + 1)B 2 (α + 1, β + 1), whence the proof follows.  2.5.3. Comparison of Molchan and Mandelbrot–Van Ness kernels Let us first prove an auxiliary lemma. Lemma 2.17.– Let α > 0 and β > 0. Then, (β + 1)B(α + 1, β − α + 1) ≥ B(α + 1, β + 1). α+β+1 Proof.– Evidently the following relations hold: (β + 1)B(α + 1, β − α + 1) (α + β + 1)B(α + 1, β + 1) =

(β + 1)Γ(α + 1)Γ(β − α + 1)Γ(β + α + 2) (α + β + 1)Γ(β + 2)Γ(β + 1)Γ(α + 1)

=

Γ(β − α + 1)Γ(β + α + 1) ≥ 1. Γ(β + 1)2

The last inequality follows from the fact that log Γ(x) is a convex function.  Theorem 2.12.– Let H ∈ (1/2, 1), a(s) = asβ , where β > 0, s ≥ 0, a > 0. t  &t and B H = t z(t, s)dWs be the representations of Let BtH = −∞ z1 (t, s)dW t 0 fBm via the Mandelbrot–Van Ness kernel z1 , defined by formula [1.9], and the

Distance Between fBm and Subclasses of Gaussian Martingales

133

Molchan kernel z, defined by [1.7], and involving the corresponding underlying Wiener processes. Then,  t  t 2 2



&s ≥ max E BtH − max E BtH − a(s)dW a(s)dWs . 0≤t≤1

0≤t≤1

0

0

Proof.– Let t > 0. Consider the representation with the Mandelbrot–Van Ness kernel  t 2

˜ := E B H − & a(s)d W h(t) s t = t2H + a2

0



t 0

= t2H + a2



s2β ds − 2acH

t 0

uβ (t − u)α du

t2β+1 − 2acH B(α + 1, β + 1)tα+β+1 . 2β + 1

Consider the representation of fBm with the Molchan kernel. Then, we have the following equalities:  t 2

H h(t) = E Bt − a(s)dWs = t2H + a2



0

t 0

= t2H + a2

s2β ds − 2a



t

sβ z(t, u)du = (Lemma 2.20) 0

2β+1

t 2acH (β + 1)B(α + 1, β − α + 1) α+β+1 . − t 2β + 1 α+β+1

˜ We immediately obtain from Lemma 2.17 that h(t) ≥ h(t). Therefore, the approximation of fBm with the power functions is better for the representation with the Molchan kernel than for the representation with the Mandelbrot–Van Ness kernel.    2.6. fBm with the Molchan kernel and H ∈ 0, 12 , in relation to Gaussian martingales The following three lemmas will help us prove the main results. The first lemma can be proved by direct calculations. Lemma 2.18.– Let f ∈ C 3 [−2, 1], then −f (−2) + 3f (−1) − 3f (0) + f (1) =



1 −2

f  (t)g(t) dt,

134

Fractional Brownian Motion

where

⎧ 2 ⎪ ⎨0.5(t + 2) , 2 g(t) = −t − t + 0.5, ⎪ ⎩ 0.5(1 − t)2 ,

−2 ≤ t ≤ −1, −1 ≤ t ≤ 0, 0 ≤ t ≤ 1.

Note that 0 ≤ g(t) ≤ 0.75 for t ∈ [−2, 1]. Using this result, we establish the following bounds for cH . Lemma 2.19.– We have the inequalities c2H c2H

 

πα sin(απ) πα sin(απ)

2 > 2H

for 0 < H
1 Γ(1 − 2α)

Distance Between fBm and Subclasses of Gaussian Martingales

135

for 0 < H < 12 , and Γ(1 − α)3 Γ(1 + α) 0, H ∈ (0, 1) and z be the Molchan kernel defined by [1.5]. Then,  t cH (β + 1)B(α + 1, β − α + 1) α+β+1 sβ z(t, s)ds = . t α+β+1 0 Proof.– It is well known that if α > −1, β > −1 and t > 0, then 

t 0

uα (t − u)β du = B(α + 1, β + 1)tα+β+1 .

We have

t 0

 I1 =

sβ z(t, s)ds = cH (I1 − I2 ), where

t 0

tα sβ−α (t − s)α ds = (formula [2.114])

= B(β − α + 1, α + 1)tα+β+1 .

[2.114]

136

Fractional Brownian Motion

 I2 =

t

αs

β−α



0

 uα−1 (u − s)α du ds

s



t

=α

uα−1 

t



0

u 0

 sβ−α (u − s)α ds du

t

=α 0

uα+β B(α + 1, β − α + 1)du =

αB(α + 1, β − α + 1) α+β+1 , t α+β+1 

whence the proof follows.

Now, consider E(BtH − Mt )2 , 0 ≤ t ≤ 1, where B H = {BtH , t ≥ 0} is an fBm having representation [1.4] with the kernel z(t, s) from [1.5], and with Hurst index 0 < H < 12 . Let M be a square-integrable martingale of the form t Mt = a(s)dWs , and let a ∈ L2 ([0, 1]). Our goal is to find 0 mina∈L2 ([0,1]) max0≤t≤1 E(BtH − Mt )2 in some particular cases. Consider the simplest case. Let the function a(s) = a > 0 be a constant. Theorem 2.13.– The following equality holds:  t 2

c απ 2 H a(s)dWs = 1 − , min max E BtH − a>0 0≤t≤1 α + 1 sin απ 0 and the value of amin when the minimum is reached in [2.115] equals απ cH amin = . α + 1 sin απ Proof.– When a(s) = a > 0 and α < 0, we have  t 2 2

 t

a(s)dWs = E(BtH )2 + E a(s)dWs E BtH − 0

−2EBtH



t 0

2H

a(s)dWs = t =t

2H

 + 0 2

0

t



2

a (s)ds − 2

+ a t − 2a



t

z(t, s)a(s)ds 0

t

z(t, s)ds. 0

From Lemma 2.20, we have  0

t

z(t, s)ds = cH

B(1 + α, 1 − α) α+1 απ tα+1 = cH t . α+1 sin απ α + 1

[2.115]

Distance Between fBm and Subclasses of Gaussian Martingales

137

Therefore,  t

2 απ tα+1 H E Bt − a(s)dWs = t2α+1 − 2acH + a2 t =: f (t, a). sin απ α + 1 0 Differentiate function f in t and equate the derivative to zero: απ α ∂f (t, a) = t2α (2α + 1) − 2acH t + a2 = 0. ∂t sin απ

[2.116]

The discriminant of quadratic equation [2.116] equals

απ 2 − 4a2 (2α + 1) sin απ

απ 2  = (2a)2 c2H − (2α + 1) > 0 sin απ

D = 4a2 c2H

for α < 0 according to Lemma 2.19. Therefore, the roots of equation [2.116] equal  

απ απ 2 1 α 2 t = αcH − (2α + 1) ± a cH 2α + 1 sin απ sin απ

a t± > 0, (A ± A2 − 2H) =: tα = ±, 2H where A = cH sinαπαπ . For H < 1/2, we have A2 − 2H > 0, and A ± > 0.

√

A2 − 2H

Taking into account that a > 0, we obtain that for t ∈ (0, t− ), the function f (t, a) is increasing, for t ∈ (t− , t+ ), the function f (t, a) is decreasing, and for t ∈ (t+ , +∞), the function f (t, a) is increasing again. Hence, maxt∈[0,1] f (t, a) = max(f (1, a), f (t− , a)). Consider the value f (1, a) = 1 − 2acH

απ 1 + a2 . sin απ α + 1

Find the derivative of f (1, a) with respect to a and equate it to zero: cH απ = 0 at the point a0 = α+1 sin απ . Therefore,

∂f (1,a) ∂a

min f (1, a) = 1 − a>0

c απ 2 H . α + 1 sin απ

138

Fractional Brownian Motion

Consider



a 2

2aA α f (t− , a) = t− t2α (A − A2 − 2H)2 t− + a2 = t− − − α+1 2H 

2aA a − (A − A2 − 2H) + a2 α + 1 2H

1 

A (A − A2 − 2H)2 − = a 2 t− (A − A2 − 2H) + 1 . 2 (2H) H(α + 1) cH απ For a = a0 = α+1 sin απ we can verify using Mathematica that the following inequality holds:

f (t− , a0 ) < f (1, a0 ), whence min max f (t, a) = 1 − a>0 t∈[0,1]

c απ 2 H . α + 1 sin απ



2.7. Distance between the Wiener process and integrals with respect to fBm Let us now, if we may say so, turn the situation inside out, namely we will look for the distance between a Wiener process and Gaussian integrals with respect to an fBm. Similarly to the previous consideration, it is reasonable to consider the underlying Wiener process. First, let us consider briefly integration with respect to an fBm for non-random integrands. 2.7.1. Wiener integration with respect to fBm For any H ∈ (1/2, 1) consider the space LH 2 ([0, T ]) consisting of the following functions: ' f∈

LH 2 ([0, T ])

:=

  T  f : [0, T ] → R  0

(

T 0

|f (s)| |f (u)| |u − s|

2H−2

duds < ∞

equipped with the norm  =

f LH 2 ([0,T ])

 T H(2H − 1)

0

12

T 0

|f (s)| |f (u)| |u − s|

2H−2

du ds

.

Distance Between fBm and Subclasses of Gaussian Martingales

139

Each member of this space can be integrated with respect to an fBm with Hurst index H on any interval [0, t] ⊆ [0, T ], and as a result, we obtain the so-called Wiener integral with respect to an fBm that is a Gaussian process t G(t) = 0 f (s)dBsH , with zero mean and covariance:  t EG(t)G(s) = H(2H − 1)

0

t 0

f (s) f (u) |u − s|2H−2 du ds.

For H ∈ (0, 1/2) and measurable function f : [0, T ] → R of bounded t variation we can define 0 f (s)dBsH via integration by parts: 



t

f (s)dBsH

0

=

f (t)BtH

−

t

f (s)BsH df (s),

0

and again it will be a Gaussian process. For the detailed description of the Wiener integration with respect to an fBm, see [BIA 08, MIS 08,NOR 99b]. 1 Now, the problem reads as follows. Let f ∈ LH 2 ([0, T ]) and H ∈ 2 , 1 . The aim is to calculate  inf

sup E

f ∈LH 2 ([0,T ]) 0≤t≤T

2

t 0

f (s) dBsH

− Wt

,

[2.117]

where W is the Wiener process involved in the representation [1.4] so that W is the underlying process for B H . Substitute α = H − z (t, s) := f

c(2) α



1 2

and consider the kernel

t

f (u) · uα (u − s)α−1 du, s

(2)

where cα = 

t 0



(2α+1)α (1−2α)B(1−2α, α)

1/2

f (s) dBsH = (1 − 2α)1/2

. According to [LEB 98], 

t 0

z f (t, s) s−α dWs .

Since the above problem seems to be rather technically complicated, we consider a partial case where the infimum in [2.117] is evaluated with respect to power functions f (u) = kuα , k > 0, instead of the whole class LH 2 ([0, T ]). For this partial case, we find the function f at which the infimum is reached in order to evaluate the infimum itself. Note that the value of the infimum is non-zero.

140

Fractional Brownian Motion

2.7.2. Wiener process and integrals of power functions with respect to fBm For the sake of simplicity, let T = 1. Theorem 2.14.– Let f = f (u) be a function of the form f (u) = kuα , k > 0, α = H − 1/2. 1) For any H ∈ [H3 , 1), where H3 ≈ 0.849278, the minimal value  k>0 t∈[0,1]

2

t

min max E

f (s) dBsH − Wt

0

is reached at the point k = kmin = itself equals

1−

1

(1−2α) 2 c(2) α B(1−α, α)·(4α+1) . 2α(2α+1)2 ·B(α+1, 2α)

2 (2) 2 (1 − 2α) cα B (1 − α, α) (4α + 1) 2α(2α + 1)3 B(α + 1, 2α)

The minimum

.

2) For any H ∈ ( 12 , H3 ), we have the following relations:  k

t∈[0,1]

2

t

min max E 0

f (s) dBsH − Wt

= min max{f (α, k, 1), f (α, k, t1 )}, k

where  

3 (Γ(1−α))3 (Γ(1 − α)) − Γ(1 − 2α) Γ(1−2α) − ⎜ 3 Γ(3α)

t1 = ⎜ ⎝ 2kΓ(2α) α (2α + 1)Γ(α)Γ(1 − 2α) ⎛

and f (α, k, t) = 2α(2α + 1)k 2 · B(α + 1, 2α) 1

t4α+1 4α + 1

− 2(1 − 2α) 2 c(2) α B (1 − α, α) k ·

t2α+1 + t. 2α + 1

2Γ(2α) 3Γ(3α)

1  ⎞ 2α

⎟ ⎟ ⎠

,

Distance Between fBm and Subclasses of Gaussian Martingales

141

Proof.– Consider the distance between the integral with respect to fBm and the Wiener process for t ∈ [0, 1]:  t 2 H E f (s) dBs − Wt 0



2

t

=E 0

f (s) dBsH  t

= H(2H − 1)

0

− 2(1 − 2α)1/2



 − 2E

t 0 t

t 0

f (s) dBsH · Wt + t

f (s)f (u)|u − s|2H−2 duds z f (t, s)s−α ds + t.

[2.118]

0

The second integral on the right-hand side of [2.118] equals  t  t t z f (t, s) · s−α ds = c(2) f (u)uα (u − s)α−1 dus−α ds α 0

0

= c(2) α

 t 0

s u 0

(u − s)α−1 s−α dsf (u)uα du

= c(2) α B (1 − α, α) Let f (u) = k · uα , then

t 0

f (u)uα du =



t

f (u)uα du. 0

t 0

k · u2α du = k ·

t2α+1 2α+1 .

Further, the first integral on the right-hand side of [2.118] can be rewritten as follows:  t t  t s 2H−2 f (s) f (u)|u − s| duds = f (s) f (u)(s − u)2H−2 duds 0

0

0

 t

t

+ 0

0

f (s) f (u)(u − s)2H−2 duds = 2

s

 t 0

If f (u) = k · uα , k > 0, then  t t  t  2H−2 f (s)f (u)|u − s| duds = 2 0

0

= 2k

2

 t 

s

u (s − u) α

0

0

= 2k 2 B(α + 1, 2α)

2α−1

t4α+1 . 4α + 1



0

s 0

f (u) (s − u)2H−2 duf (s)ds.

s

ku (s − u) α

0

2H−2

du sα ds = 2k 2 B(α + 1, 2α)

 du ksα ds



t

s4α ds

0

[2.119]

142

Fractional Brownian Motion

Taking into account [2.119], the distance under consideration equals 

2

t

E 0

f (s) dBsH 1

− Wt

= 2α(2α + 1)k 2 · B(α + 1, 2α)

− 2(1 − 2α) 2 c(2) α B (1 − α, α) k ·

t4α+1 4α + 1

t2α+1 + t =: f (α, k, t). 2α + 1

Our aim is to evaluate mink>0 maxt∈[0,1] f (α, k, t) for any fixed H ∈ ( 12 , 1) or, equivalently, for any fixed α ∈ (0, 1/2). To find a point t ∈ [0, 1] at which the maximum is reached, we differentiate the function f (α, k, t) with respect to t: ∂f (α, k, t) ∂t 1

2α = 2α(2α + 1)k 2 B(α + 1, 2α)t4α − 2(1 − 2α) 2 c(2) + 1. α B (1 − α, α) kt

Changing the variable x := t2α , we obtain the following quadratic equation: 1

2α(2α + 1)k 2 B(α + 1, 2α)x2 − 2(1 − 2α) 2 c(2) α B (1 − α, α) kx + 1 = 0.

[2.120]

Its discriminant is given by the equality 2

2 D(0) (α) = 4(1 − 2α) c(2) (B (1 − α, α)) k 2 α − 8α(2α + 1)k 2 B(α + 1, 2α)  3 2 2Hα (Γ(1 − α)) (Γ(α)) Γ(α + 1)Γ(2α) 2 = 4k − 2α(2α + 1) Γ(1 − 2α)Γ(α) Γ(3α + 1)  3 (Γ(1 − α)) 2 Γ(2α) . [2.121] = 4k 2 (2α + 1)Γ(α + 1) − Γ(1 − 2α) 3 Γ(3α) To determine the sign of the discriminant D(0) (α) in [2.121], we consider the factor D(1) (α) =

(Γ(1 − α))3 2 Γ(2α) (Γ(1 − α))3 Γ(2α + 1) − = − , Γ(1 − 2α) 3 Γ(3α) Γ(1 − 2α) Γ(3α + 1)

in parentheses on the right-hand side of [2.121], because its sign coincides with the sign of the discriminant. On the one hand, since limα→0 D(1) (α) = 0, the behavior in a neighborhood of zero has to be studied more carefully. On

Distance Between fBm and Subclasses of Gaussian Martingales

143

2 the other hand, it is obvious that D(1) (α) → − 3Γ(3/2) as α → 1/2. In other words, the discriminant is negative if the Hurst index is sufficiently close to 1. Furthermore, the signs of D(0) (α) and D(1) (α) are the same as that of

D(2) (α) = 3 log Γ(1−α)−log Γ(1−2α)−log Γ(2α + 1) + log Γ(3α + 1).

[2.122]

According to Lemma A1.2, the logarithm of the gamma function has the following expansion into Taylor series: log Γ(1 + z) = −γz +

∞ % ζ(k) k=2

k

(−z)k ,

|z| < 1,

[2.123]

where γ = lim

n→∞

% n k=1



1 − log n k

≈ 0.5772156649

is the Euler constant, and ∞ % 1 ζ(k) = k n n=1

is the Riemann zeta function. From equations [2.122] and [A1.12] we obtain D(2) (α) = 3αγ + − =

3π 2 α2 4π 2 α2 − 2αγ − + 2αγ 12 12

9π 2 α2 4π 2 α2 − 3αγ + + o(α2 ) 12 12

π 2 α2 + o(α2 ) 3

[2.124]

as α → 0. Thus, D(2) (α), and so D(1) (α) and D(0) (α) are positive if α is close to 0. Now, we will prove that α3 ∈ (0, 12 ) exists such that the discriminant D(0) (α) is positive in the interval (0, α3 ) and negative in the interval (α3 , 12 ). This property will be proved if we establish that D(2) (α)/α2 decreases in (0, 12 ).

144

Fractional Brownian Motion

Noting that log Γ(1) = 0, we calculate the following derivative: d dα



D(2) (α) α2



= α−2 (2ψ(1 − 2α) − 3ψ(1 − α)

−2ψ(2α + 1) + 3ψ(2α + 1)) + 2α−3 (log Γ(1 − 2α) − 3 log Γ(1 − α) +2 log Γ(1) + log Γ(2α + 1) − log Γ(3α + 1)) = 2α−2 (ψ(1 − 2α) + 3ψ(1 − α) − 2ψ(2α + 1) + 3ψ(2α + 1))  3α

z 1 dz ψ(1 + z)g1 + 3 α −2α α   3α

z

z 1 1 3α  = 2 dz = dz ψ  (1 + z)g2 ψ (1 + z)g3 α −2α α α −2α α      1 3α  z z 1 = − g3− dz ψ (1 + z) g3+ + α −2α α α 2 

z  1 0 

α + − ψ  (1 + z) g3− dz < 0, ψ 1+z− α −α/2 2 α where ψ(x) = Definition A1.3,

d dx (log Γ(x))

⎧ ⎪ −x2 − 2x, ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨2x + x, g3 (x) = x, ⎪ ⎪ ⎪ −x2 + 3x, ⎪ ⎪ ⎪ ⎩0,

= Γ (x)/Γ(x) is the digamma function, see

−2 < x < −1, −1 < x < 0, 0 < x < 2, 2 < x < 3, x > 3,

g2 (x) = −g3 (x) (see the graph in Figure 2.21), g1 (x) = −g2 (x) (Figure 2.20), g3+ (x) = max{g3 (x), 0}, and g3− (x) = −min{g3 (x), 0}. It is clear that the tetragamma function ψ  (x) (see Definition A1.3) increases in the interval (0, +∞) and takes only negative values. Furthermore, g3+ (x) − g3− (x + 12 ) ≥ 0 for x ∈ (−2, 3).

Distance Between fBm and Subclasses of Gaussian Martingales

145

Figure 2.20. Graph of the function g1 (x)

Figure 2.21. Graph of the function g2 (x)

The graphs of the functions g3 (x) and g3+ (x) − g3− (x + 12 ) are shown in Figure 2.22. Therefore, D(2) (α)/α2 decreases in the interval (0, 12 ). Moreover, D(2) (α) π2 = > 0 and α→0 α2 3 lim

lim1

α→ 2

D(2) (α) = −∞. α2

[2.125]

146

Fractional Brownian Motion

Figure 2.22. Graphs of the functions g3 (x) (solid line) and g3+ (x) − g3− (x + 12 ) (dashed line, coincides with g3 (x) for x ∈ (−1, 0))

The former relation in [2.125] follows from [2.124], while the latter is easy to prove, because − log Γ(1 − α) is the only term of the right-hand side of [2.122] that is unbounded in the neighborhood of 12 . Therefore, the equation D(2) (α)/α2 = 0 has a unique solution α3 in the interval (0, 12 ). The approximate value α3 computed numerically is α3 ≈ 0.349278. If α ∈ (0, α3 ), then D(2) (α) > 0, D (1) (α) > 0, and D(0) (α) > 0. Otherwise, if α ∈ (α3 , 12 ), then D(2) (α) < 0, D(1) (α) < 0, and D(0) (α) < 0. This value of α3 corresponds to H3 = 12 + α3 ≈ 0.849278. k, t) If D(0) ≤ 0, then ∂f (α, ≥ 0 and f (α, k, t) increases with respect to t, ∂t whence maxt∈[0,1] f (α, k, t) = f (α, k, 1).

Then, 1 4α + 1 1 1 − 2(1 − 2α) 2 c(2) + 1. α B (1 − α, α) k 2α + 1

f (α, k, 1) = 2α(2α + 1)k 2 B(α + 1, 2α)

[2.126]

Now, we find the point of minimum of the expression on the right-hand side of [2.126] with respect to k: 1

kmin =

(2)

(1 − 2α) 2 cα B (1 − α, α) · (4α + 1) . 2α(2α + 1)2 · B(α + 1, 2α)

[2.127]

Distance Between fBm and Subclasses of Gaussian Martingales

147

For H ∈ [H3 , 1), where H3 ≈ 0.849278 min max f (α, k, t) = k

2 (2) 2 (1 − 2α) cα B (1 − α, α) (4α + 1)

t

−

2α(2α + 1)3 B(α + 1, 2α)

2 (2) 2 (1 − 2α) cα B (1 − α, α) (4α + 1)

+1 α(2α + 1)3 B(α + 1, 2α) 2

(2) 2 B (1 − α, α) (4α + 1) (1 − 2α) cα . =1− 2α(2α + 1)3 B(α + 1, 2α) [2.128] If D(0) > 0, the roots of the quadratic equation [2.120] are given by  

3 (Γ(1−α))3 (Γ(1 − α)) − Γ(1 − 2α) Γ(1−2α) − ⎜ 3Γ(3α)

t1 = ⎜ ⎝ 2kΓ(2α) α(2α + 1)Γ(α)Γ(1 − 2α) ⎛

 

3 (Γ(1−α))3 (Γ(1 − α)) + Γ(1 − 2α) Γ(1−2α) − ⎜ 3Γ(3α)

t2 = ⎜ ⎝ 2kΓ(2α) α(2α + 1)Γ(α)Γ(1 − 2α) ⎛

2Γ(2α) 3Γ(3α)

2Γ(2α) 3Γ(3α)

1  ⎞ 2α

⎟ ⎟ ⎠

,

1  ⎞ 2α

⎟ ⎟ ⎠

.

Note that the roots t1 and t2 are positive for k > 0. Indeed, the leading coefficient and the free term in [2.120] are positive, and thus the roots are of the same sign. Moreover, the coefficient for t2α is negative implying that the roots are positive. Then: a)

∂f (α, k, t) ∂t

> 0 and f (α, k, t) increases for 0 < t < t1 ;

b)

∂f (α, k, t) ∂t

< 0 and f (α, k, t) decreases for t1 < t < t2 ;

c)

∂f (α, k, t) ∂t

> 0 and f (α, k, t) increases for t > t2 .

148

Fractional Brownian Motion

Thus: a) if t1 > 1, then f (α, k, t) increases in [0, 1] and maxt∈[0,1] f (α, k, t) = f (α, k, 1); b) if t1 < 1, then maxt∈[0,1] f (α, k, t) = max {f (α, k, 1), f (α, k, t1 )}; c) if t1 = 1, then maxt∈[0,1] f (α, k, t) = f (α, k, 1) = f (α, k, t1 ). Therefore, if H ∈ ( 21 , H3 ), where H3 ≈ 0.849278, then min max f (α, k, t) = min k

k

t∈[0,1]

max{f (α, k, 1), f (α, k, t1 )}, for t1 < 1, f (α, k, 1), for t1 ≥ 1.

