Probability and Its Applications
Published in association with the Applied Probability Trust
Editors: J. Gani, C.C. Heyde, P. Jagers, T.G. Kurtz
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Probability and Its Applications
Anderson: Continuous-Time Markov Chains.
Azencott/Dacunha-Castelle: Series of Irregular Observations.
Bass: Diffusions and Elliptic Operators.
Bass: Probabilistic Techniques in Analysis.
Berglund/Gentz: Noise-Induced Phenomena in Slow-Fast Dynamical Systems.
Biagini/Hu/Øksendal/Zhang: Stochastic Calculus for Fractional Brownian
Motion and Applications.
Chen: Eigenvalues, Inequalities and Ergodic Theory.
Choi: ARMA Model Identification.
Costa/Fragoso/Marques: Discrete-Time Markov Jump Linear Systems.
Daley/Vere-Jones: An Introduction to the Theory of Point Processes Volume I:
Elementary Theory and Methods, Second Edition.
de la Peña/Giné: Decoupling: From Dependence to Independence.
Durrett: Probability Models for DNA Sequence Evolution.
Galambos/Simonelli: Bonferroni-type Inequalities with Applications.
Gani (Editor): The Craft of Probabilistic Modelling.
Grandell: Aspects of Risk Theory.
Gut: Stopped Random Walks.
Guyon: Random Fields on a Network.
Kallenberg: Foundations of Modern Probability, Second Edition.
Last/Brandt: Marked Point Processes on the Real Line.
Leadbetter/Lindgren/Rootzén: Extremes and Related Properties of Random
Sequences and Processes.
Molchanov: Theory of Random Sets.
Nualart: The Malliavin Calculus and Related Topics.
Rachev/Rüschendorf: Mass Transportation Problems. Volume I: Theory.
Rachev/Rüschendorf: Mass Transportation Problems. Volume II: Applications.
Resnick: Extreme Values, Regular Variation and Point Processes.
Schmidli: Stochastic Control in Insurance.
Shedler: Regeneration and Networks of Queues.
Silvestrov: Limit Theorems for Randomly Stopped Stochastic Processes.
Thorisson: Coupling, Stationarity and Regeneration.
Todorovic: An Introduction to Stochastic Processes and Their Applications.
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Francesca Biagini
Bernt Øksendal
Yaozhong Hu
Tusheng Zhang
Stochastic Calculus
for Fractional Brownian
Motion and Applications
123
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Francesca Biagini, PhD
Mathematisches Institut, LMU München
Theresienstr. 39 D 80333
Munich, Germany
Yaozhong Hu, PhD
Department of Mathematics
University of Kansas, 405 Snow Hall
Lawrence, Kansas 66045-2142 USA
and
Center of Mathematics for Applications (CMA)
Department of Mathematics
University of Oslo, Box 1053 Blindern
N-0316, Oslo, Norway
Bernt Øksendal, PhD
Department of Mathematics
University of Oslo, Box 1053 Blindern N-0316,
Oslo
and
Norwegian School of Economics and Business
Administration (NHH)
Helleveien 30, N-5045, Bergen, Norway
Tusheng Zhang, PhD
Department of Mathematics
University of Manchester, Oxford Road
Manchester M13 9PL
and
Center of Mathematics for Applications (CMA)
Department of Mathematics
University of Oslo, Box 1053 Blindern
N-0316, Oslo, Norway
Series Editors:
J. Gani
Stochastic Analysis Group, CMA
Australian National University
Canberra ACT 0200, Australia
P. Jagers
Mathematical Statistics
Chalmers University of Technology
SE-412 96 Göteberg, Sweden
C.C. Heyde
Stochastic Analysis Group, CMA
Australian National University
Canberra ACT 0200, Australia
T.G. Kurtz
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706, USA
ISBN: 978-1-85233-996-8
e-ISBN: 978-1-84628-797-8
DOI: 10.1007/978-1-84628-797-8
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2008920683
Mathematics Subject Classification (2000): 60G05; 60G07; 60G15; 60H05; 60H10; 60H40; 60H07; 93E20
c Springer-Verlag London Limited 2008
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To Thilo and to my family,
F.B.
To Jun and to Ruilong,
Y.H.
To Eva,
B.Ø.
To Qinghua,
T.Z.
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Preface
Fractional Brownian motion (f Bm) appears naturally in the modeling of
many situations, for example, when describing
1.
2.
3.
4.
5.
6.
The widths of consecutive annual rings of a tree,
The temperature at a specific place as a function of time,
The level of water in a river as a function of time;
The characters of solar activity as a function of time,
The values of the log returns of a stock,
Financial turbulence, i.e. the empirical volatility of a stock, and other
turbulence phenomena,
7. The prices of electricity in a liberated electricity market.
In cases 1 to 5 the corresponding f Bm has Hurst coefficient H > 1/2, which
means that the process is persistent. In cases 6 and 7 the corresponding f Bm
has Hurst coefficient H < 1/2, which means that the process is anti-persistent.
For more information about some of these examples we refer to [209].
In addition to the above, it is a mathematically tractable fact that f Bm
represents a natural one-parameter extension (represented by the Hurst parameter H) of classical Brownian motion. Therefore, it is natural to ask if
a stochastic calculus for f Bm can be developed. This is not obvious since
f Bm is not a semimartingale (except when H = 1/2, which corresponds to
the classical Brownian motion case). Moreover, it is not a Markov process either; so the most useful and efficient classical mathematical machineries and
techniques for stochastic calculus are not available in the f Bm case. Therefore, it is necessary to develop these techniques from scratch for the f Bm. It
turns out that this can be done by exploiting the fact that f Bm is a Gaussian
process.
The purpose of this book is to explain this in detail and to give applications
of the resulting theory. More precisely, we will investigate the main approaches
used to develop a