[2.129]

Formula [2.129] can be simplified to some extent. Let k∗ be the value of k at which the function maxt∈[0,1] f (α, k, t) reaches its minimum. Recall that the minimum of the function f (α, k, 1) is reached at k = kmin (see [2.127]). Let t1 (kmin ) and t1 (k∗ ) denote the values of t1 for k = kmin and k = k∗ , respectively. Now, we need to verify that t1 (kmin ) < 1:

(Γ(1 − α))3 3Γ(3α) 2α + 1 2α

(t1 (kmin )) < = < 1. 2kmin Γ(2α) α(2α + 1)Γ(α)Γ(1 − 2α) 4α + 1 Then, we can prove that t1 (k∗ ) ≤ 1. Indeed, if t1 (k∗ ) > 1, then t1 (k) > 1 and thus maxt∈[0,1] f (α, k, t) = f (α, k, 1) in a neighborhood of the point k∗ . The  ∂  necessary condition for the minimum ∂k f (α, k, 1) k=k∗ implies that k∗ = kmin , t1 (k∗ ) < 1 and we obtain a contradiction. The equalities min max f (α, k, t) = max f (α, k∗ , t) = max{f (α, k∗ , 1), f (α, k∗ , t1 (k∗ ))}, k

t∈[0,1]

t∈[0,1]

max f (α, k, t) ≤ max{f (α, k, 1), f (α, k, t1 (k))}

t∈[0,1]

imply that min max f (α, k, t) = min max{f (α, k, 1), f (α, k, t1 (k))}. k

t∈[0,1]

k

Thus, the theorem is proved. Corollary 2.7.– Equality [2.128] implies that

2 (2) 2 (1 − 2α) cα B (1 − α, α) (4α + 1) 0, k∈R

(2)

where kmin isdefined  by [2.127]. This implies the following upper bound for cα for any H ∈ 12 , 1 :  2α(2α + 1)3 B(α + 1, 2α) (2) (3) cα < 2 =: cα . (1 − 2α)(4α + 1)B (1 − α, α) (2)

(3)

Figure 2.23 depicted   using Mathematica shows that the bound cα < cα holds for all H ∈ 12 , 1 . (3)

(2)

The graph of the difference cα − cα shown in Figure 2.24 supports this conclusion.

(2)

(3)

Figure 2.23. Plots of cα (solid line) and cα (dashed line)

Also, inequality [2.130] implies the following bound for the gamma function  for H ∈ 12 , 1 : 3

2 (2α + 1)2 Γ(2α) (Γ(1 − α)) < . Γ(1 − 2α) 3 (4α + 1) Γ(3α)

150

Fractional Brownian Motion

(3)

(2)

Figure 2.24. Plot of the difference cα − cα

2.8. Exercises Exercise 2.1.– Study the proof of Theorem 2.1 and then do the following. 1) Calculate the limits lim cH ,

H→1/2

lim c1 (H), lim cH , lim c1 (H).

H→1/2

H→1

H→1

2) Using Mathematica or other suitable software, plot the graph of ρ2 (B H , M(K1 )) = T 2H (1 − c21 (H)), as a function of H, for T = 1, T = 2. 3) Calculate the following values: min max E(BtH − aW (t) − bt)2 , min

a,b∈R 0≤t≤T

max E(BtH − aW (t) − bt)2 .

a,b∈R T /2≤t≤T

Exercise 2.2.– Study the proof of Theorem 2.2. Then, using Mathematica or other suitable software, plot the graph of   c2 ρ2 (B H , M(K2 )) = T 2H 1 − H , 2H as a function of H, for T = 1, T = 2. Exercise 2.3.– Clarify the proof of Theorem 2.3 by establishing that log Γ(x) is a convex function on (0, +∞).

Distance Between fBm and Subclasses of Gaussian Martingales

151

Exercise 2.4.– Having read Theorem 2.4, calculate the value f (k, H) = max E(BtH − Mt )2 , 0≤t≤T

t

where Mt = k 0 es sα dWs , α = H−1/2. Plot the graphs of f (k, H) as a function of k, for H = 0.6; 0.75; 0.8 and as a function of H, for k = 1; 2; 3. Exercise 2.5.– Study the proof of Theorem 2.5. Then, calculate the value f (k, H, β) = max E(BtH − Mt )2 , 0≤t≤T

where Mt = k

t 0

s−β dWs , β > 0, k > 0. Plot the graphs of f (k, H, β):

i) as a function of k, for H = 0.6; 0.75; 0.8, β = H/2; ii) as a function of H, for k = 1; 2; 3, β = 1 − H; iii) as a function of β ∈ (0, 1/2), for k = 1, H = 0.75. Exercise 2.6.– We use the notation presented in section 2.1.6. Let H ∈ ( 12 , H1 ]. Refine inequality [2.82] and prove that f (T, k1∗ , k2∗ ) ≤ ρ2 (B H , M(K7 )) ≤ f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ).

3 Approximation of fBm by Various Classes of Stochastic Processes

In Chapters 1 and 2 we studied the projections of fractional Brownian motion (fBm) on various subspaces, the distance from which to fBm was non-zero. So, in such subspaces, we cannot approximate fBm. However, a very interesting problem is still to approximate fBm using stochastic processes of comparatively simple structure or alternatively using the series of specially selected structure. The chapter is organized as follows. In section 3.1, fBm is represented as uniformly convergent series of Lebesgue integrals. Section 3.2 is devoted to the semimartingale approximation of fBm and to the approximation of the pathwise integral with respect to fBm by the integrals with respect to semimartingales. In section 3.3, we construct smooth processes that converge to fBm in the certain Besov-type space. This allows us to approximate the stochastic integral with respect to fBm by the integrals with respect to absolutely continuous processes. Section 3.4 contains a construction of absolute continuous approximations for the so-called multifractional Brownian motion, which is a generalization of fBm for the case of a time-dependent Hurst index. 3.1. Approximation of fBm by uniformly convergent series of Lebesgue integrals In this section, we consider representation of fBm via the uniformly  convergent series of special Lebesgue integrals. Let B H = BtH , t ∈ [0, T ] be

Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

154

Fractional Brownian Motion

an fBm with H ∈ ( 12 , 1). Consider the following supplementary Gaussian t process: Yt = 0 s−α dBsH , which exists for α = H − 1/2, due to results from section 2.7.1 (see also Exercise 3.1). Introduce also a Gaussian martingale M H (the so-called Molchan martingale) that equals MtH

 = CH

t 0

(t − s)−α s−α dBsH ,

where  CH =

cos(πH)21−2H Γ(2 − H) π 3/2 H(1/2 − H) Γ(3/2 − H)

1/2 .

Obviously, the following relations between these processes hold (for more details, see [NOR 99b, MIS 08]): dYt = t−α dBtH ,  t  t H α α−1 s Ys ds = sα dYs , Bt = t Yt − α 0

 Yt = cH

t 0

0

(t − u)

α

dMuH ,

and

MtH

 =

[3.1] t

0

where W is an underlying Wiener process, cH =

s−α dWs , 

[3.2]

2HΓ(3/2−H) Γ(2−2H)Γ(H+1/2)

1/2

.

Let us show that process Yt can be represented as the following series. t Theorem 3.1.– Let H ∈ (1/2, 1). Process Yt = 0 s−α dBsH admits the following representation:  1 ∞ 

n−α α H Yt = αcH t sn Mts ds, [3.3] n 0 n=0 where the series and uniformly.

∞ n−α  1 n=0

n

0

H sn Mts ds with probability 1 converges absolutely

Proof.– Let us integrate by parts the right-hand side of the representation [3.2] for Y :  Yt = cH

t 0

 t t  (t − u)α dMuH = cH (t − u)α · MuH  − cH MuH d(t − u)α 0

0

Approximation of fBm by Various Classes of Stochastic Processes

 = cH

t 0

= αcH t

MuH α(t

α



1 0

− u)

α−1

 du = αcH

1 0

155

H α−1 Mts t (1 − s)α−1 d(ts)

H (1 − s)α−1 Mts ds.

[3.4]

Recall the binomial expansion of (1 − x)α−1 for |x| < 1: (1 − α)(2 − α) 2 x 2!    ∞ 

n−α n n−α n x , + ··· + x + ··· = n n n=0

(1 − x)α−1 =1 + (1 − α)x +

where

n−α n

= (1 − α) . . . (n − α)/n!,

−α 0

[3.5]

= 1.

Let us substitute [3.5] into [3.4]: Yt = αcH t

α

 ∞ 

n−α n

n=0

1 0

H sn Mts ds.

[3.6]

Such representation holds almost surely (a.s.), and also the convergence is uniform, because the process M is continuous consequently bounded on [0, t] with probability 1. Let us estimate the remainder term of the series [3.6]:    ∞  1 

n − α n H   α s Mts ds αcH t   n 0 n=N

≤ αcH tα sup |M (u)| u∈[0,t]

n=N

= αcH tα sup |M (u)| u∈[0,t]

By Lemma A1.4, ∞

n=N

n−α n

 1 ∞ 

n−α sn ds n 0 ∞

n=N

  1 n−α . n+1 n

∼ n−α /Γ(1 − α), n → ∞ when α < 1, therefore

  1 n−α ≤ CN −α , n+1 n

with some constant C > 0.



156

Fractional Brownian Motion

After replacement ts → u in [3.3], we obtain  ∞  t

n − α  u n H  u α Mu d Yt = αcH t t t n n=0 0 = αcH

 ∞ 

n−α n

n=0

α−n−1

t



t 0

MuH un du.

[3.7]

Theorem 3.2.– The fBm B H = {BtH , t ∈ [0, T ]} admits the representation as the series ∞

H Bt = m(t) + t2α−n−1 mn (t), n=0  where m(t) = −CH

t

s2α−1 MsH ds,   t α(n + 1 − α) n − α mn (t) = cH MsH sn ds, n + 1 − 2α n 0

and  CH = α 2 cH

0

  1 n−α = α2 cH B(1 − 2a, a). n + 1 − 2α n n=0 ∞

Proof.– Substitute the formula for Yt from [3.7] into the left-hand side of the equality [3.1] and obtain the equalities  t BtH = tα Yt − α sα−1 Ys ds 0

= tα αcH

∞ 

n=0

− α 2 cH



t 0

∞ 

n−α n

sα−1



tα−n−1



t 0

 ∞ 

n−α n=0

n

 MuH un du

sα−n−1



s 0

  t n−α = αcH t2α−n−1 MuH un du n 0 n=0   s  t 2α−n−2 H n Mu u duds s −α 0

0

∞ 

  t n−α = αcH t2α−n−1 MuH un du n 0 n=0   t  t H n 2α−n−2 Mu u s dsdu −α 0

u

 MuH un du ds

Approximation of fBm by Various Classes of Stochastic Processes

157

  t ∞ 

n−α 2α−n−1 t = αcH MuH un du n 0 n=0   t n 2α−n−1 − u2α−1 Hu t Mu −α du 2α − n − 1 0     t ∞ 

n−α α 2α−n−1 H n t Mu u du 1 − = αcH 2α − n − 1 n 0 n=0   t α H 2α−1 + Mu u du 2α − n − 1 0   t ∞ 

n−α 2α−n−1 α − n − 1 t = αcH MuH un du 2α − n − 1 n 0 n=0   t  t α  u2α−1 MuH du = −CH u2α−1 MuH du + 2α − n − 1 0 0   t ∞ 

α(n + 1 − α) n − α 2α−n−1 t + cH MuH un du, n + 1 − 2α n=0 n 0 where  CH

  1 n−α = α2 cH B(1 − 2a, a). = α cH n + 1 − 2α n n=0 ∞

2



3.2. Approximation of fBm by semimartingales This section is devoted to the approximation of a fBm in Lp (Ω) by semimartingales. We show that pathwise stochastic integral with respect to fBm can be approximated by the stochastic integrals with respect to semimartingales. The section is based on the paper of N. T. Dung [DUN 11]. However, the results are presented here in a more rigorous and detailed manner. 3.2.1. Construction and convergence of approximations   Let B H = BtH , Ft , t ∈ [0, T ] be an fBm with Hurst index H ∈ (0, 1). Recall that according to [1.4], it can be represented in the form BtH =



t 0

z(t, s) dWs ,

158

Fractional Brownian Motion

where W = {Wt , t ∈ [0, T ]} is a standard Wiener process and the Molchan kernel z(t, s) is defined by [1.5]. The processes B H and W generate the same filtration, and we can assume that F = {Ft , t ∈ [0, T ]} is the filtration generated by any of these processes. For every ε > 0, we introduce the following stochastic process: BtH,ε

 =

t 0

z(t + ε, s) dWs .

[3.8]

Proposition 3.1.– 1) For every ε > F-semimartingale with the decomposition  t  t H,ε Bt = z(s + ε, s) dWs + ϕεs ds, 0

where ϕεs =



s 0

0, the process B H,ε is an

[3.9]

0

∂1 z(s + ε, u) dWu ,

∂1 z(t, s) =

∂z(t, s) . ∂t

[3.10]

2) For any p > 0, BtH,ε converges to BtH in Lp (Ω) as ε → 0 uniformly with respect to t ∈ [0, T ]. Remark 3.1.– 1) The partial derivative of the Molchan kernel [1.5] equals

[3.11] ∂1 z(t, s) = cH H − 12 tH−1/2 s1/2−H (t − s)H−3/2 . 2) The integrals in [3.8]–[3.10] are well defined due to Lemma A1.11 in Appendix 1. Proof of Proposition 3.1.– 1) By the stochastic Fubini theorem, 

t 0

ϕεs

ds =

 t 

=

0

s 0

∂1 z(s + ε, u) dWu ds =

t 0

0

z(t + ε, u) − z(u + ε, u) dWu

= BtH,ε − whence [3.9] follows.

 t

 0

t

z(u + ε, u) dWu ,

t u

∂1 z(s + ε, u) ds dWu

Approximation of fBm by Various Classes of Stochastic Processes

159

2) By the Burkholder–Davis–Gundy inequality [A1.35], for any p > 0, p  t p   

  z(t + ε, s) − z(t, s) dWs  E BtH,ε − BtH  = E  0

 ≤ Cp

t

0

z(t + ε, s) − z(t, s)

2

 p2 ds

,

where Cp > 0 is a constant. Further,  t

2 z(t + ε, s) − z(t, s) ds 0

 =  ≤

0

t



2

z (t + ε, s) ds − 2

t+ε 0

t 0

2

z (t + ε, s) ds − 2

2  H = E Bt+ε − BtH  = ε2H ,

 z(t + ε, s)z(t, s) ds +



t∧(t+ε) 0

t 0

z 2 (t, s) ds

z(t + ε, s)z(t, s) ds +



t 0

z 2 (t, s) ds

where the last equality follows from [1.3]. Hence, p    E BtH,ε − BtH  ≤ Cp εpH . 

This completes the proof.

3.2.2. Approximation of an integral with respect to fBm by integrals with respect to semimartingales In this section, we prove that the pathwise generalized Lebesgue–Stieltjes stochastic integral with respect to fBm can be approximated by the stochastic integrals with respect to semimartingales BtH,ε introduced in the previous section. Assume that a stochastic process u = {ut , t ∈ [0, T ]} satisfies the conditions: i) u is progressively measurable with respect to {Ft , t ∈ [0, T ]}; ii) the trajectories of u belong to C 1−H+δ [0, T ] a.s. with some δ > 0, i.e. sup |ut | +

0≤t≤T

sup

0≤s 12 . Assume that u = {ut , t ∈ [0, T ]} is a stochastic process satisfying conditions i)-iii). Then,  T  T H,ε P us dBs − → us dBsH . 0

0

Proof.– For every ε > 0, we consider uεt =

n

i=1

where n =

T

ε

uti−1 1[ti−1 ,ti ) (t),  + 1 , ti =

iT n

uεT = uT ,

, i = 0, . . . , n.

For any t ∈ [0, T ], t belongs to some interval [ti−1 , ti ) for some i, then condition [3.12] implies that   1−H+δ 1−H+δ |uεt − ut | ≤ ut − uti−1  ≤ K(ω) |t − ti−1 | ≤ K(ω) |ti − ti−1 | ≤ K(ω)ε1−H+δ

a.s.

[3.14]

Approximation of fBm by Various Classes of Stochastic Processes

We have      T  T   T    H,ε H ε H us dBs − us dBs  ≤  (us − us ) dBs    0   0  0      T   T H,ε

   ε H,ε  ε H  + (us − us ) dBs  +  u d B s − Bs   0   0 s 

161

[3.15]

Let us consider the first  term on the  right-hand side of [3.15]. Fix a parameter α ∈ 1 − H, min 12 , 1 − H + δ . By [A1.42], we see that     T    ε H (us − us ) dBs  ≤ C(α) uε − u2,α B H 1,1−α    0

a.s.

[3.16]

for some constant C(α) > 0, where  · 1,1−α and  · 2,α are defined by [A1.39] and [A1.40], respectively. Further,  T ε  T t ε |us − us | |ut − ut − uεs + us | ε u − u2,α = ds + ds dt sα (t − s)α+1 0 0 0  T t ε |ut − ut − uεs + us | T 1−α ε ds dt ≤ sup |us − us | + 1 − α 0≤s≤T (t − s)α+1 0 0  T t ε |ut − ut − uεs + us | T 1−α ≤ ds dt. K(ω)ε1−H+δ + 1−α (t − s)α+1 0 0 [3.17] Note that for every fixed t ∈ [0, T ], i exists such that t ∈ [ti−1 , ti ). Then,  t ε |ut − ut − uεs + us | ds (t − s)α+1 0    t  i−1  tk 

uti−1 − ut − utk−1 + us  uti−1 − ut − uti−1 + us  ds + ds = (t − s)α+1 (t − s)α+1 tk−1 ti−1 k=1

≤

i−1 

k=1

tk

2K(ω)ε1−H+δ ds + (t − s)α+1 tk−1



t

1−H+δ

K(ω) |t − s| (t − s)α+1 ti−1

ds

=

 2K(ω)ε1−H+δ  K(ω) 1−H−α+δ (t − ti−1 )−α − t−α + (t − ti−1 ) α 1−H −α+δ

≤

2K(ω)ε1−H+δ K(ω)ε1−H−α+δ (t − ti−1 )−α + . α 1−H −α+δ

162

Fractional Brownian Motion

Therefore,  T 0

t 0

≤ =

n

|uεt − ut − uεs + us | ds dt = (t − s)α+1 i=1 n  ti 1−H+δ

2K(ω)ε α

i=1

ti−1

n  ti 1−H+δ

2K(ω)ε α

i=1

ti−1





ti ti−1

t 0

|uεt − ut − uεs + us | ds dt (t − s)α+1

(t − ti−1 )−α dt +

K(ω)ε1−H−α+δ T 1−H −α+δ

(t − ti−1 )−α dt +

K(ω)ε1−H−α+δ T 1−H −α+δ

[3.18]

Then, we can estimate n 

i=1

ti ti−1

(t − ti−1 )−α dt = =

n

n

1

1

(ti − ti−1 )1−α = 1 − α i=1 1 − α i=1

 1−α T n

T 1−α nα T 1−α [T /ε + 1]α T (1 + ε/T )α ε−α = ≤ , 1−α 1−α 1−α

and from [3.17] and [3.18], we obtain uε − u2,α ≤

T 1−α 2K(ω)ε1−H−α+δ T (1 + ε/T )α K(ω)ε1−H+δ + 1−α α(1 − α) +

K(ω)ε1−H−α+δ T → 0, 1−H −α+δ

as ε → 0.

[3.19]

Recall that B H has (H − γ)-Hölder continuous paths for all γ ∈ (0, H), i.e. a finite random variable Kγ (ω) exists such that for all t, s ∈ [0, T ],   H Bt − BsH  ≤ Kγ (ω) |t − s|H−γ

a.s.

For 0 < γ < α − 1 + H, we have      BtH − BsH   t BuH − BsH   H B  + du = sup 2−α 1,1−α (t − s)1−α 0≤s φε (s).

[3.25]

Approximation of fBm by Various Classes of Stochastic Processes

By virtue of the property of isometry, we have 2  E ΔBtH,ε − ΔBsH,ε ⎛   φε (t) 2 ⎝ 1−2H = CH u φε (s)

 +

s φε (t)

 +

t s

s

u1−2H

u1−2H

φ−1 ε (u)





t s

t u

2 (v − u) 3

H− 32 H− 12

v

1

(v − u)H− 2 v H− 2 dv

(v − u)

H− 32

v

H− 12

du

2

2 dv

dv

du ⎞ du⎠

=: I1 + I2 + I3 . We estimate each of these three integrals separately. For fBm, we have |t − s|

2H



2 = E BtH − BsH   t 2  t 3 1 2 = CH u1−2H (v − u)H− 2 v H− 2 dv du . s

u

Hence, I1 ≤

2 CH

I2 ≤

2 CH

2 I3 = CH

  

s

u

0 s

u

0 t s

1−2H



t

s

1−2H

u1−2H

 

t

s t

u

(v − u)

H− 23 H− 12

(v − u)

H− 23 H− 12

v v

3

2 dv

du ≤ |t − s|

2H

du ≤ |t − s|

2H

2 dv

1

(v − u)H− 2 v H− 2 dv

2 du ≤ |t − s|

2H

, , .

Therefore, 2  2H 2(H−δ) E ΔBtH,ε − ΔBsH,ε ≤ 3 |t − s| ≤ 3 |t − s| (t − φε (t))2δ ≤ 3 |t − s|

2(H−δ)

fε2δ .

167

168

Fractional Brownian Motion

Thus, estimate [3.25] is proved in case 1. Case 2: t > φε (t) > s > φε (s). In this case, we have 2  E ΔBtH,ε − ΔBsH,ε ⎛   s 2 ⎝ = CH u1−2H φε (s)

 +

φε (t)

u

s

 +

t φε (t)

1−2H



u1−2H

s

φ−1 ε (u)

u



φ−1 ε (u)

t

u

2 (v − u)

H− 32

v

H− 12

dv

du

2 (v − u)

(v − u)

H− 32

v

H− 32 H− 12

v

H− 12

2 dv

dv

du

⎞ du⎠

=: J1 + J2 + J3 . We estimate each of these three integrals as follows: J1 ≤

2 CH





s φε (s)

φ−1 ε (u) u

2H−1 

φ−1 ε (u) s

2 (v − u)

H− 32

dv

du

−2  s   2 1 2 H− 12 H− 12 ≤ CH (φ−1 K 2H−1 H − (u) − u) − (s − u) du ε 2 φε (s) −2  s 

2H−1 −1 1 2 2H−1 H− du φε (u) − s ≤ CH K 2 φε (s) 2

2H

2H K 2H−1 −1 CH = C1 (H) φ−1

φε (s) − s ε (s) − s 1 2 2H H − 2

2δ

2(H−δ) −1 2(H−δ) 2δ ≤ C2 (H) |t − s| fε , φε (s) − s = C1 (H) φ−1 ε (s) − s

≤

since φ−1 ε (s) < t. Further,  J2 ≤ C3 (H)  ≤ C3 (H)

φε (t) s φε (t) s



2H−1 du φ−1 ε (u) − u

(t − u)

2(H−δ)−1

−1

2δ φε (u) − u du

Approximation of fBm by Various Classes of Stochastic Processes

 ≤ C3 (H)

φε (t) s

≤ C4 (H) |t − s|

(t − u)

2(H−δ)

2(H−δ)−1

169

du · sup (t − φε (t))2δ t∈[0,T ]

fε2δ .

Finally, J3 ≤

2 CH



t φε (t)

2H−1 

≤ C1 (H) (t − φε (t))

1 H− 2

2H

−2 

t φε (t)

≤ C1 (H) |t − s|

(t − u)2H−1 du

2(H−δ)

(t − φε (t))

2δ

,

since s < φε (t) < t. Thus, estimate [3.25] is also proved in case 2. Note that, since ΔBtH,ε − ΔBsH,ε has a normal distribution, it follows from [3.25] that, for all p > 0, δ ∈ (0, H − γ), and s, t ∈ [0, T ], we have p    (H−δ)p δp E ΔBtH,ε − ΔBsH,ε  ≤ CH,p |t − s| fε ,

[3.26]

where CH,p is a certain constant. For what follows, we need an analog of estimate [3.25] uniform in t and s. It follows from the Garsia–Rodemich–Rumsey inequality [A1.32] that, for any p > 0 and α > 1/p, the following inequality is true:

sup

    H,ε ΔBt − ΔBsH,ε  |t − s|α−1/p

t,s∈[0,T ]

 ≤ Cα,p ξα,p ,

[3.27]

 where Cα,p is a certain deterministic constant and

ξα,p =

  T 0

T 0

  ΔBxH,ε − ΔByH,ε p |x − y|

αp+1

1/p dx dy

.

170

Fractional Brownian Motion

Take an arbitrary θ ∈ (γ, H) and put p = 2/(H − θ), α = (θ + H)/2, δ = (H − θ)/4. Then, p  T T  p E ΔBxH,ε − ΔByH,ε  E ξα,p = dx dy αp+1 |x − y| 0 0  T T (H−δ−α)p−1 |x − y| dx dy = CH,p,δ fεδp . ≤ CH,p fεδp 0

0

Taking [3.27] into account, we obtain

E sup

 p   H,ε ΔBt − ΔBsH,ε 

t,s∈[0,T ]

|t − s|pθ

H,θ fε(H−θ)p/4 , ≤C

H,θ is a certain constant. where C It follows from the latter estimate that, for any θ ∈ (γ, H) and κ ∈ (0, 1), a constant Cκ exists such that the probability of the event      θ Aε := ΔBtH,ε − ΔBsH,ε  ≤ Cκ |t − s| fε(H−θ)/4 for all s, t ∈ [0, T ] is not less than 1 − κ. On the set Aε , for all s, t ∈ [0, T ], we obtain    t θ−γ θ−γ−1 (H−θ)/4 |t − s| + |u − s| du Δs,t ≤ Cκ fε s

 −1 θ−γ |t − s| , = Cκ fε(H−θ)/4 1 + (θ − γ) whence   H,ε B − BH 

1,γ

 −1 T θ−γ . ≤ Cκ fε(H−θ)/4 1 + (θ − γ)

Then, for any a > 0, we have   lim P B H,ε − B H 1,γ ≥ a ≤ κ,

ε→0+

because  −1 T θ−γ < a. Cκ fε(H−θ)/4 1 + (θ − γ)

Approximation of fBm by Various Classes of Stochastic Processes

171

for sufficiently small ε. Thus, as κ → 0+, we obtain   lim P B H,ε − B H 1,γ ≥ a = 0. ε→0+



The theorem is proved. Remark 3.2.– Androshchuk [AND 06] proved that, for any β ∈ (0, 1 − H),   H P B − B H,ε  − → 0, ε → 0 + . 2,β He also established the convergence of the corresponding integrals 

T 0

P us dBsH,ε − →



T 0

us dBsH

[3.28]

for a stochastic process u belonging to C 1/2+δ [0, T ], where δ > 0. A stronger result, namely Theorem 3.4, was obtained in [RAL 11b]. In view of [A1.42], this theorem implies that the convergence [3.28] holds for any stochastic process u with trajectories from W2α [0, T ], where α > 1 − H. In particular, by [A1.41], it holds for u with (1 − H + δ)-Hölder continuous sample paths, where δ > 0. Mention also that in [RAL 11b], Theorem 3.4 was applied to the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with fBm. 3.4. Approximation of multifractional Brownian motion by absolutely continuous processes In this section, we consider another construction of absolutely continuous approximations. We propose the approach, which is suitable not only for fBm, but also for its generalization, the so-called multifractional Brownian motion, where the Hurst parameter H may depend on time t. 3.4.1. Definition and examples Assume that the function H : [0, T ] → with index γ > 12 , namely |Ht1 − Ht2 | ≤ C1 |t1 − t2 |

γ

for all t1 , t2 ∈ [0, T ] and some C1 > 0.

1

2, 1

satisfies the Hölder condition

[3.29]

172

Fractional Brownian Motion

There are several generalizations of fBm to the case where the Hurst index H is varying with time. Example 3.1.– Multifractional Brownian motion is introduced in [PEL 95]. The definition in [PEL 95] is based on the Mandelbrot–Van Ness representation for fBm (see [1.8]). According to [PEL 95], multifractional Brownian motion is defined by Yt = BtHt , where for t ≥ 0 and H ∈ ( 12 , 1),  0 ! 1 H− 12 H− 12

(t − s) dWs − (−s) BtH = Γ H + 12 −∞ "  t 1 [3.30] + (t − s)H− 2 dWs , 0

and where W is a Wiener process. Example 3.2.– In the papers [BOU 10, RAL 10], the authors define the so-called multifractional Volterra-type Brownian motion based on the representation of fBm in terms of the Molchan martingale (see [1.4]). The multifractional Volterra-type Brownian motion is the process Yt = BtHt , where  t H Bt = zH (t, s) dWs , t ≥ 0, [3.31] 0

W is a Wiener process, and zH (t, s) is the Molchan kernel defined by [1.5], namely  t 1 3 1 −H 1 2 zH (t, s) = cH (H − 2 )s (v − s)H− 2 v H− 2 dv. s

Example 3.3.– Consider another generalization, called the harmonizable multifractional Brownian motion (see, for example, [BEN 97, COH 99]). Let W (·) be a complex random measure on R such that: 1) for all A, B ∈ B(R) EW (A)W (B) = λ(A ∩ B), where λ is the Lebesgue measure; 2) for an arbitrary sequence {A1 , A2 , . . . } ⊂ B(R) such that Ai ∩ Aj = ∅ for all i = j, we have  W

# i≥1

 Ai

=



W (Ai ),

i≥1

(here {W (Ai ), i ≥ 1} are centered normal random variables);

Approximation of fBm by Various Classes of Stochastic Processes

173

3) for all A ∈ B(R) W (A) = W (−A); 4) for all θ ∈ R  d  iθ e W (A), A ∈ B(R) = {W (A), A ∈ B(R)} . We define the multifractional Brownian motion in this case by Yt = BtHt , where BtH =

 R

eitx − 1 1

|x| 2

+H

W (dx).

[3.32]

In what a generalization of fBm defined by Yt = BtHt ,  H follows, we consider 1  where Bt , t ∈ [0, T ], H ∈ 2 , 1 is a family of random variables such that:

  i) for a fixed H ∈ 12 , 1 , the process BtH , t ∈ [0, T ] is an fBm with Hurst parameter H; ii) for all t ∈ [0, T ],  2 E BtH1 − BtH2 ≤ C2 (δ)(H1 − H2 )2 , where H1 , H2 ∈ constant.

1 2

+ δ, 1 − δ

for some δ ∈

[3.33]

1

0, 4 , and where C2 (δ) is a

The above conditions are satisfied, for instance, by each of the generalizations described in Examples 3.1–3.3, since conditions i) and ii) hold for representations [3.30]–[3.32] (see [COH 99, PEL 95, RAL 10]). 3.4.2. Hölder continuity   We denote Hmin := min γ, mint∈[0,T ] Ht . Conditions i) and ii) imply that the trajectories of the process Yt = BtHt are continuous a.s. Indeed, the process BtHt with Ht = const is an fBm and thus  2 2H E BtHt − BsHt = |t − s| t

174

Fractional Brownian Motion

for all t, s ∈ [0, T ]. Now we use inequalities [3.29] and [3.33] and obtain the following bound for the second moment   2 2   2 Ht Ht Ht Hs E (Yt − Ys ) ≤ 2 E Bt − Bs + E Bs − B s  2H 2γ . ≤ 2 |t − s| t + C12 C2 |t − s|

[3.34]

Note that it suffices to show that the process Yt is continuous on every interval [a, b] ⊂ [0, T ] such that b − a < 1. Hence, we can assume |t − s| < 1 without loss of generality. Then,

2 2H E (Yt − Ys ) ≤ 2 1 + C12 C2 |t − s| min . Since Yt is a Gaussian process, the latter bound implies that, given an arbitrary r > 0, C > 0 exists such that r

E (|Yt − Ys | ) ≤ C |t − s|

rHmin

for all t, s ∈ [0, T ]. By the Kolmogorov continuity theorem, this means that the process Y is continuous a.s. Moreover, it is Hölder continuous of the order Hmin − ε for all ε ∈ (0, Hmin ). In the following lemma, we prove that the Hölder coefficient has finite moments of all orders. This is a generalization of a similar result for fBm [NUA 02, Lemma 7.4]. Lemma 3.1.– For every 0 < ε < Hmin , a random variable ηε > 0 exists such q that E (|ηε | ) < +∞ for all q > 0 and sup

s,t∈[0,T ]

|Ys − Yt | |s − t|

Hmin −ε

≤ ηε .

Proof.– Putting α = Hmin −ε/2 and p = 2/ε in the Garsia–Rodemich–Rumsey inequality [A1.32], we show that |Ys − Yt | ≤ CHmin ,ε |s − t|

Hmin −ε

ξ,

[3.35]

for all s, t ∈ [0, T ], where ξ=

  T 0

T 0

 2ε

2

|Yx − Yy | ε |x − y|

2Hmin ε

dx dy

.

Approximation of fBm by Various Classes of Stochastic Processes

175

First, we assume that q > 2/ε. Then, q

E(|ξ| ) ≤

  T 0

T 0

|x − y|

 qε 2

2

q

(E (|Yx − Yy | )) qε 2Hmin ε

dx dy

.

[3.36]

Considering bounds [3.4], we obtain that  2 2H E |Yx − Yy | ≤ cT |x − y| min , whence q

E (|Yx − Yy | ) ≤ cT,q |x − y|

qHmin

,

where cT and cT,q are some constants. Then, inequality [3.36] implies that q

E(|ξ| ) ≤ cT,q T qε < +∞ q

for q > 2/ε. By Lyapunov’s inequality, this means that E(|ξ| ) < +∞ for all q > 0. Choosing ηε = CHmin ,ε ξ and using [3.35], we prove Lemma 3.1.



3.4.3. Construction and convergence of approximations Consider the following approximations: BtHt ,ε :=

1 φt (ε)



t+φt (ε) t

BsHs ds =

1 φt (ε)



φt (ε) 0

H

u+t Bu+t du

[3.37]

for the multifractional Brownian motion, where φt (ε) = φ(t, ε) : [0, T ] × R+ → R+ is a family of measurable functions such that: 1) ψε := supt∈[0,T ] φt (ε) → 0 as ε → 0+; 2) for all t, s ∈ [0, T ] and for all ε > 0    φs (ε) − φt (ε)   ≤ C3 |t − s|Hmin .    φs (ε) Here, C3 is a constant that does not depend on ε.

[3.38]

176

Fractional Brownian Motion

We can take, for example, φt (ε) = ϕ(t)ε, where ϕ(t) is bounded away from zero and satisfies the Hölder condition with index Hmin . Our goal is to prove the almost sure convergence of approximations [3.37] to the multifractional Brownian motion in the norm  · 1,β defined by [A1.39]. To this end, we first establish the convergence of similar approximations for deterministic functions. Lemma 3.2.– Let f be a Hölder continuous function with index κ ∈ (0, Hmin ), i.e. κ

|f (t) − f (s)| ≤ K |t − s| ,

t, s ∈ [0, T ].

We define 1 fε (t) = φt (ε)



φt (ε) 0

f (u + t) du.

Then, for any β ∈ (0, κ), a constant C(K, κ, β) exists such that fε − f 1,β ≤ C(K, κ, β)ψεκ−β .

[3.39]

where ψε := supt∈[0,T ] φt (ε). Proof.– Write |fε (t) − f (t) − fε (s) + f (s)| ≤ I1 (t, s) + I2 (t, s) + I3 (t, s),

[3.40]

where  |φs (ε) − φt (ε)| φt (ε) I1 (t, s) = |f (u + t) − f (t)| du, φs (ε)φt (ε) 0    1  φt (ε)  |f (u + t) − f (t)| du , I2 (t, s) =   φs (ε)  φs (ε) I3 (t, s) =

1 φs (ε)



φs (ε) 0

|f (u + t) − f (t) − f (u + s) + f (s)| du.

Let us estimate each of three terms I1 (t, s) ≤

|φs (ε) − φt (ε)| φs (ε)φt (ε)

 0

φt (ε)

Kuκ du = K

κ+1

|φs (ε) − φt (ε)| (φt (ε)) · φs (ε)φt (ε) κ+1

.

Approximation of fBm by Various Classes of Stochastic Processes

177

Taking into account [3.38], we obtain I1 (t, s) ≤

KC3 H |t − s| min ψεκ . κ+1

Similarly,   |φt (ε) − φs (ε)| K  φt (ε) κ  H u du ≤ Kψεκ I2 (t, s) ≤ ≤ KC3 |t − s| min ψεκ .   φs (ε)  φs (ε) φs (ε) Hence, I1 (t, s) + I2 (t, s) ≤ 2KC3 |t − s|

Hmin

ψεκ .

[3.41]

In order to estimate I3 (t, s), we consider two cases. Case 1. φs (ε) ≤ |t − s|. I3 (t, s) ≤

1 φs (ε)

≤

2K φs (ε)

 

φs (ε) 0 φs (ε) 0

(|f (u + t) − f (t)| + |f (u + s) − f (s)|) du uκ du =

2K 2K (φs (ε))κ ≤ ψκ . κ+1 κ+1 ε

[3.42]

Case 2. φs (ε) > |t − s|. I3 (t, s) ≤

1 φs (ε)



φs (ε) 0

(|f (u + t) − f (u + s)| + |f (t) − f (s)|) du

κ

≤ 2K |t − s| .

[3.43]

Hence, in both cases, we have β

I3 (t, s) ≤ 2K |t − s| ψεκ−β . Let us estimate the norm fε − f 1,β =

[3.44]



|fε (t) − f (t) − fε (s) + f (s)| (t − s)β 0≤s0

Further, we rewrite the second term in the form   t  fε (u) − f (u) − fε (s) + f( s) du (u − s)1+β s  t  t I1 (u, s) + I2 (u, s) I3 (u, s) du + du. ≤ 1+β (u − s) (u − s)1+β s s

[3.47]

By [3.41],  t  t I1 (u, s) + I2 (u, s) κ du ≤ 2KC ψ (u − s)Hmin −1−β du 3 ε (u − s)1+β s s 2KC3 ψ κ (t − s)Hmin −β Hmin − β ε   2KC3 ≤ ψεκ−β max 1, T Hmin −β max ψεβ . ε>0 Hmin − β

=

[3.48]

Using [3.42] and [3.43], we obtain  t  s+φs (ε)  t I3 (u, s) I3 (u, s) I3 (u, s) du = du + du 1+β 1+β (u − s) (u − s) (u − s)1+β s s s+φs (ε)   s+φs (ε) du 2K κ t κ−1−β (u − s) du + ≤ 2K ψε 1+β κ+1 s s+φs (ε) (u − s)

2K 2K (φs (ε))κ−β + ψ κ (t − s)−β − (φs (ε))−β κ−β (κ + 1)(−β) ε   1 1 ψεκ−β . ≤ 2K [3.49] + κ−β β(κ + 1)

=

Combining [3.45]–[3.49], we obtain [3.39].



Now we are ready to prove the convergence of approximations [3.37]. To make expressions simpler, we put Yt = BtHt , Ytε = BtHt ,ε .

Approximation of fBm by Various Classes of Stochastic Processes

179

Theorem 3.5.– For any β ∈ (0, Hmin ), Y ε − Y 1,β → 0

a.s. as ε → 0 + .

Proof.– By Lemma 3.1, for any δ ∈ (0, Hmin ), |Yt − Ys | ≤ ηδ |t − s|

Hmin −δ

Hmin −β , 2

Hence, choosing δ =

. by Lemma 3.2, we obtain

Yε − Y 1,β ≤ C(ηδ , Hmin − δ, β)ψεδ → 0 a.s. as ε → 0 + .



Remark 3.3.– 1) A weaker form of Theorem 3.5 (with convergence in probability instead of almost sure convergence) was proved in the paper [RAL 11a], where it was applied for the approximation of solutions to stochastic differential equations driven by multifractional Brownian motion. 2) Similar approximations for anisotropic fractional and multifractional random fields in the plane were considered in [RAL 12]. 3) Similarly to Remark 3.2, we see that Theorem 3.5 implies the convergence of the integrals  0

T

us dBsHs ,ε →

 0

T

us dBsHs

a.s. as ε → 0+

for any stochastic process u with trajectories from C 1−Hmin +δ [0, T ], where δ > 0. Note that the case of Ht = H = const corresponds to the usual fBm.   Corollary 3.1.– Let BtH , t ∈ [0, T ] be an fBm with Hurst parameter H ∈ ( 21 , 1), and let the functions φt (ε) satisfy the following conditions: 1) supt∈[0,T ] φt (ε) → 0 as ε → 0+; 2) for all t, s ∈ [0, T ] and for all ε > 0    φs (ε) − φt (ε)   ≤ C |t − s|H ,    φs (ε) where C is a constant that does not depend on ε.

180

Fractional Brownian Motion

Then, the approximations BtH,ε

1 = φt (ε)



t+φt (ε) t

BsH ds

converge, namely  H,ε  B − B H 1,β → 0

a. s. as ε → 0+

for all β ∈ (0, H). 3.5. Exercises Exercise 3.1.– Using the inequality (see [MIS 08, Theorem 1.9.1]) 1/2   T

0

T

0

f (u)f (v) |u − v|

2H−2

du dv

≤ C(H) f L1/H ([0,T ]) ,

where C(H) is some positive constant, prove that for H > 1/2, well defined for β > −H.

T 0

sβ dBsH is

Exercise 3.2.– Prove the existence of the following Wiener integrals: T i) 0 eκs dBsH , where H ∈ (0, 1), κ ∈ R; T ii) 0 sin(κs) dBsH , where H ∈ (0, 1), κ ∈ R; T iii) 0 f (BsH ) dBsH , where H > 12 , f satisfies the Lipschitz condition, i.e. a constant C exists such that for all x, y ∈ R: |f (x) − f (y)| ≤ C |x − y| .

[3.50]

Exercise 3.3.– Let α < 1. Derive the asymptotic relationship (from the proof of Theorem 3.1)   n−α n−α ∼ as n → ∞ n Γ(1 − α) from Stirling’s formula: √ Γ(x) ∼ 2πxx−1/2 e−x

as

x → ∞.

[3.51]

Exercise 3.4.– Assume that H > 12 , the function f satisfies the Lipschitz condition [3.50]. Prove that the process us = f (BsH ) satisfies conditions i)-iii) of Theorem 3.3. (Hint: in order to verify condition iii), use Lemma A1.10 in Appendix 1.)

Appendix 1 Auxiliary Results from Mathematical, Functional and Stochastic Analysis

A1.1. Special functions Definition A1.1.– The gamma function is defined by the integral 

∞

Γ(x) =

sx−1 e−s ds,

[A1.1]

0

which converges for any x > 0. Here are some basic properties of the gamma function: i) the gamma function Γ(x) is positive, continuous and has continuous derivatives of all orders on (0, ∞); ii) Γ(x + 1) = x · Γ(x), x > 0;

  √ iii) Γ(1) = Γ(2) = 1, Γ(n) = (n − 1)!, n ∈ N, Γ 12 = π; iv) the gamma function is logarithmically convex on (0, +∞), i.e. its logarithm is convex on (0, +∞); v) the Euler reflection formula is Γ(x) · Γ(1 − x) =

π , 0 < x < 1; sin xπ

[A1.2]

vi) Legendre’s formula is 1  1  22x− 2 Γ(2x) = Γ(x)Γ x + 1 , x > 0; 2 (2π) 2

Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

[A1.3]

182

Fractional Brownian Motion

vii) Gauss’s formula for logarithmic derivative of gamma function has the following form:  1 Γ  (x) 1 − tx−1 = dt − γ, x > 0, [A1.4] Γ (x) 1−t 0 where γ is the Euler constant, γ = lim

 n

n→∞

k=1



1 − log n k

≈ 0.5772156649.

Definition A1.2.– The beta function, also called the Euler integral of the first kind, is defined by the integral 

1

B(x, y) = 0

sx−1 (1 − s)y−1 ds, x > 0, y > 0.

[A1.5]

Here are some basic properties of the beta function: i) for any x > 0, y > 0: B(x, y) = B(y, x); ii) B(x, 1) = x1 , x > 0; iii) the beta function is related to the gamma function by the formula: B(x, y) =

Γ(x) · Γ(y) . Γ(x + y)

[A1.6]

For the next definition, recall that the gamma function is strictly positive and infinitely differentiable on (0, ∞); therefore, the same can be said about its logarithm. Definition A1.3.– The digamma function, or the logarithmic derivative of d the gamma function, is defined as ψ(x) = dx (log Γ(x)) = Γ (x)/Γ(x); the trigamma function is defined as the derivative ψ  (z) and the tetragamma function is defined as ψ  (x). Now we give some auxiliary results that are used in the main chapters. Lemma A1.1.– For 0 < α < 12 , we have 3

Γ(α + 1) · (Γ(1 − α)) < Γ(1 − 2α).

[A1.7]

Appendix 1

183

Proof.– Inequality [A1.7] becomes an equality for α = 0:  3 Γ(α + 1) (Γ(1 − α)) 

α=0

  = Γ(1 − 2α)

= 1, α=0

or, in a logarithmic form, 

  log Γ(α + 1) + 3 log (Γ(1 − α)) 

α=0

  = log Γ(1 − 2α)

= 0. α=0

[A1.8]

Now, note that for two functions ϕ, ψ : [x0 , x1 ] −→ R, such that ϕ(x0 ) = ψ(x0 ) and ϕ (x) < ψ  (x), x ∈ (x0 , x1 ), we conclude that ϕ(x) < ψ(x), x ∈ (x0 , x1 ). Let ϕ(x) = log Γ(x + 1) + 3 log Γ(1 − x), ψ(x) = log Γ(1 − 2x), x0 = 0, x1 = 21 . x0 = 0 in [A1.8] implies that ϕ(x0 ) = ψ(x0 ) = 0. Let us calculate the derivatives: ϕ (x) =

Γ  (x + 1) Γ  (1 − x) −3 , Γ (x + 1) Γ (1 − x)

ψ  (x) = −2

Γ  (1 − 2x) . Γ (1 − 2x)

Using the Gauss formula [A1.4], we obtain 

 1 1 − tx 1 − t−x dt − C − 3 dt + 3C 1−t 0 1−t 0  1 −x 3t − tx − 2 dt, = 2C + 1−t 0  1 1 − t−2x dt + 2C. ψ  (x) = −2 1−t 0 ϕ (x) =

1

  Hence, it is sufficient to prove that for t ∈ (0, 1) and x ∈ 0, 12 3t−x − tx < 2t−2x . Let t−x =: y. Then, inequality [A1.9] is equivalent to the inequality 3y 2 − 1 < 2y 3 ,

[A1.9]

184

Fractional Brownian Motion

or which is equivalent for the positive y, (y − 1)2 (2y + 1) > 0. However, this inequality holds for all non-negative y = 1, in particular, for y = t−x , x > 0 and t ∈ (0, 1). Therefore, we have [A1.7].  The next result is based on Exercise 11.3, Chapter 2, from [OLV 74]. Lemma A1.2.– The logarithm of the gamma function admits the following expansion into Taylor series: log Γ(1 + z) = −γz +

∞  ζ(k)

k

k=2

(−z)k ,

|z| < 1,

[A1.10]

where γ is the Euler constant, and ζ(k) =

∞  1 k n n=1

is the Riemann zeta function. Proof.– This expansion can be obtained from infinite product representation of the gamma function (see [OLV 74, Chapter 2, equation (1.06)]): ∞  z  −z/n  1 e 1+ , = zeγz Γ(z) n n=1

Γ(1 + z) = e

−γz

[A1.11]

 ∞  z −1 z/n 1+ , e n n=1

log Γ(1 + z) = −γz +

 z  . − log 1 + n n

∞   z n=1

These equalities are correct for all such z that Γ(z) or, respectively, Γ(z + 1) is defined. Now assume that |z| < 1. Applying the expansion log(1 + x) =

∞  −(−x)k k=1

k

,

|x| < 1.

Appendix 1

185

We have  z  − log 1 + n n n=1

 ∞ ∞  z  −1  z k − = −γz + − n k n n=1

log Γ(1 + z) = −γz +

∞   z

k=1

= −γz +

∞  ∞  n=1 k=2

(−z)k . knk

For −1 < z ≤ 0, all items of the double series are non-negative and the double series is convergent. Hence, for all z, |z| < 1, the double series is absolutely convergent, which justifies summation interchange. Thus, log Γ(1 + z) = −γz +

∞  ∞  (−z)k knk n=1

k=2

= −γz +

∞ ∞ ∞   (−z)k  1 (−z)k = −γz + ζ(k). k k n=1 n k

k=2

k=2

We have ∞ proved that the expansion [A1.10] is correct for all z, |z| < 1. Since ζ(2) = n=1 n−2 = π 2 /6, log Γ(1 + z) = −γz +

π2 2 z + o(z 2 ) 12

as z → 0.

[A1.12]

The next two results describe the asymptotic behavior of the digamma function and of the binomial coefficients, respectively. Lemma A1.3.– The digamma function has the following asymptotics at infinity: lim (ψ(x) − log x) = 0.

x→+∞

[A1.13]

Proof.– The digamma function satisfies the functional equation (see [OLV 74], Chapter 2, Exercise 2.1): 1 ψ(z + 1) = ψ(z) + , z

z = 0, −1, −2, . . .

[A1.14]

186

Fractional Brownian Motion

Furthermore, the digamma function admits the following representation (see [OLV 74, Chapter 2, Exercise 2.2]): ∞

ψ(z) = −γ −

1  + z n=1



1 1 − n n+z

 z = 0, −1, −2, . . . ,

,

[A1.15]

This representation can be derived from the equality [A1.11] by taking the 1 logarithmic derivative of Γ(z) . Differentiating [A1.15], we obtain ψ  (z) =

∞ 

1 , (s + z)2 n=1

ψ  (z) =

∞ 

−2 . (s + z)3 n=1

[A1.16]

The function ψ  (z), which is the trigamma function, attains only positive values for z > 0. Hence, a well-known convexity of log Γ(z) on (0, +∞) follows. The tetragamma function ψ  (z) is negative and increasing on (0, +∞). Since ψ(z) satisfies equation [A1.14], we conclude that ψ(z) = ψ({z} + 1) +

1 1 1 1 + + ... + + , {z} + 1 {z} + 2 z−2 z−1

[A1.17]

for all z > 4, where {z} = z − z is the fractional part of z. From [A1.17] and boundedness of ψ(z) on interval [1, 2], the following asymptotic relation follows: ψ(x) ∼ log x

as x → +∞.

Moreover, let us establish that ψ(x) − log x → 0

as

x → +∞.

In order to do this, note that for n ∈ N and N ∈ N 1  −γ − + n N

k=1



1 1 − k k+n

 = −γ +

n−1  k=1

1  1 − . k k+n N

k=1

Taking limit as N → ∞ and using the representation [A1.15], we obtain ∞

1  ψ(n) = −γ − + n

k=1



1 1 − k k+n

 = −γ +

n−1  k=1

1 , k

Appendix 1

187

whence it follows that limn→∞ ψ(n) − log(n − 1) = 0, and so limn→∞ ψ(n) − log n = 0. Since the function ψ(x) is increasing, for all x > 1, ψ( x ) − log x − log

x = ψ( x ) − log x ≤ ψ(x) − log x x

≤ ψ( x ) − log x = ψ( x ) − log x + log

x , x

where x and x are the closest to x integers that are less than or equal to or, respectively, greater than or equal to x. Taking limit as x → +∞, we obtain [A1.13].  For n ∈ N ∪ {0} and x ∈ R, a binomial coefficient is defined as   x x(x − 1) · · · (x − n + 1) = . n n! Lemma A1.4.– For α ∈ R \ N, 

n−α n

 ∼

n−α Γ(1 − α)

as

n → ∞.

Proof.– We start from the equality 

n−α n

 =

(1 − α)(2 − α) · · · (n − α) . n!

Using functional equation for the gamma function, we obtain (1 − α)(2 − α) · · · (n − α) =

Γ(n − α + 1) . Γ(1 − α)

We also use that n! = Γ(n + 1). Hence 

n−α n

 =

Γ(n − α + 1) . Γ(1 − α)Γ(n + 1)

[A1.18]

For all n > min(−1, α − 1), by the Lagrange mean value theorem, θn exists such that log Γ(n + 1) − log Γ(n − α + 1) = αψ(n + θn ).

188

Fractional Brownian Motion

Here, θn lies between 1 − α and 1. As a result, all θn ’s make a bounded sequence. Using asymptotics [A1.13], we obtain log Γ(n + 1) − log Γ(n − α + 1) = α log(n + θn ) + o(1) = α log n + o(1) as

n → ∞.

[A1.19]

Due to [A1.18] and [A1.19],   n−α exp(log Γ(n − α + 1) − log Γ(n + 1)) = Γ(1 − α) n =

exp(−α log n + o(1)) Γ(1 − α)

∼

exp(−α log n) n−α = Γ(1 − α) Γ(1 − α)

as

n → ∞.



The next lemma is useful for comparison of various constants, introduced in Chapter 2. Lemma A1.5.– For any H ∈ (0, 1), the following equality holds: Γ( 32 − H) sin(πH)Γ(2H) . = Γ(2 − 2H) Γ(H + 12 ) Proof.– We use Euler’s reflection formula Γ(x)Γ(1 − x) =

π sin(πx) , x

i) Let 0 < H < 1/2. Then, on the one hand,  Γ

         1 3 1 1 1 = −H Γ H + −H Γ −H Γ H + 2 2 2 2 2 π(1/2 − H) π(1/2 − H) = . sin(π(1/2 − H)) cos(πH)

=

On the other hand, sin(πH)Γ(2 − 2H)Γ(2H) = sin(πH)(1 − 2H)Γ(1 − 2H)Γ(2H) =

π(1/2 − H) sin(πH)π(1 − 2H) = , sin(2πH) cos(πH)

and the proof follows.

∈ (0, 1).

Appendix 1

189

ii) Let 1/2 < H < 1. Then, on the one hand,  Γ

         1 3 3 1 1 = H− Γ −H Γ H + −H Γ H − 2 2 2 2 2 =

π(H − 1/2) π(H − 1/2) =− . sin(π(H − 1/2)) cos(πH)

On the other hand, sin(πH)Γ(2 − 2H)Γ(2H) = sin(πH)(2H − 1)Γ(2 − 2H)Γ(2H − 1) =

π(H − 1/2) sin(πH)π(2H − 1) =− , sin(π(2H − 1) cos(πH) 

and the proof follows. Now we establish the upper bound for the constant cH from [1.6]. Lemma A1.6.– For 0 < H < 1, the following inequality holds c2H < 2H,

[A1.20]

with equality if and only if H = 1/2. Proof.– Note that log Γ(x) is a strictly convex function for x ∈ (0, +∞). Therefore, for any x > 0, y > 0, the Jensen inequality holds: log Γ

x + y 2


0 for all k ∈ R.  For any H > 1/2, let us consider the constants, introduced via the equalities [2.2] and [2.49], respectively: c1 (H) =

α B(1 − α, α)cH , α+1

Appendix 1

191

and c3 (H) := c1 (H)(α + 1) = αB(1 − α, α)cH . Evidently, c3 (H) = αΓ(1 − α)Γ(α)cH =

πα cH , sin πα

whence c3 (H) > cH . Denote k20 :=

[A1.24]

c23 (H)−2H 2(c3 (H)−cH ) .

Lemma A1.8.– For any H > 1/2, the following inequality holds: k20 < 0. Proof.– It follows immediately from inequality [A1.7] that (Γ(1 − α))3 · Γ(α + 1) < 1. Γ(1 − 2α) Therefore 2

2

(c3 (H)) = (α Γ(1 − α)Γ(α)cH ) = (Γ(1 − α)Γ(α + 1)cH ) = 2H ·

2

(Γ(1 − α))3 · Γ(α + 1) < 2H. Γ(1 − 2α)

It means that the numerator of k20 is negative. From [A1.24], it follows that the denominator is positive. Therefore, k20 < 0.  A1.2. Slope functions: monotone rational function In section 2.1, we use such well-known facts from mathematical analysis: Definition A1.4.– Let f : R → R be a function of one variable. Function (x2 ) , x1 = x2 , x1 ,x2 ∈ R is called the slope function. g(x1 , x2 ) = f (xx11)−f −x2 Proposition A1.1.– [FIK 65] If f is strictly convex, then the function g strictly increases with respect to one variable for a fixed second.

192

Fractional Brownian Motion

Corollary A1.1.– If the function f is strictly convex and g is the slope function for fixed x1 and x2 , x1 = x2 , then g(x1 + α, x2 + α) strictly increases by α > 0. Now let us prove the result that is used in the proof of Lemma 2.10. Lemma A1.9.– If T > 0, for α ∈ (0, 1/2), the function ζ(t) :=

(T 2α+1 − t2α+1 )(T − t) (T α+1 − tα+1 )2

[A1.25]

is strictly monotonically decreasing on (0, T ) with respect to t. Proof.– It is sufficient to check that the derivative of the right-hand side of [A1.25] with respect to t is negative. Let us calculate this derivative:   − (2α + 1)t2α (T − t) − (T 2α+1 − t2α+1 ) (T α+1 − tα+1 )2 ∂ζ(t) + = ∂t (T α+1 − tα+1 )4 +

2(T 2α+1 − t2α+1 )(T − t)(T α+1 − tα+1 )(α + 1)tα . (T α+1 − tα+1 )4

[A1.26]

Since (T α+1 − tα+1 )4 > 0, it is sufficient to prove that the numerator of the right-hand side of [A1.26] is negative. We prove this fact by performing the following transformations: − (2α + 1)t2α T α+2 − T 3α+2 + (2α + 2)t2α+1 T α+1 + (2α + 1)t3α+1 T + T 2α+1 tα+1 − (2α + 2)t3α+2 + 2(α + 1)tα T 2α+2 − 2(α + 1)t3α+1 T − 2(α + 1)tα+1 T 2α+1 + 2(α + 1)t3α+2 = (−2α − 1)tα+1 T 2α+1 − t3α+1 T − (2α + 1)t2α T α+2 − T 3α+2 + (2α + 2)t2α+1 T α+1 + 2(α + 1)tα T 2α+2 = (2α + 1)tα T 2α+1 (T − t) − (2α + 1)t2α T α+1 (T − t) + tα T 2α+2 + t2α+1 T α+1 − t3α+1 T − T 3α+2 = (2α + 1)tα T α+1 (T α − tα )(T − t) − (T α − tα )T 2α+2 + t2α+1 T (T α − tα )   = T (T α − tα ) (2α + 1)tα T α (T − t) − T 2α+1 + t2α+1 .

Appendix 1

193

Since T (T α − tα ) > 0, it is sufficient to prove that (2α + 1)tα T α (T − t) − T 2α+1 + t2α+1 < 0. Divide [A1.27] by T 2α+1 . By substitution inequality to the following one:

[A1.27] t T

x2α+1 − (2α + 1)xα+1 + (2α + 1)xα − 1 < 0,

= x, we transform this

x ∈ (0, 1).

[A1.28]

Since at the point x = 0, expression [A1.28] is equal to −1, and at the point x = 1 it is equal to zero, it is sufficient to check that the function is increasing on [0, 1], i.e. its derivative is positive. Indeed, take the derivative of [A1.28] with respect to x and divide it by 2α + 1 to obtain x2α − (α + 1)xα + αxα−1 ,

x ∈ (0, 1).

[A1.29]

Divide [A1.29] by xα−1 ∈ (0, 1): xα+1 − (α + 1)x + α,

x ∈ (0, 1).

[A1.30]

Since at the point x = 0 the value of [A1.30] is equal to α, and at the point x = 1 it is equal to zero, it is sufficient to prove that the function is decreasing on (0, 1), i.e. its derivative is negative. Differentiate [A1.30] with respect to x: (α + 1)xα − (α + 1) < 0 for xα < 1, and the lemma is complete.  A1.3. Convex sets and convex functionals Let X be a real topological vector space. The dual of X is the vector space X ∗ = {continuous linear maps from X into R}. The following result is a version of the Hahn–Banach theorem from [SCH 97, Section 28.4, Theorem (HB20)]. Theorem A1.1.– (Separation of points from convex sets.) Let B be a nonempty closed convex subset of a real, locally convex topological vector space X. Let a ∈ X \ B. Then, Λ ∈ X ∗ exists such that Λ(a) < inf b∈B Λ(b). If X is a Hilbert space, then every element Λ ∈ X ∗ can be uniquely represented in the form Λ(x) = x, h for some h ∈ X, by the Riesz

194

Fractional Brownian Motion

representation theorem. Therefore, in the case X = L2 ([0, 1]), Theorem A1.1 can be rewritten as follows. Corollary A1.2.– Let B be a non-empty closed convex subset of L2 ([0, 1]) and a ∈ L2 ([0, 1]) \ B. Then, h ∈ X exists such that a, h < inf b∈B b, h. The following result is taken from [BAS 03, Proposition 2.3]. Proposition A1.2.– Let G be a closed convex subset of a Hilbert space. If f is a continuous and convex functional on G satisfying f (x) → ∞

whenever

x → ∞,

[A1.31]

then a point x0 ∈ G exists, satisfying f (x0 ) = minx∈G f (x). In particular, each continuous and convex functional f on L2 ([0, 1]) satisfying [A1.31] attains its minimal value. A1.4. The Garsia–Rodemich–Rumsey inequality The following result is called the Garsia–Rodemich–Rumsey inequality [GAR 71, GAR 74]. Theorem A1.2.– Let p > 0, α > p1 and f ∈ C([a, b]), then for all s, t ∈ [a, b], a constant Cα,p > 0 exists such that p

αp−1

|f (t) − f (s)| ≤ Cα,p |t − s|

 b a

b a

|f (x) − f (y)| αp+1

|x − y|

p

dx dy.

[A1.32]

Inequality [A1.32] is useful for studying the continuity of paths of Gaussian processes. In particular, the following bound for the Hölder constant of fractional Brownian motion (fBm) can be proved using [A1.32] (see [NUA 02, Lemma 7.4]).

 Lemma A1.10.– Let BtH , t ≥ 0 be an fBm of the Hurst index H ∈ (0, 1). Then, for every 0 < ε < H and T > 0, a positive random variable ηε,T exists p such that E |ηε,T | < ∞ for all p ∈ [1, ∞) and for all s, t ∈ [0, T ]   H Bt − BsH  ≤ ηε,T |t − s|H−ε

a.s.

Appendix 1

195

A1.5. Theorem on normal correlation Let X = (X1 , . . . , Xm ) and Y = (Y1 , . . . , Yn ) be two Gaussian vectors with non-singular covariance matrices. Assume that (X1 , . . . , Xm , Y1 , . . . , Yn ) is also a Gaussian vector. Denote mX = EX,

mY = EY,

ΣXX = Cov(X, X),

ΣXY = Cov(X, Y),

ΣY Y = Cov(Y, Y).

Theorem A1.3.– The conditional distribution of X given Y is Gaussian with parameters E(X | Y) = mX + ΣXY Σ−1 Y Y (Y − mY ),  Cov(X, X | Y) = ΣXX − ΣXY Σ−1 Y Y ΣXY .

For the proof, see, for example, [LIP 01, Th. 13.2]. A1.6. Martingales and semimartingales We propose some basic definitions related to stochastic processes, especially martingales. All stochastic processes are assumed to be real-valued. For more detailed acquaintance with the question, see, for example, [LIP 89, MIS 17, PRO 05]. Let (Ω, F, P) be a complete probability space. Definition A1.5.– A real-valued stochastic process on (Ω, F, P) parameterized by R+ is a set of random variables of the form X := {Xt , t ≥ 0}, Xt (ω) : R+ × Ω → R. Definition A1.6.– A set of σ-algebras F = (Ft )t≥0 is called a filtration if it satisfies the following assumptions: i) for any 0 ≤ s ≤ t Fs ⊂ Ft ⊂ F; ii) for any t ≥ 0 Fs = ∩t>s Ft ; iii) F0 contains all the sets from F of zero P-measure. Definition A1.7.– A stochastic process X = {Xt , t ≥ 0} is adapted to the filtration F = (Ft )t≥0 if Xt is Ft -measurable for any t ≥ 0.

196

Fractional Brownian Motion

Definition A1.8.– A stochastic process X = {Xt , t ≥ 0} is a martingale with respect to a filtration F = (Ft )t≥0 if it satisfies the following assumptions: i) X is adapted to the filtration F; ii) E|Xt | < ∞ for any t ≥ 0; iii) for any t ≥ s ≥ 0 E (Xt |Fs ) = Xs P-a.s. Definition A1.9.– A stochastic process X = {Xt , t ≥ 0} is called a Gaussian process if all its finite-dimensional distributions are Gaussian. Definition A1.10.– A Wiener process, or Brownian motion, is a stochastic process W = {Wt , t ≥ 0} satisfying the following assumptions: i) W0 = 0 a.s.; ii) W has independent increments; iii) for any t > 0, s ≥ 0, Wt+s − Ws have a normal distribution with zero mean and variance t. Proposition A1.3.– The definition A1.10 of the Wiener process W = {Wt , t ≥ 0} is equivalent to the following one: i) W is a Gaussian process; ii) EWt = 0, t ∈ [0, ∞); iii) Cov(Wt , Ws ) = min{t, s}, where t, s ∈ [0, ∞). Definition A1.11.– A stochastic process X = {Xt , t ≥ 0}, adapted to the filtration F = (Ft )t≥0 , is called a semimartingale, if it admits the representation (possibly, not unique) Xt = X0 + At + Mt , t ≥ 0,

[A1.33]

where M is an F-martingale, M0 = 0 a.s. and A is an F-adapted process with trajectories a.s. of bounded variation on any interval [0, t], t > 0, A0 = 0 a.s. A1.7. Integration with respect to Wiener process and fractional Brownian motions A1.7.1. Wiener integrals The term Wiener integration refers to the case where the integrand is a non-random function. Let W = {Wt , t ≥ 0} be a Wiener process and f =

Appendix 1

197

f (s) : [0, T ] → R be a non-random function. Let the function f be stepwise with respect to some partition 0 = t1 < . . . < tn = T of [0, T ], i.e. f (0) = f0 , f (s) = fk

for tk−1 < s ≥ tk ,

Then, we define the respective integral as Wtk−1 ).

k = 1, 2, . . . , n.

T

f (s) dWs :=

0

n k=1

fk (Wtk −

Let L2 ([0, T ]) = L2 ([0, T ], S, λ1 ) be a measurable space with the σ-algebra S of Lebesgue-measurable sets and the Lebesgue measure λ1 . Also, let f ∈ L2 ([0, T ]). Then, a sequence of stepwise functions {fn , n≥ 1} exists, which converges to the function  to the standard isometric  f in L2 ([0, T ]). Then, due T property, a sequence 0 fn (s) dWs , n = 1, 2, . . . converges in L2 (Ω, F, P). Therefore, we can define the Wiener integral 

T 0

 f (s) dWs = L2 (Ω, F, P) − lim

n→∞

T 0

T 0

f (s) dWs as the limit

fn (s) dWs .

[A1.34]

Now, let us define the Wiener integral as a function  t on an interval. Let f : [0, +∞) → R be a measurable function such that 0 f 2 (s) ds < ∞ for any t > 0 (alternatively, for any t ∈ [0, T ]). Then, the integral [A1.34], when we replace T for t, exists for any t > 0 and it can be proved that it is continuous in t modification (for the details of Wiener and general stochastic integration, see, for example, [MIS 17]). The integral



T

E 0

T 0

f (t)dWt is a centered Gaussian random variable with 2

f (s)dWs



T

=

f 2 (s)ds.

0

Moreover, the following Burkholder–Davis–Gundy inequality holds: for any p > 0, a constant Cp > 0 exists such that

 p  t   E sup  f (s)dWs  ≤ Cp 0≤t≤T 0

T

p/2 2

f (s)ds

.

[A1.35]

0

We can define the Wiener integral with respect to an fBm in the following way [NOR 99b]. For H > 12 , let us define an integral transform Γ:  Γf (t) := H(2H − 1)

∞ 0

f (s)|s − t|2H−2 ds,

198

Fractional Brownian Motion

and also define a scalar product with index Γ:  ∞ ∞ f, gΓ := (f, Γg) = H(2H − 1) f (s)g(t)|s − t|2H−2 dsdt, 0

0

where (·, ·) is the inner product in L2 ([0, ∞)). Let LΓ2 be a space of equivalent classes of measurable functions f such that f, f Γ < ∞. Mapping BtH → 1[0,t) can be extended to an isometry between the smallest closed linear subspace L2 (Ω, F, P) that contains {BtH , t ≥ 0}, and ∞ the LΓ2 space. It is possible now to define for f ∈ LΓ2 the integral 0 f (t)dBtH as an image of the function f in this isometry, and 

t 0

 f (s)dBsH

∞

= 0

f (s)1[0,t] (s)dBsH .

It can also be proved that continuous in t.

t 0

f (s)dBsH has a modification that is a.s.

A1.7.2. Elements of fractional calculus: respect to fBm

pathwise integration with

This section is devoted to the construction of the pathwise integral with respect to fBm. This approach was developed by Zähle [ZÄH 98, ZÄH 99]. It uses the following notions of fractional integrals and derivatives. Definition A1.12.– Let f ∈ L1 (a, b). The Riemann–Liouville left- and rightsided fractional integrals of order α > 0 are defined for a.a. x ∈ (a, b) by  x 1 α Ia+ f (x) := (x − y)α−1 f (y) dy, [A1.36] Γ(α) a  (−1)−α b α f (x) := (y − x)α−1 f (y) dy, Ib− Γ(α) x respectively, where (−1)−α = e−iπα . Definition A1.13.– For a function f : [a, b] → R, the Riemann–Liouville leftand right-sided fractional derivatives of order α (0 < α < 1) are defined by  x f (y) d 1 α Da+ f (x) := 1(a,b) (x) dy, Γ(1 − α) dx a (x − y)α  b f (y) (−1)1+α d α f (x) := 1(a,b) (x) dy. Db− Γ(1 − α) dx x (y − x)α

Appendix 1

199

α α Denote by Ia+ (Lp ) (respectively Ib− (Lp )) the class of functions f that can α α be presented as f = Ia+ ϕ (respectively f = Ib− ϕ) for ϕ ∈ Lp (a, b). α α (Lp ) (respectively f ∈ Ib− (Lp )), p ≥ 1, Proposition A1.4.– For f ∈ Ia+ the corresponding Riemann–Liouville fractional derivatives admit the following Weyl representation    x f (x) − f (y) f (x) 1 α Da+ f (x) = +α dy , α+1 Γ(1 − α) (x − a)α a (x − y)

  b f (x) − f (y) f (x) (−1)α α +α dy , Db− f (x) = α+1 Γ(1 − α) (b − x)α x (y − x)

where the convergence of the integrals holds pointwise for a.a. x ∈ (a, b) for p = 1 and in Lp (a, b) for p > 1. Let f, g : [a, b] → R. Assume that the limits f (u+) := lim f (u + δ) and δ↓0

g(u−) := lim g(u − δ) δ↓0

exist for a ≤ u ≤ b. Denote fa+ (x) = (f (x) − f (a+))1(a,b) (x), gb− (x) = (g(x) − g(b−))1(a,b) (x). 1−α α (Lp ), gb− ∈ Ib− (Lq ) for some Definition A1.14.– Assume that fa+ ∈ Ia+ 1 1 + ≤ 1, 0 < α < 1. The generalized (fractional) Lebesgue–Stieltjes integral p q of f with respect to g is defined by



b a



b

f (x) dg(x) :=(−1)α a

1−α α Da+ fa+ (x) Db− gb− (x) dx+

  + f (a+) g(b−) − g(a+) .

[A1.37]

Note that this definition is correct, i.e. independent of the choice of α ([ZÄH 98, Prop. 2.1]). If αp < 1, then [A1.37] can be simplified to 

b a



b

f (x) dg(x) := (−1)α a

1−α α Da+ f (x) Db− gb− (x) dx.

[A1.38]

200

Fractional Brownian Motion

In particular, Definition A1.14 allows us to integrate Hölder continuous functions. Let C λ ([a, b]), λ ∈ (0, 1], denote the space of Hölder continuous functions equipped with the norm f λ := f ∞ + sup

|f (t) − f (s)| λ

|s − t|

s,t∈[a,b] s=t

f ∞ = sup |f (t)|.

,

t∈[a,b]

Theorem A1.4.– [ZÄH 98, Th 4.2.1] Let f ∈ C λ ([a, b]), g ∈ C μ ([a, b]) with λ + μ > 1. Then, the assumptions of Definition A1.14 are satisfied with any α ∈ (1 − μ, λ) and p = q = ∞. Moreover, the generalized Lebesgue–Stieltjes b integral a f (x) dg(x) defined by [A1.37] coincides with the Riemann–Stieltjes integral 

b

f (x) dg(x) := lim



|π|→0

a

f (x∗i )(g(xi+1 ) − g(xi )),

i

where π = {a = x0 ≤ x∗0 ≤ x1 ≤ . . . ≤ xn−1 ≤ x∗n−1 ≤ xn = b}, |π| = maxi |xi+1 − xi |. Let T > 0, β ∈ (0, 1). Denote by W1β [0, T ] the space of measurable functions f : [0, T ] → R such that  f 1,β :=

sup

0≤s 0. Then, by Theorem A1.4, the integral 

t



t

u dB H := (−1)α

0

0

1−α H α (D0+ u)(x)(Dt− Bt− )(x)dx,

[A1.43]

exists for any t ∈ [0, T ] and coincides with the Riemann–Stieltjes integral; here, we may choose arbitrary α ∈ (1 − H, 1). A1.8. Existence of integrals of the Molchan kernel and its derivatives

 Let B H = B H , t ≥ 0 be fBm of Hurst index H ∈ (0, 1). According to t [1.4], it admits the integral representation BtH = 0 z(t, s) dWs , where z(t, s) is the Molchan kernel [1.5] and W is a Wiener process. In the next lemma, we prove the existence of integrals from Proposition 3.1. Recall the notation  s ϕεs = ∂1 z(s + ε, u) dWu , 0

∂z(t, s) ∂1 z(t, s) = = cH ∂t

 H−

1 2



tH−1/2 s1/2−H (t − s)H−3/2 ,

see [3.10]–[3.11]. Lemma A1.11.– Let t ∈ [0, T ], ε > 0. The functions z(t + ε, ·), z(· + ε, ·) and t ∂1 z(t + ε, ·) are square integrable on [0, t] and the integral 0 |ϕεs | ds is finite a.s. Proof.– 1. For the function z(t + ε, s), we have 

t 0

z 2 (t + ε, s) ds ≤

 0

t+ε

 H 2 z 2 (t + ε, s) ds = E Bt+ε = (t + ε)2H < ∞. [A1.44]

2. In order to prove the square integrability of z(s + ε, s), let us consider two cases. If H > 12 , then the function z(t, s) is non-decreasing in t and we see that  t  t z 2 (s + ε, s) ds ≤ z 2 (t + ε, s) ds < ∞, 0

0

202

Fractional Brownian Motion

by [A1.44]. For H < 12 , we write z(s + ε, s) = cH (s + ε)H−1/2 s1/2−H εH−1/2    s+ε 1 1/2−H + cH uH−3/2 (u − s)H−1/2 du −H s 2 s =: z  (s + ε, s) + z  (s + ε, s),

s > 0.

Then, the first term is square integrable:  t  t 2 (z  (s + ε, s)) ds = c2H ε2H−1 (s + ε)2H−1 s1−2H ds 0

0

≤ c2H ε2H−1 t1−2H



t 0

(s + ε)2H−1 ds < ∞.

Using the substitution u = s(y + 1), we can bound the second term as follows:    s+ε 1 z  (s + ε, s) = cH uH−3/2 (u − s)H−1/2 du − H s1/2−H 2 s    ε/s y H−1/2 1 = cH dy ≤ C1 sH−1/2 , − H sH−1/2 3/2−H 2 (y + 1) 0 where 

1 −H 2



∞

y H−1/2 dy (y + 1)3/2−H 0     1 1 = cH − H B H + , 1 − 2H . 2 2

C1 = cH

Therefore,  0

t

2

(z  (s + ε, s)) ds ≤ C12

Hence, the integral

t 0



t 0

s2H−1 ds < ∞.

z 2 (s + ε, s) ds is finite.

Appendix 1

203

3. For ∂1 z(t + ε, s), by [3.11], we obtain 

t

∂1 z(t + ε, s)

0

=

c2H



1 H− 2

2

ds

2 (t + ε)

t

s

1−2H

0



t 0

By the substitution s = 

2H−1

(t+ε)v v+1 ,

2H−3

(t + ε − s)

s1−2H (t + ε − s)2H−3 ds.

we obtain

ds = (t + ε)

−1



t/ε

v 1−2H dv =

0

t2−2H ε2H−2 . (2 − 2H)(t + ε)

Then, 

t

0

∂1 z(t + ε, s)

2

ds =

 2 c2H H − 12 (t + ε)2H−2 t2−2H ε2H−2 . 2 − 2H

[A1.45]

4. Using [A1.45], we can estimate for 0 ≤ s ≤ t 1/2  2 = E |ϕεs | ≤ E |ϕεs |  = 0

  E 

s

∂1 z(s + ε, u)

2

s 0

2 1/2  ∂1 z(s + ε, u) dWu  1/2

du

= C2 s1−H (s + ε)H−1 εH−1 ,

  where C2 = cH H − 21 (2 − 2H)−1/2 . Consequently, 

t

E 0

|ϕεs | ds ≤ C2 t1−H εH−1

Therefore, the integral

t 0



t

(s + ε)H−1 ds =

0

|ϕεs | ds is finite a.s.

C2 1−H H−1 ε (t + ε)H < ∞. t H 

Appendix 2 Evaluation of the Chebyshev Center of a Set of Points in the Euclidean Space

The Chebyshev center of a set is defined in section 1.8.2 in Chapter 1 (see Definition 1.2). To find the Chebyshev center of a finite set Z = {zi , i = 1, . . . , m} ⊂ Rn is to minimize maxz∈Z x − z = maxi=1,...,m x − zi  over x ∈ Rn . Note that the criterion function maxz∈Z x − z and its square are convex. Before we present the algorithm for finding the Chebyshev center, we state auxiliary optimization problems and present subroutines for finding their solutions. They are: the problem of the hypersphere of minimal radius that contains a given finite set and two constrained least squares problems. Then, we find necessary and sufficient conditions for the point to be the Chebyshev center of a finite set and formulate the dual problem. This immediately reduces the Chebyshev center problem to the fully constrained least squares (FCLS) problem. Then, we present two algorithms for finding the Chebyshev center of a finite set of affinely independent points. Each of the Chebyshev center algorithms can be used in section 1.8 for finding the best approximation by alternating minimization method. Each algorithm is presented twice: first, as a sequence of steps where the algorithm is mentioned, and second, in pseudocode at the end of Appendix 2. A2.1. Circumcenter of a finite set Definition A2.1.– Consider points z1 , . . . , zm in the Euclidean space Rn . A point x is called the circumcenter of {z1 , . . . , zm } if: 1) point x is equidistant from z1 , . . . , zm , i.e. x − z1  = x − z2  = . . . = x − zm ;

Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

206

Fractional Brownian Motion

2) point x is the closest to {z1 , . . . , zm } among all equidistant points, i.e. x − zm  =

min

y∈Rn :y−z1 =...=y−zm 

y − zm .

In other words, the circumcenter is the solution to the optimization problem  x − zm 2 → min, provided that x − z1  = x − z2  = . . . = x − zm .

[A2.1]

The circumcenter may exist and may not exist, because the constraints in the optimization problem can be incompatible. However, if it exists, it is unique, as shown in the following lemma. Lemma A2.1.– If the constraints in optimization problem [A2.1] are compatible, problem [A2.1] has a unique solution. Proof.– A point x satisfies the constraints in optimization problem [A2.1] if and only if x − zi  = x − zm  for all

i = 1, . . . , m − 1,

which is equivalent to (x − zm ) − (zi − zm )2 = x − zm 2 , 2

i = 1, . . . , m − 1;



x − zm  − 2(zi − zm ) (z − zm ) + zi − zm 2 = x − zm 2 , i = 1, . . . , m−1; 2(zi − zm ) (x − zm ) = zi − zm 2 ,

i = 1, . . . , m − 1.

[A2.2]

Thus, the constraint set is defined by a system of linear equations and is an affine subspace in Rn . Since the constraint set is closed and non-empty and the criterion function x − zm 2 is continuous and tends to +∞ as x → ∞, the constrained minimum exists. As the constraint set is convex and the criterion function x − zm 2 is strictly convex, the constrained minimum is reached at a unique point.  Definition A2.2.– Let z1 , . . . , zm be the points in Rn . The sphere {z ∈ Rn : z − a = R} of the minimal radius on which all the points zi are located is called the circumsphere of {z1 , . . . , zm }. The circumsphere exists if and only if the circumcenter exists. The circumcenter is the center of the circumsphere.

Appendix 2

207

Now find the circumcenter of {z1 , . . . , zm } under the assumption that the points z1 , . . . , zm are affinely independent, i.e. for w1 ∈ R, . . . , wm ∈ R, the m m equalities w z = 0 and w = 0 imply that wi = 0 for all i. i=1 i i i=1 i Let x be the solution to the problem [A2.1]. Then, x minimizes x − zm 2 under constraints that will be used in form [A2.2]. The Lagrange function is x − zm 2 −

m−1 

  wi 2(zi − zm ) (x − zm ) − zi − zm 2 ,

i=1

where wi ∈ R are the Lagrange multipliers. The Lagrange equation is x − zm =

m−1 

wi (zi − zm ).

[A2.3]

j=1

Substitute [A2.3] into constraints [A2.2]: 2

m 

(zi − zm ) (zj − zm )wi = zi − zm 2 ,

i = 1, . . . , m − 1.

[A2.4]

j=1

Denote n × (m − 1) matrix Z and m−1-dimensional vector w: ˇ Z = [z1 − zm , . . . , zm−1 − zm ],

[A2.5]

w ˇ = (w1 , . . . , wm−1 ) . Note that zi − zm 2 is a diagonal element of the matrix Z  Z. With this notation, equation [A2.4] can be rewritten as 2Z  Z w ˇ = diag(Z  Z ),

[A2.6]

where diag(Z  Z ) is a vector created of diagonal elements of the matrix Z  Z, and equation [A2.3] can be rewritten as x = zm + Z w. ˇ

[A2.7]

Affine independence of the vectors zi implies that the matrix Z is of full rank (this seems to be obvious; however, the formal proof is presented in section A2.2) and equation [A2.6] has a unique solution w ˇ=

1  −1 (Z Z ) diag(Z  Z ). 2

[A2.8]

208

Fractional Brownian Motion

Theorem A2.1.– If m ≥ 2 and the points z1 , . . . , zm are affinely independent, then the circumcenter of {z1 , . . . , zm } exists and is equal to 1 x = zm + Z(Z  Z )−1 diag(Z  Z ), 2

[A2.9]

where the matrix Z is defined in [A2.5]. Proof.– First, we prove that the constraints in optimization problem [A2.1] are compatible. Vector w ˇ defined in [A2.8] and vector x defined in [A2.7] satisfy equations [A2.3], [A2.4] and [A2.6] and constraints in problem [A2.1]. Thus, constraints in [A2.1] are compatible. By Lemma A2.1, the solution to optimization problem [A2.1] exists. By [A2.6] and [A2.7], it satisfies [A2.9].  Due to Theorem A2.1, the following algorithm finds the circumcenter of points z1 , . . . , zm as long as z1 , . . . , zm are affinely independent. We present the algorithm in two formats: here as a sequence of steps, and in section A2.5 in pseudocode (see Algorithm A2.1). Algorithm for finding the circumcenter of a finite set Z: Step 1. If m = 1, then take the trivial solution z1 as the circumcenter and stop. Step 2. Create n × (m − 1) matrix Z = [z1 − zm , . . . , zm−1 − zm ]. Step 3. Set w ˇ = 12 (Z  Z )−1 diag(Z  Z ). ˇ Take x as the circumcenter. Step 4. Set x = zm + Z w. A2.2. Constrained least squares The problem of finding a minimal-norm point in the convex hull of a finite number of affinely independent points is studied in [WOL 76], together with the algorithm for solving the problem. Another version of the algorithm is presented in [CHA 03, Section 10.3]. A point in the convex hull of a finite set {x1 , . . . , xm } can be represented  m as m i=1 wi xi with non-negative coefficients wi , which satisfy the constraint i=1 wi = 1. Finding the coefficients wi of the minimal-norm element is regarded as the FCLS (see the definition later in this section, equation [A2.15]).

Appendix 2

209

A2.2.1. Two constrained least squares problems: SCLS and FCLS Let X be an n × m matrix with affinely independent columns. The affine independence of the columns means that  rank



X 1 ... 1

= m.

[A2.10]

Hence, m ≤ n + 1. Let y ∈ Rn be a vector. Definition A2.3.– Sum-constrained least squares (SCLS) problem is the problem of finding the vector w ∈ Rm that minimizes Xw − y under the m constraint i=1 wi = 1: Xw − y → min

provided that

m 

wi = 1.

[A2.11]

i=1

Here, Xw − y is the norm in Euclidean space, so that the norm of the   n 2 1/2 . vector y equals y = i=1 yi Definition A2.4.– The FCLS problem is the problem of finding the vector w ∈ m R m with non-negative elements that minimizes Xw − y under the constraint i=1 wi = 1:  Xw − y → min provided that w1 ≥ 0,

...,

wm ≥ 0,

m

i=1

wi = 1.

[A2.12]

Denote x1 , x2 , . . . , xm the columns of the matrix X: X = [x1 , x2 , . . . , xm ]. If w is the solution to the SCLS problem [A2.11], then Xw is the projection of the point y onto the affine subspace spanned by the points x1 , x2 , . . . , xm . If w is the solution to the FCLS problem [A2.12], then Xw is the point from the convex hull of x1 , x2 , . . . , xm that is nearest to y. Denote X1 = X − y(1, . . . , 1) = [x1 − y, . . . , xm − y]. Then, under the constraints of either [A2.11] or [A2.12], we have Xw − y = X1 w.

[A2.13]

210

Fractional Brownian Motion

Hence, the SCLS problem [A2.11] is equivalent to X1 w → min

provided that

m 

wi = 1,

i=1

and the FCLS problem [A2.12] is equivalent to X1 w → min

provided that

w1 ≥ 0,

...,

wm ≥ 0,

m 

wi = 1.

i=1

Thus, without loss of generality, we can assume that y = 0. Therefore, consider the SCLS and FCLS problems with y = 0: Xw → min

provided that

m 

wi = 1;

[A2.14]

i=1

Xw → min wm ≥ 0,

m 

provided that

w1 ≥ 0,

...,

wi = 1.

[A2.15]

i=1

Theorem A2.2.– Provided that the columns of the matrix X are affinely independent, both the SCLS and FCLS problems have a unique solution. Proof.– First, prove the theorem for the case y = 0. m On the constraint sets {w ∈ Rm : i=1 wi = 1} (for the SCLS problem)  m and m {w ∈ [0, 1]m : i=1 wi = 1} (for the FCLS problem), the equality i=1 wi = (1, . . . , 1)w = 1 holds true, whence 

   X Xw w= . 1 ... 1 1

Hence, the alternative criterion function       2 X X X − 1 = w Q1 (w) = w−1 w 1 ... 1 1 ... 1 1 ... 1 coincides with the squared criterion function Xw2 on the constraint sets. Since Q1 (w) is continuous and tends to +∞ as w → ∞, and the constraint sets for the SCLS and FCLS problems are closed and non-empty, the solutions to the SCLS [A2.14] and FCLS [A2.15] problems exist. Since Q1 (w) is a strictly

Appendix 2

211

convex set and the constraint sets are convex, the solution to any of the SCLS and FCLS problems is unique. Let us establish that the assumption y = 0 does not lead to loss of generality. m Under the constraint i=1 wi = 1, we have that Xw − y = X1 w, Xw − y2 = X1 w2 , where the matrix X1 is defined in [A2.13]. Therefore, we can transform the optimization problem to an equivalent one with y = 0 by changing the matrix X. It remains to prove that if the columns of the matrix X are affinely independent, then the columns of the matrix X1 are also affinely independent. The rest of the proof is devoted to the latter implication. Because of the equality        X1 X − y(1, . . . , 1) I −y X = = 1 ... 1 1 ... 1 0 1 1 ... 1 (where I is the n × n identity matrix, so ranks of matrices in [A2.16] are equal:     X1 X rank = rank . 1 ... 1 1 ... 1

I

−y 0 1



[A2.16]

is a non-singular matrix), the

We have the following equivalent properties: the matrix X has affinely independent columns if and only if   X rank = m, 1 ... 1 which holds true if and only if   X1 rank = m, 1 ... 1 which in turn holds true if and only if the matrix X1 has affinely independent columns.  A2.2.2. SCLS algorithm Now, construct the solution to the SCLS problem [A2.14]. Denote m×(m−1) matrix T and m-dimensional vector em : ⎞ ⎛ ⎞ ⎛ 0 1 0 ⎟ ⎜ .. ⎟ ⎜ . . ⎟ ⎜ ⎟ ⎜ . em = ⎜ . ⎟ . T =⎜ ⎟, ⎝0⎠ ⎝0 1⎠ 1 −1 . . . −1

212

Fractional Brownian Motion

Then w = Tw ˇ + em ,

w ˇ ∈ Rm−1 ,

[A2.17]

m is a representation of the vector w that satisfies the constraint i=1 wi = 1 from [A2.14]. Equation [A2.17] establishes one-to-one correspondence between m Rm−1 and the constraint set {w ∈ Rm : i=1 wi = 1}. Affine independence of the columns of the matrix X implies the linear independence of the columns of the matrix XT . Indeed, we have that (1, . . . , 1)T = 0, whence 

XT 0



 =

 X T. 1 ... 1

[A2.18]

The factors on the right-hand side of [A2.18] are matrices with linearly independent columns. Therefore, their product is also a matrix with linearly independent columns, and  rank(XT ) = rank

XT 0

 = m − 1.

Thus, XT is a full-rank matrix with linearly independent columns. Substitute [A2.17] into [A2.14]. The resulting unconstrained minimization problem XT w ˇ + Xem  → min

[A2.19]

has a solution w ˇ = −(T  X  XT )−1 T  X  Xem . Denote X2 = XT = [x1 − xm , . . . , xm−1 − xm ] and note that Xem = xm . Then, the minimizer in [A2.19] is equal to w ˇ = −(X2 X2 )X2 xm .

[A2.20]

Lemma A2.2.– If w ∈ Rn is the minimizer in the SCLS problem [A2.14], then all the elements of the vector X  X w are equal.

Appendix 2

213

Proof.– Here, the notations w ˇ and T are the same as above. Vector w ˇ minimizes XT w ˇ + Xem , whence ˇ + Xem ) = 0. T  X  (XT w The minimizer w is represented as w = T w ˇ + em , whence ˇ + em ) = T  X  (XT w ˇ + Xem ) = 0. T  X  X w = T  X  X(T w The null space of the matrix T  consists of m-dimensional vectors with all elements equal. Thus, all the elements of the vector X  Xw are equal.  Now we present the algorithm for solving problem [A2.14]. If m = 1, then the constraint set consists of a single point and the solution is trivial, i.e. w = 1. Otherwise, we solve [A2.20] and [A2.17]. It is described in Algorithm A2.2, section A2.5. mAlgorithm for solving the SCLS problem to minimize Xw under constraint i=1 wi = 1 has the following steps: Step 1. If X is a one-column matrix (i.e. if m = 1), then take the trivial solution w = 1 and stop. Step 2. Construct an m−1-column matrix X2 : X2 := [x1 − xm , . . . , xm−1 − xm ], where x1 , . . . , xm are columns of the matrix X. Step 3. Construct m-dimensional vector w, whose first m − 1 elements are equal to the elements of the vector −(X2 X2 )−1 X2 xm : ⎛

⎞ w1 ⎜ .. ⎟  −1  ⎝ . ⎠ = −(X2 X2 ) X2 xm wm−1 and the last element is equal to wm = 1 −

m−1 

wi .

i=1

A2.2.3. FCLS algorithm Now we solve the FCLS problem [A2.15]. Let D be any convex set in Rm . Also let Dj = {wj : w ∈ D} be the projection of the set D onto jth axis.

214

Fractional Brownian Motion

Definition A2.5.– 1) The point w0 is called a feasible point of a minimization problem f (w) → min

provided that

w∈D

if w0 satisfies the constraints, i.e. if w0 ∈ D. 2) For the function f : Rm ⊃ D → R and for the fixed index j, 1 ≤ j ≤ m, the function B : Dj → R defined by formula B(x) =

inf

w∈D:wj =x

f (w)

is called the profile of the function f . The profile of a convex function is a convex function. Lemma A2.3.– The vector w is a solution to the FCLS problem if and only if the following holds mtrue: w is the feasible point in the sense that wi ≥ 0 for i = 1, . . . , m and i=1 wi = 1, and the elements of the vector  = X Xw satisfy the condition 

i −

 min j wi = 0

j=1,...,m

for all i = 1, . . . , m.

[A2.21]

Now we present the algorithm for finding the minimizer in the FCLS problem [A2.14]. In the algorithm, letter J means the set, J ⊂ {1, 2, . . . , m}; X[·, J] is the submatrix of the matrix X constructed of the columns of the matrix X with indices within the set J; w[J] = (wi , i ∈ J) is the vector made of the elements of vector w with indices within J. Algorithm for solving the FCLS problem [A2.15], which consists of the m minimization of Xw under constraints i=1 wi = 1 and wi ≥ 0 for i = 1, . . . , m, has the following form: Step 1. Create an m-dimensional vector w0 that satisfies the constraints [A2.15]. Step 2. Put J = {i : w0i > 0}. Step 3. In order to find the minimizer in the problem  Xw → min m provided that i=1 wi = 1 and wi = 0 for all i ∈ J,

[A2.22]

Appendix 2

215

we consider m-dimensional vector w with elements w[J] = SCLS(X[·, J]), wi = 0

for all

i ∈ J.

Step 4. If all the elements of vector w are non-negative, go to step 6. Otherwise, go to step 5. Step 5. Move w0 toward w as far as it satisfies the constraints. Make sure that after the movement, all the elements of w0 are non-negative. Also make sure that at least one element of w0 that was previously positive becomes 0. Go to step 2. Step 6. Calculate the vector  = X  Xw. Step 7. If the minimal element of the vector  is reached for some index i ∈ J, i.e. if min i = min i ,

i=1,...,m

i∈J

then fix w as the result and terminate the algorithm. Otherwise, go to step 8. Step 8. Select an index j such that wj = mini=1,...,m wi . Step 9. Set w0 = w. i = j:

Step 10. Let the set J consist of indices i for which either w0i > 0 or

J := {i : w0i > 0} ∪ {j}. Go to step 3. The FCLS algorithm is presented in section A2.5 as Algorithm A2.3. In the structured representation of the algorithm, step 5 is explained in detail in lines 8–12. Theorem A2.3.– Let X be the matrix with affinely independent columns. If all the computations are exact, then Algorithm A2.3 ends and gives the solution to problem [A2.15].

216

Fractional Brownian Motion

Proof.– First, prove that if the algorithm ends, then the vector w, which it calculates, will be the solution to the FCLS problem [A2.15]. When the algorithm ends, all the elements of vector w are non-negative, w[J] = SCLS(X[·, J]), wi = 0 for all i ∈ J, and for vector  = X  Xw, we have that min i = i∈J

min i .

i=1,...,m

[A2.23]

m  Thus, i=1 wi = i∈J wi = 1, and vector w satisfies the constraints of optimization problem [A2.15]. Due to Lemma A2.2, all elements of the vector (X[·, J]) X[·, J]w[J] are equal. Since [J] = (X[·, J]) Xw = (X[·, J]) X[·, J]w[J], all the elements of the vector (i , i ∈ J) = w[J] are equal. Due to [A2.23], i =

min j

j=1,...,m

for all i ∈ J,

whence [A2.21] follows. Thus, vector w satisfies Lemma A2.3 and minimizes [A2.15]. Now prove that the algorithm ends. On every iteration of the inner cycle, the finite set J becomes smaller. Hence, on each iteration of the outer cycle, the inner cycle is iterated only a finite number of times. Thus, it is enough to prove that the outer cycle is iterated only a finite number of times. Below we prove this fact. At lines 4–6 or at lines 14–16 of Algorithm A2.3 in section A2.5, w becomes the minimizer of problem [A2.22]. After the execution of line 1, w0 is a feasible  point. Indeed, it is ensured that all elements of w0 are non-negative, and m i=1 w0i = 1 after line 1. On execution of line 10, w0 becomes a convex combination of its previous value m and vector w, which satisfies w = 1. At line 22, we set w0 = w. Thus, i i=1 m the equality i=1 w0i = 1 never becomes false. During the execution of the algorithm, Xw0 never increases. Indeed, after line 6 or 16, w minimizes [A2.22] while w0 satisfies the constraints in [A2.22].

Appendix 2

217

Then, Xw ≤ Xw0. We set w0 = w at line 22 and w0 = (1−αmin )w0 before + line 10

αmin w at line 10. Xw0 after line

The norm Xw is a convex function in w. Hence = (1 − αmin )Xw0 before + αmin Xw 10 line 10 ≤ (1 − αmin ) Xw0 before + αmin Xw ≤ Xw0 before . line 10

line 10

Now prove that Xw0 strictly decreases on each iteration of the outer cycle starting from the second one. Note that before line 23 j ∈ J. Denote C = i for i ∈ J on line 18 (for all i ∈ J, the elements i of the vector  are equal). After line 21, j < C. For j obtained in line 21 and J obtained in line 23 and any x ∈ R, denote B(x) =

min

wj =x wi =0 for all i∈J m wi =1

Xw2 .

[A2.24]

i=1

The function B(x) is convex as the profile of a convex function. If x = 0, then the minimum in [A2.24] is reached for w = w0, so that B(0) = Xw2 = Xw02 after line 21. Later on, w is redefined in lines 4–6. With tedious computations, we can prove that the derivative of B(x) at x = 0 is equal to j − C < 0, whence arg min B(x) > 0 x∈R

and

min B(x) < B(0). x∈R

Then, after lines 4–6 wj = arg min B(x) > 0, x∈R

2

Xw = min B(x) < B(0) = Xw02 , x∈R

Xw < Xw0. If the inner cycle is skipped, then Xw0 strictly decreases on line 22. Otherwise, on the first iteration of the inner cycle j ∈ {i ∈ J : wi < 0}, αmin > 0 and Xw0 strictly decreases on line 10. Thus, Xw0 strictly decreases on each iteration of the outer cycle. The vector w can reach only a finite number of different values because there are only a finite number of sets J ⊂ {1, 2, . . . , m}. Thus, Xw can reach only

218

Fractional Brownian Motion

a finite number of different values. After line 22, w0 = w, and because Xw0 strictly decreases on each iteration of the outer cycle, Xw strictly decreases between runs of line 22. Hence, line 22 is performed only a finite number of times. The outer cycle has only a finite number of iterations, and as it was discussed earlier, the algorithm ends.  A2.2.4. Relation between solutions to the SCLS and FCLS problems Here, we consider the SCLS [A2.14] and FCLS [A2.15] problems. Lemma A2.4.– Let X be an n × m matrix with affinely independent columns. Let wSCLS and wFCLS be the solutions to the SCLS and FCLS problems, respectively. Then, one of the following statements holds true: 1) the equality wSCLS = wFCLS holds true (as a result, all the elements of the vector wSCLS are non-negative); 2) an index i exists for which wSCLS,i < 0 and wFCLS,i = 0. Proof.– Assume that statement 2) does not hold true, i.e. wSCLS,i ≥ 0

for all such i that wFCLS,i = 0.

[A2.25]

Let us prove that statement 1) holds true, i.e. wSCLS = wFCLS . Indeed, by Lemma A2.2, all elements of the vector X  XwSCLS are equal. Denote SCLS the element of X  XwSCLS . Then X  XwSCLS = (1, . . . , 1) SCLS .

[A2.26]

Denote X  XwFCLS = (1 , . . . , m ) , and min =

min i .

i=1,...,m

Then, by Lemma A2.3 (see equality [A2.21]), wFCLS,i i = wFCLS,i min

for all i = 1, . . . , m.

[A2.27]

m [A2.26] and [A2.27] in the following transformations. Since mWe use equalities w = i=1 SCLS,i i=1 wFCLS,i = 1, we have that   XwSCLS 2 = wSCLS X  XwSCLS = wSCLS (1, . . . , 1) SCLS = SCLS ,

Appendix 2

219

and  XwFCLS 2 = wFCLS X  XwFCLS

=

m 

wSCLS,i i =

i=1

m 

wSCLS,i min = min ,

i=1

whence SCLS = XwSCLS 2 = ≤

min m

w∈[0,1]m :

i=1

min  m

w∈Rm :

wi =1

i=1

wi =1

Xw2

Xw2 = XwFCLS 2 = min .

[A2.28]

 On the one hand, we can evaluate wFCLS X  XwSCLS as follows:   wFCLS X  XwSCLS = wFCLS (1, . . . , 1) SCLS = SCLS .

[A2.29]

On the other hand,  wFCLS X  XwSCLS =

m 

i wSCLS,i

i=1

=

m 

min wSCLS,i +

i=1

= min +

m 

(i − min )wSCLS,i

i=1 m 

(i − min )wSCLS,i .

[A2.30]

i=1

The inequality (i − min )wSCLS,i ≥ 0

[A2.31]

can be verified by considering two cases. If wFCLS,i > 0, then [A2.31] becomes the equality due to [A2.27]. Otherwise, if wFCLS,i = 0, then [A2.31] follows from assumption [A2.25]. It follows from [A2.28], [A2.29], [A2.30] and [A2.31] that SCLS = min . Hence, wFCLS is also a solution to the SCLS problem. By Theorem A2.2, wFCLS = wSCLS . 

220

Fractional Brownian Motion

A2.3. Dual problem The Chebyshev center of the non-empty finite set {z1 , z2 , . . . , zm } minimizes the functional Q(y) = max y − zi 2 . i=1,...,m

The Chebyshev center of a non-empty finite set exists and is unique (see Definition 1.2 and Corollary 1.6 in section 1.8.2). Theorem A2.4.– Let z1 , . . . , zm and x be the points in Rn . Then, the following statements are equivalent: I) x is the Chebyshev center of {z1 , z2 , . . . , zm }. II) Non-negative numbers w1 , . . . , wm and R exist such that: m a) wi ≥ 0 for all i = 1, . . . , m, and i=1 wi = 1; b) zi − x ≤ R for all i=1, . . . , m; c) zi − x = R for all i such that wi > 0; d) x =

m 

wi zi .

i=1

III) The set Z = {z1 , z2 , . . . , zm } can be split into two subsets X and Z \ X (which we call the set of active points and the set of passive points) as follows: a) x belongs to the convex hull of the active set X ; b) x is a center of the smallest sphere that contains the active set X ; i.e. x is the circumcenter of the active points (see the definition of circumcenter in section A2.1); c) the passive points lie closer to x than the active points, i.e. the passive points lie inside or on the circumsphere of the active points. Remark A2.1.– In Theorem A2.4, condition III(c) means that z − x ≤ x1 − x

for all z ∈ Z \ X and x1 ∈ X .

However, under condition III(b), this is equivalent to z − x ≤ x1 − x

for all z ∈ Z and x1 ∈ X .

Appendix 2

221

In order to prove Theorem A2.4, we will verify the Karush–Kuhn–Tucker conditions, implementing this verification via the following version of the Karush–Kuhn–Tucker theorem. Theorem A2.5.– Let g, f1 , . . . , fm be concave differentiable functions on Rn . Suppose the following condition holds true: there exists x ∈ Rn such that for any i = 1, 2, . . . , m : fi (x) > 0.

[A2.32]

Then, x0 ∈ Rn is the maximizer of the optimization problem  g(x) → max provided that

fi (x) ≥ 0,

[A2.33]

i = 1, 2, . . . , m,

if and only if u0 ∈ Rm exists such that for the function φ(x, u) = g(x) +

m 

ui fi (x)

[A2.34]

i=1

the following conditions hold true: ∂φ(x0 , u0 ) =0 ∂xj fi (x0 ) ≥ 0,

for all j = 1, . . . , n;

u0i ≥ 0,

and

u0i fi (x0 ) = 0

[A2.35] for all i = 1, . . . , m. [A2.36]

For the proof, we refer to [KUH 51, Theorems 1, 2, and Section 8]. See also [COT 12]. Remark A2.2.– For necessity (the “only if” part of Theorem A2.5), the concavity of the function g(x) is not needed. For sufficiency (the “if” part of Theorem A2.5), condition [A2.32] is not needed. Proof of Theorem A2.4.– The Chebyshev center is the minimizer of the function Q(y). The function Q(y) is not “everywhere” differentiable. Hence, we reduce the minimization of Q to a constrained maximization problem that involves only differentiable functions. The maximum in Q(y) = maxi=1,...,m y − zi 2 is equal to the least R2 that satisfies inequalities R2 ≥ y − zi 2 ,

i = 1, . . . , m.

[A2.37]

222

Fractional Brownian Motion

Therefore, among all pairs (R2 , x) that satisfy R2 ≥ x − zi 2 ,

i = 1, . . . , m,

the minimal component R2 is equal to min Q(y), and the corresponding x is the minimizer of Q. Thus, the problem of unconstrained minimization of Q(x) is reduced to the following constrained maximization problem:  −R2 → max provided that

R2 − x − zi 2 ≥ 0,

i = 1, . . . , m.

Here, we have n + 1 unknown variables: one is R2 ∈ R and the other n are the elements of the vector x. In order to describe the maximizer, we apply Theorem A2.5. The functions g and fi are g(R2 , x) = −R2 ,

fi (R2 , x) = R2 − x − zi 2 ;

these functions are concave and differentiable. Obviously, inequalities fi (R2 , x) > 0 (i = 1, . . . , m) hold true if x = 0 and R2 = 1 + maxi=1,...,m zi 2 . The function φ form [A2.34] has a form φ(R2 , x; w) = −R2 +

m 

wi (R2 − x − zi 2 ).

i=1

The necessary and sufficient maximality conditions are provided by Theorem A2.5. Condition [A2.35] takes the form −1 +

m 

wi = 0,

[A2.38]

i=1

−2

m 

wi (x − zi ) = 0,

[A2.39]

i=1

while condition [A2.36] takes the following form: for all i = 1, . . . , m R2 − x − zi 2 ≥ 0,

[A2.40]

wi ≥ 0,

[A2.41]

wi (R2 − x − zi 2 ) = 0.

[A2.42]

Appendix 2

223

Obviously, R2 ≥ 0. Conditions [A2.38]–[A2.42] are equivalent to the √ conditions in statement II with R = R2 . Indeed, [A2.38] and [A2.41] are equivalent to II(a), and [A2.40] is equivalent to II(b). If [A2.41] holds true, then [A2.42] is equivalent to II(c). If [A2.38] holds true, then [A2.39] is equivalent to II(d). Now prove that statements II and III are equivalent. Suppose that statement II holds true. Split the set {zi , i = 1, . . . , m} as follows. Let the active set consist of the points zi with positive weights wi : X = {zi : wi > 0}. Then, III(a) follows from II(a) and II(d), III(b) follows from II(c) and III(a), and III(c) follows from II(b) and II(c). Thus, statement III holds true. Now suppose that statement III holds true. Due to III(a), x is a convex combination of elements x=



Wz z,

z∈X



Wz = 1,

z∈X

and Wz ≥ 0 for all z ∈ X. Now choose the vector w. If all zi ’s are different, i.e. if zi = zj for all i = j, then let  wi =

0 Wzi

if zi ∈ X , if zi ∈ X .

Otherwise, if some vector z appears in {z1 , . . . , zm } several times, we should be careful not to assign its weight Wz to multiple wi ’s. In that case, we put ⎧ ⎪ ⎨0 wi = Wzi ⎪ ⎩ 0

if zi ∈ X ; if zi ∈ X and zi = zj for all j, i < j ≤ m; if zi = zj for some j, i < j ≤ m.

Then, II(a) and II(d) hold true. Point x is equidistant from all active points due to III(b). Denote this distance as R. Then, II(c) holds true. Condition II(b) follows from II(c) and III(c). Thus, statement II holds true. 

224

Fractional Brownian Motion

The dual optimization problem is to find the weights w1 , . . . , wm that maximize the functional m  i=1

2 m  wi zi  − wi zi → max 2

[A2.43]

i=1

under the constraints wi ≥ 0,

i = 1, . . . , m,

and

m 

wi = 1.

[A2.44]

i=1

If there exists a point y equidistant from zi ’s, i.e. if z1 − y = z2 − y = . . . = zm − y, then the dual problem is equivalent to minimizing the value m 2  wi zi − y

[A2.45]

i=1

under constraints [A2.44]. Theorem A2.6.– 1) A point x ∈ Rn is the Chebyshev center of {z1 , . . . , zm } if and only if non-negative numbers w1 , . . . , wn exist such that: a) w1 , . . . , wm maximize the functional [A2.43] under constraints [A2.44]; b) x =

m 

wi zi .

i=1

2) If there exists a point y equidistant from z1 , . . . , zm , then a point x ∈ Rn is the Chebyshev center of {z1 , . . . , zm } if and only if non-negative numbers w1 , . . . , wn exist such that: a) w1 , . . . , wm minimize the functional [A2.45] under constraints [A2.44]; b) x =

m  i=1

wi zi .

Appendix 2

225

In the proof of Theorem A2.6, we will use the following version of the Karush–Kuhn–Tucker theorem. Theorem A2.7.– Let g be a concave differentiable function Rn → R, and let n n f be an affine function R → R in the form f (x) = i=1 ki xi + h for some fixed k1 , . . . , kn , and h. Then, x0 is a maximizer of the problem ⎧ ⎪ ⎨g(x) → max provided that ⎪ ⎩

xi ≥ 0 for all i = 1, . . . , n and f (x) = 0

[A2.46]

if and only if u0 exists such that for the function φ(x, u) = g(x) + uf (x)

[A2.47]

the following conditions hold true ∂φ(x0 , u0 ) ≤ 0, ∂xj

∂φ(x0 , u0 ) · xj = 0 ∂xj

xj ≥ 0,

for all j = 1, . . . , n, and f (x0 ) = 0. For the proof, we again refer to [KUH 51, Theorems 1, 2, and Section 8]. Proof of Theorem A2.6.– Part 1. We apply Theorem A2.7 to obtain a system of equations and inequalities that is equivalent to constrained maximization problem [A2.43]–[A2.44]. We have g(w) =

m  i=1

2 m  wi zi  − wi zi , 2

f (x) = 1 −

i=1

m 

wi .

i=1

Note that the function g is concave and the function f is affine. Function φ from [A2.47] and its partial derivatives equal φ(w, u) =

m  i=1

2 m   m   wi , wi zi + u 1 − wi zi 2 − i=1

m 

∂φ(w, u) = zk 2 − 2zk wi zi − u. ∂wk i=1

i=1

226

Fractional Brownian Motion

The necessary and sufficient optimality conditions now have a form: ⎧ there exists u ∈ R such that for all k = 1, . . . , m ⎪ ⎪ ⎪  ⎪ ⎪ zk 2 − 2zk m ⎪ i=1 wi zi − u ≤ 0, ⎪ ⎪ ⎨ wk ≥ 0, m ⎪ (zk 2 − 2zk i=1 wi zi − u)wk = 0; ⎪ ⎪ ⎪ m ⎪  ⎪ ⎪ ⎪ wi = 0. ⎩1 −

[A2.48]

i=1

Conditions [A2.48] can be rewritten as follows: u ∈ R exists such that: m a) wi ≥ 0 for all i = 1, . . . , m and i=1 wi = 1; b) zk 2 − 2zk

m 

wi zi ≤ u for all such k = 1, . . . , m that wk = 0;

i=1

c) zk 2 − 2zk

m 

wi zi = u for all such k = 1, . . . , m that wk > 0.

i=1

Since 2 2 m m m    wi zi = zk  − 2zk wi zi + wi zi zk − i=1

i=1

i=1

m 2 with  i=1 wi zi  not depending on k, the optimality conditions (a)–(c) are equivalent to the following: C ≥ 0 exists such that: m a’) wi ≥ 0 for all i = 1, . . . , m and i=1 wi = 1; 2 m  wi zi ≤ C for all such k = 1, . . . , m that wk = 0; b’) zk − i=1

2 m  wi zi = C for all such k = 1, . . . , m that wk > 0. c’) zk − i=1

m With C = R2 and i=1 wi zi = x, conditions (a’)–(c’) are the same as in part II of Theorem A2.4, which provides necessary and sufficient conditions for the point x to be a Chebyshev center.

Appendix 2

227

Part 2. The point x minimizes the functional Q(x) = maxi=1,...,m x − zi 2 if and only if point x ˜ = x − y minimizes the functional Q(˜ x + y) = max ˜ x − (zi − y)2 . i=1,...,m

Due to part 1 of the present theorem, x ˜ minimizes Q(˜ x + y) if and only if there exist weights w1 , . . . , wm that maximize m  i=1

2 m  wi zi − y2 − wi (zi − y)

[A2.49]

i=1

under constraint [A2.44], and x ˜=

m 

wi (zi − y).

[A2.50]

i=1

Due to the equidistance condition zi − y = z1 − y for all i = 1, . . . , m m and the equality i=1 wi = 1, we have m  i=1

m 2 m 2   wi zi − y − wi (zi − y) = z1 − y − wi zi − y . 2

i=1

i=1

Hence, the constrained maximization of [A2.49] is equivalent to the constrained minimization of [A2.45]. Under the constraints, equality [A2.50], i.e. x−y =

m 

wi (zi − y),

i=1

holds true if and only if x =

m

i=1

wi zi .



A2.4. Algorithm for finding the Chebyshev center We slightly modify the algorithm presented in [BOT 93]. As a prerequisite, we will require that the input points are affinely independent. During the computation, point a will stand for the approximation to the Chebyshev

228

Fractional Brownian Motion

center, and X ⊂ Z will be the current active set. The active points will be equidistant from a, and the passive points are located not further, i.e. x − a will be the same for all x ∈ X , x − a ≥ z − a for x ∈ X and z ∈ Z. The circumcenter of the active set will be denoted by o. It will be a projection of the current approximation a onto the hyperplane spanned by the active set. However, it may or may not lie in the convex hull of the active set. In our algorithm for finding the Chebyshev center, two steps are used several times. In the deflate step, the approximation a is moved towards o as long as the largest distance between the approximation and passive points does not exceed the distance between the approximation and active points. In the other step, unnecessary points are removed from the set of active points. A2.4.1. Deflate step Recall the conditions that hold true during the process of the Chebyshev center algorithm: – Z is the fixed finite set, consisting of affinely independent points, the circumcenter of which we are evaluating; – X ⊂ Z is the set of active points; – the points o and a are equidistant from the active points (i.e. o − x is the same for all x ∈ X , and a − x is also the same for all x ∈ X ); – o is the circumcenter of X ; – o and a satisfy the inequality a − x ≥ a − z

for all

x ∈ X , z ∈ Z.

[A2.51]

In the deflate step, we try to reduce maxz∈Z a − z = a − x (where x is an arbitrary point of the set X ), keeping the conditions satisfied. The set of points equidistant from the active points is an affine subspace, which is orthogonal to the affine subspace spanned by the active points, see the proof of Lemma A2.1. As a result, all the points of the segment [o, a] = {o + λ(a − o), λ ∈ [0, 1]} are equidistant from the active points, i.e. o + λ(a − o) − x = const

for unfixed x ∈ X and fixed λ ∈ [0, 1],

Appendix 2

229

and vectors a − o and x − o are orthogonal, i.e. (x − o) (a − o) = 0 for all x ∈ X. In the following, x ∈ X is some active point. The norms x − o, x − a and o + λ(a − o) − x do not depend on x, so it does not matter which active point is chosen. Find the point anew = o+λ(a−o) from the segment [o, a] for which anew −x is minimal while inequality [A2.51] holds true. Since anew − x2 = λ(a − o) − (x − o)2 = λ2 a − o2 + x − o2 , the norm anew − x decreases as λ ≥ 0 decreases. Hence, we find the minimal λ ≥ 0 for which o + λ(a − o) − x ≥ o + λ(a − o) − z for all

z ∈ Z.

[A2.52]

Then, take anew = o + λ(a − o) as the new value for a. For a given z ∈ Z, the inequality o + λ(a − o) − x ≥ o + λ(a − o) − z is equivalent to λ(a − o) − (x − o)2 ≥ λ(a − o) − (z − o)2 , λ2 a − o2 − x − o2 ≥ λ2 a − o2 − 2λ(a − o) (z − o) + z − o2 , 2λ(a − o) (z − o) ≥ z − o2 − x − o2 .

[A2.53]

Note that due to [A2.51], inequality [A2.52] holds true for λ = 1. Therefore, inequality [A2.53] holds true for λ = 1 and for all z ∈ Z. Now solve inequality [A2.53] for λ ∈ [0, 1]. Consider three cases. If (a − o) (z − o) ≤ 0, then [A2.53] holds true for all λ ∈ [0, 1], because it holds true for λ = 1. If (a − o) (z − o) > 0 and z − o2 − x − o2 ≤ 0, then obviously [A2.53] holds true for all λ ∈ [0, 1]. Otherwise, if (a − o) (z − o) > 0 and z − o2 − x − o2 > 0, then [A2.53] holds true for all λ≥

z − o2 − x − o2 . (a − o) (z − o)

230

Fractional Brownian Motion

For fixed λ ∈ [0, 1], inequality [A2.52] is satisfied if and only if inequality [A2.53] is satisfied for all z ∈ Z, which is equivalent to λ≥

z − o2 − x − o2 (a − o) (z − o)

for all z ∈ K,

where K is a subset of Z, K = {z ∈ Z : (a − o) (z − o) > 0

and z − o2 − x − o2 > 0}.

Note that K ∩ X = ∅. If K = ∅, then the minimal λ ∈ [0, 1] for which [A2.52] holds true is 0. If K=

∅, then minimal λ for which [A2.52] holds true is λ = max z∈K

z − o2 − x − o2 . 2(a − o) (z − o)

[A2.54]

In the former case, optimal anew coincides with o. In the latter case, for anew = o + λ(a − o) with λ defined in [A2.54], for at least one z ∈ K the equality anew − z = anew − x holds true. In the deflate step, we perform the following: – move a towards o as close as possible without ruining inequality [A2.51]; – if the new point a does not coincide with o (this is the case K = ∅), include points z ∈ K for which a − z = a − x into the active set X ; – adjust the circumcenter o according to the new active set X . The algorithm of the deflate step is the following: Step 1. Find the set K := {z ∈ Z \ X : (a − o) (z − o) > 0 and z − o − r2 > 0}, where r2 = x − o2 for all x ∈ X (the value of x − o2 does not depend on the choice of x ∈ X is used). 2

Step 2. If K = ∅, put a := o and stop. Step 3. For every z ∈ K, evaluate λz :=

z − o2 − r2 . 2(a − o) (z − o)

Step 4. Put λmax := maxz∈K λz .

Appendix 2

231

Step 5. Ensure that λmax ∈ [0, 1]. Hence, if λmax > 1 (which cannot occur if all the computations were exact, but may occur due to computational errors), then put λmax := 1. Step 6. Put a := o + λmax (a − o). Step 7. Include into the active point set X the element (or the elements) z ∈ K for which λz = λmax (or λz ≥ λmax if λmax was adjusted in step 5): X := X ∪ {z ∈ K : λz ≥ λmax }. to o.

Step 8. Evaluate the circumcenter of the active set X and assign its value

Then a, X and o are the results. This is Algorithm A2.4 presented in section A2.5 in pseudocode. Note the following fact: either the active point set X does not change in the deflate step and a = o after the step, or the active point set X strictly increases in the deflate step. A2.4.2. Two algorithms for the evaluation of the Chebyshev center In this section, we present two algorithms for the evaluation of the Chebyshev center of an affinely independent set Z. The affine independence makes the evaluation of the circumcenter of the input points (or of any of their subset) well defined. Once we have Circumcenter and FCLS algorithms available, we can find the algorithm as follows. The first algorithm for finding the Chebyshev center of an affine independent set Z is as follows: Step 1. Find the circumcenter of the set Z: o := Circumcenter(Z). Step 2. Construct the matrix X := [z1 − o, . . . , zm − o], where z1 , . . . , zm are the elements of the set Z. Step 3. Choose an arbitrary vector w0 that satisfies the constraints [A2.44].

232

Fractional Brownian Motion

Step 4. Solve the FCLS problem: w := FCLS(X, w0). Step 5. Put a :=

m

i=1

wi zi . The point a is the Chebyshev center.

The weights evaluated in step 4 constitute the solution to the dual problem (see Theorem A2.6). As a result, the algorithm evaluates the Chebyshev center as expected. The drawback of this algorithm is its instability if the points zi are nearly affinely dependent. In this case, the circumcenter o can be a distant point, which may affect the precision of computation. Hence, we propose another algorithm for finding the Chebyshev center. The inputs for this algorithm are the initial approximation a and the finite set Z. The outputs are the Chebyshev center a and the final set of active points X . The second algorithm for finding the Chebyshev center of an affine independent set Z is as follows: Step 1. Initialize the set X of active points. It will consist of the point of Z, which are the most distant from the a: R := max z − a, z∈Z

X := {z ∈ Z : z − a = R}. Step 2. Evaluate the circumcenter of the active points: o := Circumcenter(X ). Step 3. Repeat the deflate step [a, X , o] := deflate(a, X , o, Z) until a = o. 

Step 4. Evaluate the weights in the convex combination o wSCLS,x x. To do this, construct the matrix Xo of the form

x∈X

Xo := [x1 − o, . . . , xp − o],

=

Appendix 2

233

where x1 , . . . , xp are the elements of the active set X and p is the number of elements in X . Then, apply the SCLS algorithm: wSCLS := SCLS(Xo ). Step 5. If all the elements of the vector wSCLS are non-negative, take a as the Chebyshev center, X as the final set of active points and stop. Step 6. Solve the FCLS problem. Construct a vector w0 of the initial approximation: max(0, wSCLS,i ) , w0i := p j=1 max(0, wSCLS,j )

i = 1, . . . , p.

Then, put wFCLS := FCLS(Xo , w0). Step 7. In Lemma A2.4, the second alternative holds true. An index i exists such that wSCLS,i < 0 and wFCLS,i = 0. (Choose one if these hold true for multiple i’s.) Remove the point x that corresponds to such i from the active set X . Go to step 2. It is Algorithm A2.5 presented in pseudocode in section A2.5. During the execution of the algorithm, the inequality x − a ≥ z − a

for all x ∈ X and z ∈ Z

holds true. When the algorithm ends, a = o is a convex combination of the active points; moreover, it is the circumcenter of the set X of the active points. By Theorem A2.4, a is the Chebyshev center of the set Z. A2.5. Pseudocode of algorithms Now we transform the algorithms previously presented as a sequence of steps into pseudocode in order to allow the reader to implement them in a structured programming language.

234

Fractional Brownian Motion

Algorithm A2.1.– Pseudocode of the algorithm for finding the center of the circumscribed sphere of a finite set (see the sequence of steps in section A2.1). function [x] = Circumcenter(Z). Data: Z = {z1 , . . . , zm } is an m-element set of n-dimensional vectors. Result: x is an n-dimensional vector. Set m to the number of vectors in the set Z, if m = 1 then return(z1 ), else set Z := [z1 − zm , . . . , zm−1 − zm ], i.e., make n × (m − 1) matrix Z with columns zk − zm , k = 1, . . . , m − 1, where z1 , . . . , zm are the vectors in the set Z, w ˇ := 12 (Z  Z )−1 diag(Z  Z ), x := zm + Z w, ˇ return(x), end end function Algorithm A2.2.– Pseudocode of the algorithm for solving the SCLS problem (see the sequence of steps in section A2.2.2). function [w] = SCLS(X). Data: X is n × m matrix with affinely independent columns. Result: w is an m-dimensional vector. Set m to the number of columns of X, if m = 1 then return(1), else set X2 := [x1 − xm , . . . , xm−1 − xm ], where x1 , . . . , xm are the columns of the matrix X. Construct an m-dimensional vector w: w1:(m−1) := −(X2 X2 )−1 X2 xm , m−1 wm := 1 − i=1 wi , return(w), end end function

Appendix 2

235

The lines of the following algorithm are referenced in the proof of Theorem A2.3. Algorithm A2.3.– Pseudocode of the algorithm for solving the FCLS problem (see the sequence of steps in section A2.2.3). function [w] = FCLS(X, w0 ). Data: X is n × m matrix with affinely independent columns, w0 is an m-dimensional vector. Result: w is an m-dimensional vector. Ensure that w0 satisfies the constraints. J := {i : w0 i > 0}, while True do 4 construct m-dimensional vector w: 5 w[J] := SCLS(X[·, J]), 6 the other elements of the vector w are set to 0, 7 while min(w) < 0 do w0i 8 for all i ∈ J such that wi < 0 set αi := −w , i 9 αmin := min (1, mini∈J:wi 0}, 14 construct m-dimensional vector w: 15 w[J] := SCLS(X[·, J]), 16 the other elements of the vector w are made equal to 0, 17 end 18  := X  Xw, 19 min := min , 20 if mini∈J i = min then return(w), 21 Select j such that j = min , 22 w0 := w, 23 J := {i : w0 i > 0} ∪ {j}, 24 end end function 1 2 3

Here, “Ensure that . . . the equality w0 i = 0 holds true” means the following. If computations are performed with errors, then set w0 i = 0. Otherwise, if all computations are exact, then do nothing.

236

Fractional Brownian Motion

Algorithm A2.4.– Pseudocode of the deflate step of the Chebyshev center algorithm (see the sequence of steps in section A2.4.1). function [a, X , o] = deflate(a, X , o, Z). Data: Z is an affinely independent set of points in Rn , a ∈ Rn , X ⊂ Z, X = ∅ such that a − z ≤ a − x for all z ∈ Z and x ∈ X , o is the circumcenter of X . Result: a ∈ Rn , X ⊂ Z such that a − z ≤ a − x for all z ∈ Z and x ∈ X , o is the circumcenter of X . Set r2 := meanx∈X x − o2 , set K := {z ∈ Z \ X : (a − o) (z − o) > 0 and z − o2 − r2 > 0}, if K = ∅ then set a := o, else z − o2 − r2 for every z ∈ K set λz := , 2(a − o) (z − o) set λmax := maxz∈K λz , ensure that λmax ∈ [0, 1] by setting λmax = min(1, λmax ), set a := o + λmax (a − o), set X := X ∪ {z ∈ K : λz ≥ λmax }, set o := Circumcenter(X ), end return(a, X , o) end function Algorithm A2.5.– Pseudocode of the second algorithm for finding the Chebyshev center (see the sequence of steps in section A2.4.2). function[a, X ] = ChebyCenter1(a, Z). Data: Z is an affinely independent set of points in Rn , a ∈ Rn is the initial approximation. Results: a ∈ Rn is the Chebyshev center of Z, X ⊂ Z is such a subset that a is the circumcenter of X . R := maxz∈Z z − a, X := {z ∈ Z : z − a = z}, while True do o := Circumcenter(X ), repeat [a, X , o] = deflate(a, X , o, Z) until a = o,

Appendix 2

construct n × p matrix Xo = [x1 − o, . . . , xp − o], where p is the number of elements in the set X and x1 , . . . , xp are the elements of X , wSCLS := SCLS(Xo ), if min(wSCLS ) ≥ 0 then return(a, X ), construct the vector w+ ∈ Rp with elements wi+ = max(0, wSCLS,i ),  −1 p + w0 := w+ , j=1 wj wFCLS := FCLS(Xo , w0), select an index i for which wSCLS,i < 0 and wFCLS,i = 0, remove i-th element xi from the set X , end end function

237

Appendix 3 Simulation of fBm

In this appendix, we describe several simulation procedures for fBm. We concentrate on the so-called exact methods. They allow us to model a sample from fBm as a Gaussian vector with a certain covariance matrix. Section A3.1 is devoted to the basic method, the Cholesky decomposition of the covariance matrix. Section A3.2 describes a faster algorithm of finding this decomposition. Section A3.3 describes a more efficient method of simulation based on the socalled circulant embedding of the covariance matrix. Section A3.4 gives a brief overview of various approximate simulation methods, which may be convenient and efficient in some particular problems. A3.1. The Cholesky decomposition method Suppose that we need to simulate the trajectory of fBm   B H = BtH , t ∈ [0, T ] . Practically, it is only possible to simulate in discrete time. Therefore, we take an equidistant partition of the interval [0, T ] and consider the following problem: how to simulate the values BTH/N , H H H B2T /N , . . . , BT (N −1)/N , BT ? These values form a centered Gaussian vector with a certain covariance matrix. Therefore, they can be simulated as a linear transform of a standard Gaussian vector (a sequence of independent standard normal random variables). This idea can be implemented in several ways, which will be discussed in this section. Note also that due to the self-similarity property, it suffices to generate the H values B1H , B2H , . . . , BN and multiply them by (T /N )H . Since the increments of fBm are stationary, it can be more convenient to proceed in the following way: H H to simulate the values X1 = B1H , X2 = B2H − B1H , . . . , XN = BN − BN −1 and then to construct a sample of an fBm via the cumulative sums, i.e. B0H = 0, k BkH = i=0 Xi , k ≥ 1. Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

240

Fractional Brownian Motion

The random variables X1 , X2 , . . . form a stationary sequence of standard Gaussian variables (which is called fractional Gaussian noise) with covariance γ(k) = EX1 Xk+1 =

 1 2H (k + 1)2H + |k − 1| − 2k 2H , 2

k ≥ 0.

In other words, X = (X1 , X2 , . . . , XN ) is a centered Gaussian vector with covariance matrix ⎛ ⎞ 1 γ(1) γ(2) . . . γ(N − 2) γ(N − 1) ⎜ γ(1) 1 γ(1) . . . γ(N − 3) γ(N − 2)⎟ ⎜ ⎟ ⎜ γ(2) γ(1) 1 . . . γ(N − 4) γ(N − 3)⎟ ⎜ ⎟ Γ = ΓN = ⎜ ⎟ .. .. .. .. .. .. ⎜ ⎟ . . . . . . ⎜ ⎟ ⎝γ(N − 2) γ(N − 3) γ(N − 4) . . . 1 γ(1) ⎠ γ(N − 1) γ(N − 2) γ(N − 3) . . . γ(1) 1 Therefore, it can be represented as X = LZ, where Z = (Z1 , . . . , ZN ) is a standard Gaussian vector, and the matrix L is such that LL = Γ. Hence, in order to simulate the Gaussian vector X, we need to find a “square root” of the covariance matrix Γ. Since Γ is a symmetric positive definite matrix, it admits a Cholesky decomposition Γ = LL , where L = (lij )i,j=1,··· ,N is a lower triangular matrix (which means lij = 0 for i < j). In order to find the elements of L, we rewrite the condition LL = Γ in the coordinate-wise form: j

lik ljk = γ(i − j),

1 ≤ j ≤ i ≤ N.

k=1

Then, the elements of lij , j ≤ i, can be calculated recursively. Indeed, for i = 1, we obtain the following equality for l11 : 2 l11 = γ(0) = 1,

For i = 2, we obtain two equations: l21 l11 = γ(1),

2 2 l21 + l22 = 1.

Appendix 3

241

 This enables us to find l21 = γ(1) and l22 = 1 − γ 2 (1). It is not hard to see that for n ≥ 2, the non-zero elements of (n + 1)th row are determined by ln+1,1 = γ(n),  ln+1,j =

ln+1,n+1

−1 ljj

γ(n + 1 − j) −

  n

 = 1 − l2 k=1

j−1

 ln+1,k ljk

,

2 ≤ j ≤ n,

k=1

n+1,k

[A3.1]

Remark A3.1.– The matrix Γ is positive definite as a correlation matrix of a stationary process. For a real symmetric positive definite matrix, the Cholesky decomposition is unique and the corresponding lower-triangular matrix is real (see, for example, [HOR 90, Corollary 7.2.9]). This implies that the expressions under square root in [A3.1] are always non-negative. For the matrix Γ, this fact 2 2 can be directly proved. Indeed, 1 − l21 (1) = 1 − γ (1) > 0, because γ(1) = n 1 2H−1 2 2 − 1 ∈ (− 2 , 1). Then, the positiveness of 1 − k=1 ln+1,k for n ≥ 2 can be obtained by induction. As soon as the matrix L is found, we can simulate the vector X from the standard normal vector Z by the formula X = LZ. Recall that the fBm is H recovered from X by the cumulative summation: B0H = 0, BkH = Bk−1 + Xk , 1 ≤ k ≤ N. Note that we can simulate the vector X = (X1 , . . . , XN ) recursively. Indeed, since the matrix L is lower-triangular, we see that Xn depends only on the first n n elements of Z = (Z1 , . . . , ZN ). More precisely, Xn = k=1 lnk Zk , where lnk are constructed by a recursive scheme as described above. This means that we can add new elements to the sample from fractional Gaussian noise if needed, the sample size need not be known in advance. This occurs, for example, when a simulation should stop at some random time. The advantage of the Cholesky method is that it is easy to implement. Moreover, it can be applied for simulation of any Gaussian vector with non-singular covariance matrix. In particular, we may directly generate the H sample Y = (BkT /N , k = 1, . . . , N ) by this method, using the Cholesky decomposition of its covariance matrix Cov(Y) = (σij )i,j=1,...,N , where   T 2H i2H + j 2H − |i − j|2H σij = . 2N 2H

242

Fractional Brownian Motion

Note that this approach leads to more precise modeling, because the approximation of an fBm via the cumulative sums of a fractional Gaussian noise involves negligible mistakes on the covariance structure (see [COE 00, Section 3.7]). However, in order to keep the same notation for all exact methods of the present appendix, we have described this algorithm for the case of the sample from the fractional Gaussian noise X. Anyway, the Cholesky method has a computational complexity of order O(N 3 ), which means that it is quite slow. However, when more than one sample is needed, we should calculate the matrix L only once. This allows us to save time (the computation of every additional sample requires order O(N 2 ) computer flops). Note that Γ is a Toeplitz (diagonal-constant) matrix. This makes it possible to calculate its square root in a more efficient way. In the next two sections, we describe two possible algorithms that are less computationally demanding than the standard Cholesky method. A3.2. The Hosking method The Hosking method (also known as the Durbin or Levinson method) for the simulation of a stationary Gaussian sequence was proposed in [HOS 84]. It generates Xn+1 given Xn , . . . , X1 recursively. Let us introduce the following n-dimensional vector cn and n×n matrix Fn : ⎞ γ(1) ⎜ γ(2) ⎟ ⎟ ⎜ cn = ⎜ . ⎟ , . ⎝ . ⎠ ⎛

⎛ 0 ⎜0 ⎜ Fn = ⎜ . ⎝ ..

... ... .. .

0 1 .. .

1

...

0

γ(n)

⎞ 1 0⎟ ⎟ .. ⎟ . .⎠ 0

Then, we have two representations for the covariance matrix Γn+1 in terms of Γn :  Γn+1 =

1 cn

c n Γn



 =

Γn c n Fn

 Fn cn . 1

[A3.2]

Now let us compute the conditional distribution of Xn+1 given Xn , . . . , X1 . Evidently, the covariance matrix of (Xn+1 , Xn , . . . , X1 ) equals Γn+1 . Using the first representation in [A3.2], we apply the theorem on

Appendix 3

243

normal correlation (Theorem A1.3). We obtain that the required conditional distribution is Gaussian with the following parameters: ⎛ ⎞ Xn  −1 ⎜ .. ⎟ μn = E(Xn+1 | Xn , . . . , X1 ) = cn Γn ⎝ . ⎠ , X1 −1 σn2 = Var(Xn+1 | Xn , . . . , X1 ) = 1 − c n Γn c n .

This result allows us to generate the values X1 , X2 , . . . , Xn subsequently, starting from X1 ∼ N (0, 1). However, taking the inverse of Γn at every step is computationally expensive. Below we describe the algorithm for recursive computation of Γ−1 n . It was originally proposed by Hosking [HOS 84]; we present it in a slightly different form taken from Dieker [DIE 02]. Let us denote dn = Γ−1 n cn .

[A3.3]

It follows from the second representation in [A3.2] by block matrix inversion that    1 σn2 Γ−1 −Fn dn n + Fn dn dn Fn Γ−1 = . [A3.4] n+1 −d 1 σn2 n Fn Using block matrix multiplication, we can easily obtain the following recursions for dn and σn2 :  dn+1 =

 dn − φn Fn dn , φn



2 σn+1

=

σn2

γ(n + 1) − τn − σn2

2 ,

[A3.5]

where  τn = d n Fn cn = cn Fn dn ,

φn =

γ(n + 1) − τn . σn2

Indeed, we have dn+1



  −Fn dn cn = γ(n + 1) 1  2 −1  1 σ n Γ n c n + Fn d n d  n Fn cn − Fn dn γ(n + 1) = 2 −d σn n Fn cn + γ(n + 1) Γ−1 n+1 cn+1

1 = 2 σn

 σn2 Γ−1 n + Fn d n d n F n  −dn Fn

244

Fractional Brownian Motion

=

1 σn2



σn2 dn + Fn dn τn − Fn dn γ(n + 1) −τn + γ(n + 1)

 =

  dn − φn Fn dn . φn

−1  Since σn2 = 1 − c n Γn cn = 1 − cn dn , we see that   dn − φ n Fn dn 2   σn2 − σn+1 − c = c d − c d = (c , γ(n + 1)) n+1 n+1 n n n n dn φn   = c n dn − φn cn Fn dn + φn γ(n + 1) − cn dn  2   γ(n + 1) − τn = φn γ(n + 1) − τn = , σn2

whence the second equality in [A3.5] follows. The relations [A3.5] allow us to simulate the process X1 , X2 , . . . by the following algorithm. In the first step, we generate X1 ∼ N (0, 1) and put μ1 = γ(1)X1 , σ12 = 1 − γ 2 (1), τ1 = γ 2 (1), d1 = (γ(1)) (one-dimensional vector). Then, in the (n + 1)th step, we generate a normal random variable Xn+1 with 2 mean μn and variance σn2 , compute τn , φn and dn+1 and find the variance σn+1   and the mean μn+1 = dn+1 (Xn+1 , . . . , X1 ) of the next element in the sample. As before, the fBm is recovered by cumulative summation. As pointed out in [DIE 02, Subsection 2.1.2], the Hosking method is equivalent to the Cholesky method in the sense that it computes implicitly the same lower-triangular matrix L. However, the Hosking algorithm is faster, because its complexity is of order O(N 2 ) for a sample of size N . At the same time, the Hosking method is also very simple and easy to implement. Similarly to the Cholesky method, the sample size does not need to be known in advance. A3.3. The circulant method This algorithm was originally proposed by Davies and Harte [DAV 87] and was later generalized by Wood and Chan [WOO 94] and Dietrich and Newsam [DIE 97]. Like the Hosking and Cholesky methods, this method tries to find a “square root” of the covariance matrix ΓN , in the sense that ΓN = SS  for

Appendix 3

245

some square matrix S. The main idea is to embed ΓN in the so-called circulant covariance matrix C of size M = 2(N − 1). More precisely, we define C by ⎛

c0

⎜cM −1 ⎜ ⎜cM −2 ⎜ C = circ(c0 , c1 , . . . , cM −1 ) = ⎜ . ⎜ .. ⎜ ⎝ c2 c1

c1 c0 cM −1 .. .

c2 c1 c0 .. .

... ... ... .. .

cM −2 cM −3 cM −4 .. .

c3 c2

c4 c3

... ...

c0 cM −1

⎞ cM −1 cM −2 ⎟ ⎟ cM −3 ⎟ ⎟ .. ⎟ , . ⎟ ⎟ c1 ⎠ c0

where  c0 = 1,

ck =

γ(k), γ(M − k),

k = 1, 2, . . . , N − 1, k = N, N + 1, . . . , M − 1.

[A3.6]

The method is based on the following result (see, for example, [BAR 90, Section 13.2]). Theorem A3.1.– The circulant matrix C has a representation C = QΛQ∗ , where Λ = diag(λ0 , λ1 , . . . , λM −1 ),

λk =

M −1

j=0



jk cj exp −2πi M

 [A3.7]

is the diagonal matrix of eigenvalues of C, and the matrix Q is defined by −1 Q = (qjk )M j,k=0 ,

  1 jk , qjk = √ exp −2πi M M

Q∗ denotes the conjugate transpose of Q. Note that Q is unitary: Q∗ Q = QQ∗=IM, the identity matrix. Consequently, 1/2 1/2 1/2 C = SS ∗ , where S = QΛ1/2 Q∗ , Λ1/2 = diag λ0 , λ1 , . . . , λM −1 . In the case of fractional Gaussian noise, the matrix C is positive definite (see [CRO 99, PER 02]), so all the eigenvalues λk are positive; as a result, the matrix S is real. Thus, in order to simulate the fractional Gaussian noise, we need to generate a vector (Z1 , Z2 , . . . , ZM ) of standard normal variables, multiply it by S and take the first N coordinates of the resulting vector. The practical realization of this approach requires the computation of discrete Fourier transform (DFT), direct and inverse. Indeed, the vector of

246

Fractional Brownian Motion

eigenvalues (λ0 , λ1 , . . . , λM −1 ) is the DFT of (c0 , c√1 , . . . , cM −1 ). The multiplication by matrix Q acts, up to the constant 1/ M , as taking the DFT, namely √ M Qx =

M −1

k=0

M −1  jk xk exp −2π M j=0

for x = (x0 , x1 , .√. . , xM −1 ) ∈ CM . Similarly, the multiplication by Q∗ is, up to the constant M , taking the inverse DFT, i.e. ⎞ ⎛  M −1  M −1

1 1 jk ⎠ √ Q∗ x = ⎝ xj exp 2π . M j=0 M M k=0

In view of this, it is recommended to choose N = 2q + 1 for some q ∈ N so that M = 2(N − 1) is a power of 2. In this case, DFT computation can be done efficiently by the so-called fast Fourier transform (FFT) algorithm, which has the complexity of order O(N log N ). The literature contains several similar simulation algorithms based on circulant embedding (see, for example, [COE 00, DIE 02, KRO 15]). Below we give a version taken from [SHE 15]. 1) Set N = 2q + 1 and M = 2q+1 . 2) Calculate γ(1), . . . , γ(N − 1) and set c0 , c1 , . . . , cM −1 according to [A3.6]. 3) Take FFT to obtain λ0 , λ1 , . . . , λM −1 . Theoretically, we should obtain real numbers. However, since all computer calculations are imprecise, the resulting values will have tiny imaginary parts, so we need to take the real part of the result. 4) Generate independent standard Gaussian Z1 , . . . , ZM . 5) Take the real part of inverse FFT of Z1 , . . . , ZM to obtain the vector

√1 Q∗ (Z1 , . . . , ZM ) . M

6) Multiply the last element-wise by

 √ √ λ0 , λ1 , . . . , λM −1 .

7) Take the FFT of the result to obtain (Y1 , . . . , YM ) =

√ 1 M QΛ1/2 √ Q∗ (Z1 , . . . , ZM ) = S(Z1 , . . . , ZM ) . M

8) Take the real part of Y1 , . . . , YM to obtain the fractional Gaussian noise X1 , . . . , XN .

Appendix 3

247

Note that if we need several realizations of the sample, we should repeat Steps 4–8; the values of λ0 , λ1 , . . . , λM −1 (steps 1–3) are computed only once. The MATLAB codes for this algorithm can be found in [SHE 15]. The main advantage of the circulant method is its speed (the algorithm has complexity of order O(N log N )). The algorithm is probably most efficient among exact methods. Remark A3.2.– Mathematical software usually has ready-to-use functions for FFT computation. However, we should be aware that FFT can be implemented with different normalization parameters. For example, MATLAB offers two functions, fft and ifft, for direct and inverse FFT, respectively, where y=fft(x) is defined as yk =

  (j − 1)(k − 1) , xj exp −2πi n j=1

n

and x=ifft(y) is defined as xk =

  n 1 (j − 1)(k − 1) . yj exp 2πi n j=1 n

At the same time, in Mathematica, the functions Fourier and InverseFourier are implemented so that y=Fourier[x] means that   n 1 (j − 1)(k − 1) yk = √ , xj exp 2πi n n j=1 and, respectively, x=InverseFourier[y] is defined as   n 1 (j − 1)(k − 1) xk = √ . yj exp −2πi n n j=1 The default normalization in Mathematica can be changed using the FourierParameters option. It is worth mentioning that for the case 0 < H < 1/2, a different circulant method was proposed in [LOW 99] (see also [BAR 03, Section 2.4] for its improved version).

248

Fractional Brownian Motion

A3.4. Approximate methods The literature contains a number of approximate methods of simulation of an fBm. Some of them can be faster than the exact methods, which is their main advantage. However, the speed (as well as accuracy) usually depends on the choice of some parameters. The stochastic representation method is based on the Mandelbrot–Van Ness representation [1.8], according to which an fBm can be represented as an integral with respect to the two-sided Wiener process. This integral can be approximated by Riemann-type sums to simulate the process. More precisely, we can consider the following discrete-time process as an approximation of fBm:  0 n 

 c H H B = H (n − k)H−1/2 − (−k)H−1/2 Zk N N k=−aN  n

H−1/2 n = 1, 2, . . . , N − 1, + (n − k) Zk , k=0

where Zk are independent standard normal variables, aN is a sequence such that aN → ∞ as N → ∞. The various improvements of this method can be found in [MAN 69], [SAM 94, Section 7.11] and [VOS 88]. Similarly, we can construct an approximation by discretization in the following spectral representation of fBm:  ∞   ∞ cos(λt) − 1 sin(λt) aH H Bt = √ dξ(λ) − , t ∈ [0, T ], 2H+1 2H+1 dη(λ) π λ 2 λ 2 0 0 where W1 (λ), W2 (λ) are two independent standard Wiener processes,  aH =

2 π



∞ 0

1 − cos(λt) dλ λ2H+1

−1/2

 =

Γ(1 − 2H) cos(πH) πH

−1/2 .

For the points 0 = λ0 < λ1 < . . . < λM = Λ, we construct the following model of B H : M −1  M −1

sin(λi t) aH cos(λi t) − 1 H  , t ∈ [0, T ], M ∈ N, Xi − B (t) = √ 2H+1 2H+1 Yi π i=1 2 λi 2 i=1 λi where Xi , Yi , i = 1, 2, . . . , M − 1, are independent normal random variables with EXi = EYi = 0,

EXi2 = EYi2 = λi+1 − λi .

Appendix 3

249

In [KOZ 18], the conditions for the parameters M ∈ N, 0 < λ1 < . . . < λM = Λ for simulation B H in Lp ([0, T ]) with given accuracy and reliability are established. Several algorithms of spectral simulation are proposed in [DIE 03, FED 88, PAX 97, VOS 88]. They generate an approximation of fractional Gaussian noise using the DFT, starting with the spectral density of the process at Fourier frequencies. Then, a random complex sequence with this spectral density is simulated and an approximation of the process at times t = k/N , k = 0, 1, . . . , N − 1 is obtained using inverse FFT. Wavelet-based simulation is an approach that is closely related to spectral simulation. It was introduced and developed in [ABR 96, MEY 99, SEL 95]. The method is based on the wavelet expansion of fBm and provides approximations that converge to fBm almost surely and uniformly on compact intervals. We refer to [PIP 04, PIP 05] for a detailed discussion of this method and its improvements. Among other approximate approaches, we mention conditionalized random midpoint displacement [NOR 99a], aggregation of renewal processes [WIL 03] and aggregation of autoregressive processes [GRA 80]. For an overview of these and several other methods and for concrete simulation algorithms, we refer to the surveys [BAR 03, COE 00, DIE 02] and references cited therein.

Solutions

Solutions to exercises from Chapter 1 Solution to Exercise 1.1. Let us establish existence and uniqueness of the point of minimum. 1) Existence of the point of minimum follows from section 1.4, because the particular form of the kernel z(t, s) was never used in the proof. 2) Similarly, uniqueness can be obtained in the same way as it was done in section 1.6. However, we put a short proof of uniqueness here. Note that f (a) ≥ 0 for all a ∈ L2 ([0, 1]). Hence, the functionals f (a) and (f (a))2 reach their minima on the same set of points. Suppose that f (a) reaches minimum at points a1 and a0 with essentially different functions a1 and a0 , which means that a1 − a0  > 0; here, all norms are in L2 ([0, 1]). Then, f 2 (a1 ) = ga21 (1) = (min f )2 and f 2 (a0 ) = ga20 (1) = (min f )2 . If is easy to show that f 2 (a) is a convex function of a, and ga2 (1) = a − z(1, · )2 is a strictly convex function of a. The convexity of f 2 (a) follows from the facts that square function is convex and increasing in the range of f (a) and the functional f (a) is convex due to Lemma 1.4. The strict convexity of ga2 (1) = a − z(1, · )2 follows from Remark 1.8. From the convexity and strong convexity, it follows that for any point at = ta1 + (1 − t)a0 , t ∈ (0, 1), we have (min f )2 ≤ f 2 (at ) ≤ tf 2 (a1 ) + (1 − t)f 2 (a0 ) = (min f )2 , ga2t (1) < tga21 (1) + (1 − t)ga20 (1) = (min f )2 ,

Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

252

Fractional Brownian Motion

whence ga2t (1) < f 2 (at ) = (min f )2 . Thus, for every point at , t ∈ (0, 1), the function f reaches its minimum at at while gat (1) = f (at ). This is contrary to the assumption that for every function a where f (a) reaches minimum, ga (1) = f (a). Solution to Exercise 1.2. We give the solution only for the second statement. 2) Note that the kernel of the Riemann–Liouville process is (β−1)homogeneous, namely for any c > 0 z(ct, cs) =

(t − c)β−1 (ct − cs)β−1 10≤s 1. Prove the following property of semicontinuity: lim sup ga (t) ≤ ga (1). t→1+

[S.1]

Indeed, for t > 0  ga (t) = a − z(t, ·)L2 [0,t] ≤ a − z(1, ·)L2 [0,t] + z(t, ·) − z(1, ·)L2 [0,t]  = a − z(1, ·)L2 [0,1] + E(Rβ (t) − Rβ (1))2   = ga (1) + E(Rβ (t) − Rβ (1))2 . Due to the mean-square continuity of the Riemann–Liouville process, E(Rβ (t) − Rβ (1))2 tends to 0 as t → 1. Hence lim sup t→1+

  ga (t) ≤ ga (1),

Solutions

253

whence [S.1] follows. Recall our assumption ga (1) < f (a). Since lim sup ga (t) < f (a), t→1+

which holds true due to [S.1], and sup ga (t) = f (a),

t∈[0,1]

which is the definition of f (a), c > 1 exists such that sup ga (t) = f (a).

[S.2]

t∈[0,c]

Define a function b(s) = c1−β a(cs). Then  gb (t) = 

t 0 t

= 0

(b(s) − z(t, s)2 )2 ds (c1−β a(cs) − c1−β z(ct, cs))2 ds

= c2−2β = c1−2β

 

t 0

(a(cs) − z(ct, cs))2 ds

ct 0

(a(u) − z(ct, u))2 du = c1−2β ga (ct).

With [S.2], we have f (b) = sup gb (a) = sup c1−2β ga (ct) t∈[0,1]

=c

1−2β

t∈[0,1]

sup ga (u) = c1−2β f (a).

u∈[0,c]

However, the functional f reaches its minimal value at point a. Hence c1−2β f (a) = f (b) ≥ f (a).

[S.3]

Since f (a) ≥ 0 and c1−2β < 0, inequality [S.3] is possible only if f (a) = 0. However, as ga (1) ≥ 0, this contradicts to assumption ga (1) < f (a).

254

Fractional Brownian Motion

We have verified the assumption from item 2 of Exercise 1.1. Now, apply Exercise 1.1 to prove the desired uniqueness. Remark S.1.– In the solution above, the function ga (t) is in fact continuous on [0, +∞). Solution to Exercise 1.3. First, let us prove the inequality (x2 − x1 )2 ≤ max(x21 , x22 ). 4

[S.4]

Indeed, apply inequality [1.15]: (x2 − x1 )2 x2 x2 ≤ 2 + 1 ≤ max(x21 , x22 ). 4 2 2 Now, for N = 3, create a matrix ⎞ ⎛ 100 K = ⎝3 1 0⎠ 211

[S.5]

and verify that the functional F reaches its minimum at multiple points. Hence, let us prove that F ((2, 1, 1) ) = F ((2, 1, 2) ) = min3 F (a). a∈R

Indeed, for every a ∈ R3 F (a) = max((a1 − k11 )2 , (a1 − k21 )2 + (a2 − k22 )2 , (a1 − k31 )2 + (a2 − k32 )2 + (a3 − k33 )2 ) = max((a1 − 1)2 , (a1 − 3)2 + (a2 − 1)2 , (a1 − 2)2 + (a2 − 1)2 + (a3 − 1)2 ) ≥ max((a1 − 1)2 , (a1 − 3)2 ) ≥

((a1 − 3) − (a1 − 1))2 = 1. 4

Thus, 1 is a lower bound for F (a): F (a) ≥ 1

for all a ∈ R3 .

[S.6]

Solutions

255

Now evaluate F (a) for a = (2, 1, 1) and for a = (2, 1, 2) : F ((2, 1, 1) ) = max((2 − 1)2 , (2 − 3)2 + (1 − 1)2 , (2 − 2)2 + (1 − 1)2 + (1 − 1)2 ) = max(1, 1, 0) = 1,

[S.7]

F ((2, 1, 2) ) = max((2 − 1)2 , (2 − 3)2 + (1 − 1)2 , (2 − 2)2 + (1 − 1)2 + (2 − 1)2 ) = max(1, 1, 1) = 1. From [S.6] and [S.7], it follows that mina∈R3 F (a) = 1. Finally, F ((2, 1, 1) ) = F ((2, 1, 2) ) = min3 F (a) = 1, a∈R

and the minimum of F is reached at two different points. Remark S.2.– For the matrix K defined in [S.5], the set of all points, where the minimum of F is reached, is a line segment in R3 : arg min F = {(a1 , a2 , a3 ) ∈ R3 : a1 = 2, a2 = 1, 0 ≤ a3 ≤ 2}. Remark S.3.– The N × N matrix K for which the functional F reaches minimum at multiple points can be constructed for any N ≥ 3. For N = 1, the minimum of F is reached at the unique point a = (k11 ). For N = 2, the minimum of F is reached at the unique point

 21 a = k11 +k , k22 . 2 Solutions to exercises from Chapter 2 Solution to Exercise 2.6. We have obvious inequalities f (T, k1 , k2 ) ≤ max f (T, k1 , k2 ), t∈[0,T ]

min f (T, k1 , k2 ) ≤ min max f (T, k1 , k2 ).

k1 ,k2

k1 ,k2 t∈[0,T ]

On the left-hand side, the minimum is reached for k1 = k1∗ , k2 = k2∗ (see the proof of Theorem 2.7). The right-hand side is equal to ρ2 (B H , M(K7 )). Thus, f (T, k1∗ , k2∗ ) ≤ ρ2 (B H , M(K7 )).

256

Fractional Brownian Motion

In section 2.1.6.2, we showed that if H ∈ ( 12 , H1 ], then f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ) ≥ f (T, k1∗ , k2∗ ). In this case max(f (T, k1∗ , k2∗ ), f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ )) = f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ), and inequality [2.83] becomes ρ2 (B H , M(K7 )) ≤ f (t1 (k1∗ , k2∗ ), k1∗ , k2∗ ). Solutions to exercises from Chapter 3 Solution to Exercise 3.3. By Stirling’s formula [3.51], Γ(n + 1 − α) (n + 1 − α)n+1/2−α e−n−1+α ∼ Γ(n + 1) (n + 1)n+1/2 e−n−1 −1/2−α  n+1  α α −α 1− 1− = (n + 1) eα n+1 n+1 ∼ (n + 1)−α ∼ n−α ,

as n → ∞,

−1/2−α n+1

α α → 1 and 1 − n+1 → e−α as n → ∞. Now the since 1 − n+1 required result follows from the representation 

n−α n

see [A1.18].

=

Γ(n − α + 1) . Γ(1 − α)Γ(n + 1)

References

[ABR 96] Abry P., Sellan F., “The wavelet-based synthesis for fractional Brownian motion proposed by Sellan F. and Meyer Y.: Remarks and fast implementation”, Applied and Computational Harmonic Analysis, vol. 3, no. 4, pp. 377–383, 1996. [AND 06] Androshchuk T.O., “Approximation of a stochastic integral with respect to fractional Brownian motion by integrals with respect to absolutely continuous processes”, Theory of Probability and Mathematical Statistics, vol. 73, pp. 19–29, 2006. [AYA 03] Ayache A., Taqqu M.S., “Rate optimality of wavelet series approximations of fractional Brownian motion”, Journal of Fourier Analysis and Applications, vol. 9, no. 5, pp. 451–471, 2003. [BAN 08] Banna O., Mishura Y., “Approximation of fractional Brownian motion with associated Hurst index separated from 1 by stochastic integrals of linear power functions”, Theory of Stochastic Processes, vol. 14, nos 3–4, pp. 1–16, 2008. [BAN 11] Banna O.L., Mishura Y.S., “A bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval”, Theory of Probability and Mathematical Statistics, vol. 83, pp. 13–25, 2011. [BAN 15] Banna O.L., Mishura Y.S., Shklyar S.V., “Approximation of a Wiener process by integrals with respect to the fractional Brownian motion of power functions of a given exponent”, Theory of Probability and Mathematical Statistics, vol. 90, pp. 13–22, 2015. [BAR 90] Barnett S., Matrices: Methods and Applications, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1990. [BAR 03] Bardet J.-M., Lang G., Oppenheim G. et al., “Generators of longrange dependent processes: A survey”, Theory and Applications of Long-range Dependence, pp. 579–623, Birkhäuser Boston, Boston, MA, 2003.

Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

258

Fractional Brownian Motion

[BAS 03] Bashirov A.E., Partially Observable Linear Systems Under Dependent Noises, Systems & Control: Foundations & Applications, Birkhäuser Verlag, Basel, 2003. [BEN 97] Benassi A., Jaffard S., Roux D., “Elliptic Gaussian random processes”, Revista Mathemática Iberoamericana, vol. 13, no. 1, pp. 19–90, 1997. [BIA 08] Biagini F., Hu Y., Øksendal B. et al., Stochastic Calculus for Fractional Brownian Motion and Applications, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2008. [BOT 93] Botkin N.D., Turova-Botkina V.L., “An algorithm for finding the Chebyshev center of a convex polyhedron”, System Modelling and Optimization, vol. 197 of Lecture Notes in Control and Information Sciences, pp. 833–844, Springer, 1993. [BOU 10] Boufoussi B., Dozzi M., Marty R., “Local time and Tanaka formula for a Volterra-type multifractional Gaussian process”, Bernoulli, vol. 16, no. 4, pp. 1294–1311, 2010. [CHA 03] Chang C.-I., Hyperspectral Imaging: Techniques for Spectral Detection and Classification, Kluwer, New York, 2003. [CHA 15] Chan N.H., Wong H.Y., Simulation Techniques in Financial Risk Management, Wiley, 2015. [CHE 12] Chen C., Sun L., Yan L., “An approximation to the Rosenblatt process using martingale differences”, Statistics & Probability Letters, vol. 82, no. 4, pp. 748–757, 2012. [COE 00] Coeurjolly J.-F., “Simulation and Identification of the Fractional Brownian Motion: A bibliographical and comparative study”, Journal of Statistical Software, vol. 5, pp. 1–53, 2000. [COH 99] Cohen S., “From self-similarity to local self-similarity: The estimation problem”, Fractals: Theory and Applications in Engineering, pp. 3–16, Springer, London, 1999. [COT 12] Cottle R.W., “William Karush and the KKT theorem”, Grötschnel M. (ed.), Optimization Stories, pp. 255–269, Deutschen MathematikerVereinigung, Berlin, Documenta Mathematica, Extra Volume, 2012. [COU 88] Courant R., Differential and Integral Calculus, vol. 1, Wiley, Hoboken, NJ, 1988. [CRO 99] Crouse M., Baraniuk R.G., “Fast, exact synthesis of Gaussian and nonGaussian long-range-dependent processes”, IEEE Transactions on Information Theory, 1999. [DAV 87] Davies R.B., Harte D.S., “Tests for Hurst effect”, Biometrika, vol. 74, no. 1, pp. 95–101, 1987. [DAV 05] Davis M.H.A., “Martingale representation and all that”, Advances in Control, Communication Networks, and Transportation Systems, Systems Control Found. Appl., pp. 57–68, Birkhäuser Boston, Boston, MA, 2005.

References

259

[DIE 97] Dietrich C.R., Newsam G.N., “Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix”, SIAM Journal on Scientific Computing, vol. 18, no. 4, pp. 1088–1107, 1997. [DIE 02] Dieker T., Simulation of Fractional Brownian Motion, Master’s thesis, Vrije Universiteit, Amsterdam, 2002. [DIE 03] Dieker A.B., Mandjes M., “On spectral simulation of fractional Brownian motion”, Probability in the Engineering and Informational Sciences, vol. 17, no. 3, pp. 417–434, 2003. [DOR 13] Doroshenko V., Mishura Y., Banna O., “The distance between fractional Brownian motion and the subspace of martingales with “similar” kernels”, Theory of Probability and Mathematical Statistics, vol. 87, pp. 41–49, 2013. [DUN 11] Dung N.T., “Semimartingale approximation of fractional Brownian motion and its applications”, Computers & Mathematics with Applications, vol. 61, no. 7, pp. 1844–1854, 2011. [FED 88] Feder J., Fractals, Physics of Solids and Liquids, Plenum Press, New York, 1988. [FIK 65] Fikhtengol’ts G.M., The Fundamentals of Mathematical Analysis. Vol. II, vol. 73 of International Series of Monographs in Pure and Applied Mathematics, Pergamon Press, 1965. [GAR 71] Garsia A.M., Rodemich E., Rumsey Jr. H., “A real variable lemma and the continuity of paths of some Gaussian processes”, Indiana University Mathematics Journal, vol. 20, pp. 565–578, 1970/1971. [GAR 74] Garsia A.M., Rodemich E., “Monotonicity of certain functionals under rearrangement”, Annales de l’Institut, vol. 24, no. 2, pp. vi, 67–116, 1974, Colloque International sur les Processus Gaussiens et les Distributions Aléatoires (Colloque Internat. du CNRS, No. 222, Strasbourg, 1973). [GOR 78] Gorodecki˘ı V.V., “On convergence to semi-stable Gaussian processes”, Theory of Probability and its Applications, vol. 22, no. 3, pp. 498–508, 1978. [GRA 80] Granger C.W.J., “Long memory relationships and the aggregation of dynamic models”, Journal of Econometrics, vol. 14, no. 2, pp. 227–238, 1980. [HOR 90] Horn R.A., Johnson C.R., Matrix Analysis, Cambridge University Press, Cambridge, 1990, Corrected reprint of the 1985 original. [HOS 84] Hosking J.R.M., “Modeling persistence in hydrological time series using fractional differencing”, Water Resources Research, vol. 20, no. 12, pp. 1898–1908, 1984. [KOZ 18] Kozachenko Y.V., Pashko A.O., Vasylyk O.I., “Simulation of fractional Brownian motion in the space Lp ([0, T ])”, Theory of Probability and Mathematical Statistics, vol. 97, 2018.

260

Fractional Brownian Motion

[KRO 15] Kroese D.P., Botev Z.I., “Spatial process simulation”, Stochastic Geometry, Spatial Statistics and Random Fields, vol. 2120 of Lecture Notes in Math., pp. 369–404, Springer, Cham, 2015. [KUB 17] Kubilius K., Mishura Y., Ralchenko K., Parameter Estimation in Fractional Diffusion Models, vol. 8 of Bocconi & Springer Series, Bocconi University Press, [Milan]; Springer, Cham, 2017. [KUH 51] Kuhn H.W., Tucker A.W., “Nonlinear programming”, Neyman J. (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492, University of California, Berkeley, 1951. [LEB 98] Le Breton A., “Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion”, Statistics & Probability Letters, vol. 38, no. 3, pp. 263–274, 1998. [LIP 89] Liptser R.S., Shiryayev A.N., Theory of Martingales, vol. 49 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1989. [LIP 01] Liptser R.S., Shiryaev A.N., Statistics of Random Processes. II. Applications, vol. 6 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 2001. [LOW 99] Lowen S.B., “Efficient generation of fractional Brownian motion for simulation of infrared focal-plane array calibration drift”, Methodology and Computing in Applied Probability, vol. 1, no. 4, pp. 445–456, 1999. [MAN 68] Mandelbrot B.B., Van Ness J.W., “Fractional Brownian motions, fractional noises and applications”, SIAM Reviews, vol. 10, pp. 422–437, 1968. [MAN 69] Mandelbrot B.B., Wallis J.R., “Computer experiments with fractional Gaussian noises: Parts 1, 2, 3”, Water Resources Research, vol. 5, pp. 228–267, 1969. [MEY 99] Meyer Y., Sellan F., Taqqu M.S., “Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion”, Journal of Fourier Analysis and Applications, vol. 5, no. 5, pp. 465–494, 1999. [MIS 08] Mishura Y.S., Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008. [MIS 09] Mishura Y.S., Banna O.L., “Approximation of fractional Brownian motion by Wiener integrals”, Theory of Probability and Mathematical Statistics, vol. 79, pp. 107–116, 2009. [MIS 17] Mishura Y., Shevchenko G., Theory and Statistical Applications of Stochastic Processes, ISTE Ltd and John Wiley & Sons, Inc., 2017. [MIS 18] Mishura Y., Zili M., Stochastic Analysis of Mixed Fractional Gaussian Processes, ISTE Press Ltd, London and Elsevier Ltd, Oxford, 2018.

References

261

[MUR 11] Muravlëv A.A., “Representation of a fractional Brownian motion in terms of an infinite-dimensional Ornstein–Uhlenbeck process”, Russian Math. Surveys, vol. 66, no. 2, pp. 439–441, 2011. [NIE 04] Nieminen A., “Fractional Brownian motion and martingale-differences”, Statistics & Probability Letters, vol. 70, no. 1, pp. 1–10, 2004. [NOR 99a] Norros I., Mannersalo P., Wang J.L., “Simulation of fractional Brownian motion with conditionalized random midpoint displacement”, Advances in Performance Analysis, vol. 2, no. 1, pp. 77–101, 1999. [NOR 99b] Norros I., Valkeila E., Virtamo J., “An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions”, Bernoulli, vol. 5, no. 4, pp. 571–587, 1999. [NOU 12] Nourdin I., Selected Aspects of Fractional Brownian Motion, vol. 4 of Bocconi & Springer Series, Springer, Milan; Bocconi University Press, Milan, 2012. [NUA 02] Nualart D., Răşcanu A., “Differential equations driven by fractional Brownian motion”, Collectanea Mathematica, vol. 53, no. 1, pp. 55–81, 2002. [NUA 03] Nualart D., “Stochastic integration with respect to fractional Brownian motion and applications”, Stochastic Models (Mexico City, 2002), vol. 336 of Communications in Contemporary Mathematics, pp. 3–39, American Mathematical Society, Providence, RI, 2003. [OLV 74] Olver F. W.J., Asymptotics and Special Functions, Academic Press, San Diego, CA, 1974. [PAX 97] Paxson V., “Fast, approximate synthesis of fractional Gaussian noise for generating self-similar network traffic”, ACM SIGCOMM Computer Communication Review, vol. 27, no. 5, pp. 5–18, 1997. [PEL 95] Peltier R.-F., Lévy Véhel J., “Multifractional Brownian motion: Definition and preliminary results”, INRIA Research Report, vol. 2645, 1995. [PER 02] Perrin E., Harba R., Jennane R. et al., “Fast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises”, IEEE Signal Processing Letters, vol. 9, no. 11, pp. 382–384, 2002. [PIP 04] Pipiras V., “On the usefulness of wavelet-based simulation of fractional Brownian motion”, Preprint. Available at: http://www.stat.unc.edu/faculty/ pipiras, 2004. [PIP 05] Pipiras V., “Wavelet-based simulation of fractional Brownian motion revisited”, Applied and Computational Harmonic Analysis, vol. 19, no. 1, pp. 49– 60, 2005. [PRO 05] Protter P.E., Stochastic Integration and Differential Equations, vol. 21 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2005. [RAL 10] Ralchenko K.V., Shevchenko G.M., “Path properties of multifractal Brownian motion”, Theory of Probability and Mathematical Statistics, vol. 80, pp. 119–130, 2010.

262

Fractional Brownian Motion

[RAL 11a] Ralchenko K.V., “Approximation of multifractional Brownian motion by absolutely continuous processes”, Theory of Probability and Mathematical Statistics, vol. 82, pp. 115–127, 2011. [RAL 11b] Ralchenko K.V., Shevchenko G.M., “Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations”, Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1460–1475, 2011. [RAL 12] Ralchenko K.V., Shevchenko G.M., “Smooth approximations for fractional and multifractional fields”, Random Operators and Stochastic Equations, vol. 20, no. 3, pp. 209–232, 2012. [SAM 94] Samorodnitsky G., Taqqu M.S., Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, 1994, Stochastic models with infinite variance. [SAM 06] Samorodnitsky G., “Long range dependence”, Foundations and Trends(R) in Stochastic Systems, vol. 1, no. 3, pp. 163–257, 2006. [SCH 97] Schechter E., Handbook of Analysis and Its Foundations, Academic Press, Inc., San Diego, CA, 1997. [SEL 95] Sellan F., “Synthèse de mouvements browniens fractionnaires à l’aide de la transformation par ondelettes”, C. R. Acad. Sci. Paris Sér. I Math., vol. 321, no. 3, pp. 351–358, 1995. [SHE 15] Shevchenko G., “Fractional Brownian motion in a nutshell”, Analysis of Fractional Stochastic Processes, vol. 36 of International Journal of Modern Physics: Conference Series, Pageid. 1560002, World Science Publishing, Hackensack, NJ, 2015. [SHE 16] Shen G., Yin X., Yan L., “Approximation of the Rosenblatt sheet”, Mediterranean Journal of Mathematics, vol. 13, no. 4, pp. 2215–2227, 2016. [SHK 14] Shklyar S., Shevchenko G., Mishura Y. et al., “Approximation of fractional Brownian motion by martingales”, Methodology and Computing in Applied Probability, vol. 16, no. 3, pp. 539–560, 2014. [TAQ 75] Taqqu M.S., “Weak convergence to fractional Brownian motion and to the Rosenblatt process”, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 31, pp. 287–302, 1974/75. [VOS 88] Voss R.F., “Fractals in nature: from characterization to simulation”, The Science of Fractal Images, pp. 21–70, Springer, 1988. [WIL 03] Willinger W., Paxson V., Riedi R.H. et al., “Long-range dependence and data network traffic”, Theory and Applications of Long-range Dependence, pp. 373–407, Birkhäuser Boston, Boston, MA, 2003. [WOL 76] Wolfe P., “Finding the nearest point in a polytope”, Mathematical Programming , vol. 11, pp. 128–149, 1976.

References

263

[WOO 94] Wood A. T.A., Chan G., “Simulation of stationary Gaussian processes in [0, 1]d ”, Journal of Computational and Graphical Statistics, vol. 3, no. 4, pp. 409–432, 1994. [YAN 15] Yan L., Li Y., Wu D., “Approximating the Rosenblatt process by multiple Wiener integrals”, Electronic Communications in Probability, vol. 20, no. 11, pp. 1–16, 2015. [ZÄH 98] Zähle M., “Integration with respect to fractal functions and stochastic calculus. I”, Probability Theory and Related Fields, vol. 111, no. 3, pp. 333–374, 1998. [ZÄH 99] Zähle M., “On the link between fractional and stochastic calculus”, Stochastic dynamics (Bremen, 1997), pp. 305–325, Springer, New York, 1999.

Index

A, B, C affinely independent columns, 209 points, 207 approximation of fBm absolute continuous, 164 by semimartingales, 157 by series of integrals, 153 multifractional Brownian motion, 171 beta function, 182 binomial coefficient, 187 Brownian motion, 196 fractional, 2 Chebyshev center, 34 algorithm, 205 Cholesky decomposition, 32, 240 circulant matrix, 245 method, 244 circumcenter, 205 circumsphere, 206 constrained least squares, 208 criterion function, 34 D, E, F deflate step, 228 digamma function, 182 Euler constant, 182

integral of the first kind, 182 reflection formula, 181 feasible point, 214 filtration, 195 fractional Brownian motion, 2 discrete-time, 31 calculus, 198 derivative, 198 Gaussian noise, 240 integral, 198 fully constrained least squares, 209 algorithm, 213 G, H, I gamma function, 181 Gauss formula, 182 Gaussian process, 196 Hosking method, 242 Hurst index, 2 inequality Burkholder–Davis–Gundy, 197 Garsia–Rodemich–Rumsey, 194 integral fractional, 198 generalized Lebesgue–Stieltjes, 199 Wiener, 196

Fractional Brownian Motion: Approximations and Projections, First Edition. Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

266

Fractional Brownian Motion

L, M, P, R Legendre formula, 181 Mandelbrot–Van Ness representation, 3 martingale, 196 Molchan kernel, 3 representation, 3 multifractional Brownian motion, 171 principal functional, 2 profile, 214 Riemann zeta function, 184 Riemann–Liouville process, 51 S, T, W self-similar, 2 semimartingale, 196

“similar” functions, 118 Simpson formula, 125 slope function, 191 stochastic process, 195 adapted, 195 sum-constrained constrained least squares, 209 sum-constrained least squares algorithm, 211 tetragamma function, 182 theorem Hahn–Banach, 193 Karush–Kuhn–Tucker, 225 on normal correlation, 195 trigamma function, 182 Weyl representation, 199 Wiener integration, 138, 196 process, 196

Other titles from

in Mathematics and Statistics

2019 GANA Kamel, BROC Guillaume Structural Equation Modeling with lavaan VOTSI Irene, LIMNIOS Nikolaos, PAPADIMITRIOU Eleftheria, TSAKLIDIS George Earthquake Statistical Analysis through Multi-state Modeling (Statistical Methods for Earthquakes Set – Volume 2)

2018 AZAÏS Romain, BOUGUET Florian Statistical Inference for Piecewise-deterministic Markov Processes IBRAHIMI Mohammed Mergers & Acquisitions: Theory, Strategy, Finance PARROCHIA Daniel Mathematics and Philosophy

2017 CARONI Chysseis First Hitting Time Regression Models: Lifetime Data Analysis Based on Underlying Stochastic Processes (Mathematical Models and Methods in Reliability Set – Volume 4) CELANT Giorgio, BRONIATOWSKI Michel Interpolation and Extrapolation Optimal Designs 2: Finite Dimensional General Models CONSOLE Rodolfo, MURRU Maura, FALCONE Giuseppe Earthquake Occurrence: Short- and Long-term Models and their Validation (Statistical Methods for Earthquakes Set – Volume 1) D’AMICO Guglielmo, DI BIASE Giuseppe, JANSSEN Jacques, MANCA Raimondo Semi-Markov Migration Models for Credit Risk (Stochastic Models for Insurance Set – Volume 1) GONZÁLEZ VELASCO Miguel, del PUERTO GARCÍA Inés, YANEV George P. Controlled Branching Processes (Branching Processes, Branching Random Walks and Branching Particle Fields Set – Volume 2) HARLAMOV Boris Stochastic Analysis of Risk and Management (Stochastic Models in Survival Analysis and Reliability Set – Volume 2) KERSTING Götz, VATUTIN Vladimir Discrete Time Branching Processes in Random Environment (Branching Processes, Branching Random Walks and Branching Particle Fields Set – Volume 1) MISHURA YULIYA, SHEVCHENKO Georgiy Theory and Statistical Applications of Stochastic Processes NIKULIN Mikhail, CHIMITOVA Ekaterina Chi-squared Goodness-of-fit Tests for Censored Data (Stochastic Models in Survival Analysis and Reliability Set – Volume 3)

SIMON Jacques Banach, Fréchet, Hilbert and Neumann Spaces (Analysis for PDEs Set – Volume 1)

2016 CELANT Giorgio, BRONIATOWSKI Michel Interpolation and Extrapolation Optimal Designs 1: Polynomial Regression and Approximation Theory CHIASSERINI Carla Fabiana, GRIBAUDO Marco, MANINI Daniele Analytical Modeling of Wireless Communication Systems (Stochastic Models in Computer Science and Telecommunication Networks Set – Volume 1) GOUDON Thierry Mathematics for Modeling and Scientific Computing KAHLE Waltraud, MERCIER Sophie, PAROISSIN Christian Degradation Processes in Reliability (Mathematial Models and Methods in Reliability Set – Volume 3) KERN Michel Numerical Methods for Inverse Problems RYKOV Vladimir Reliability of Engineering Systems and Technological Risks (Stochastic Models in Survival Analysis and Reliability Set – Volume 1)

2015 DE SAPORTA Benoîte, DUFOUR François, ZHANG Huilong

Numerical Methods for Simulation and Optimization of Piecewise Deterministic Markov Processes DEVOLDER Pierre, JANSSEN Jacques, MANCA Raimondo Basic Stochastic Processes LE GAT Yves Recurrent Event Modeling Based on the Yule Process (Mathematical Models and Methods in Reliability Set – Volume 2)

2014 COOKE Roger M., NIEBOER Daan, MISIEWICZ Jolanta Fat-tailed Distributions: Data, Diagnostics and Dependence (Mathematical Models and Methods in Reliability Set – Volume 1) MACKEVIČIUS Vigirdas Integral and Measure: From Rather Simple to Rather Complex PASCHOS Vangelis Th Combinatorial Optimization – 3-volume series – 2nd edition Concepts of Combinatorial Optimization / Concepts and Fundamentals – volume 1 Paradigms of Combinatorial Optimization – volume 2 Applications of Combinatorial Optimization – volume 3

2013 COUALLIER Vincent, GERVILLE-RÉACHE Léo, HUBER Catherine, LIMNIOS Nikolaos, MESBAH Mounir Statistical Models and Methods for Reliability and Survival Analysis JANSSEN Jacques, MANCA Oronzio, MANCA Raimondo Applied Diffusion Processes from Engineering to Finance SERICOLA Bruno Markov Chains: Theory, Algorithms and Applications

2012 BOSQ Denis Mathematical Statistics and Stochastic Processes CHRISTENSEN Karl Bang, KREINER Svend, MESBAH Mounir Rasch Models in Health DEVOLDER Pierre, JANSSEN Jacques, MANCA Raimondo Stochastic Methods for Pension Funds

2011 MACKEVIČIUS Vigirdas Introduction to Stochastic Analysis: Integrals and Differential Equations MAHJOUB Ridha Recent Progress in Combinatorial Optimization – ISCO2010 RAYNAUD Hervé, ARROW Kenneth Managerial Logic

2010 BAGDONAVIČIUS Vilijandas, KRUOPIS Julius, NIKULIN Mikhail Nonparametric Tests for Censored Data BAGDONAVIČIUS Vilijandas, KRUOPIS Julius, NIKULIN Mikhail Nonparametric Tests for Complete Data IOSIFESCU Marius et al. Introduction to Stochastic Models VASSILIOU PCG Discrete-time Asset Pricing Models in Applied Stochastic Finance

2008 ANISIMOV Vladimir Switching Processes in Queuing Models FICHE Georges, HÉBUTERNE Gérard Mathematics for Engineers HUBER Catherine, LIMNIOS Nikolaos et al. Mathematical Methods in Survival Analysis, Reliability and Quality of Life JANSSEN Jacques, MANCA Raimondo, VOLPE Ernesto Mathematical Finance

2007 HARLAMOV Boris Continuous Semi-Markov Processes

2006 CLERC Maurice Particle Swarm Optimization

WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA.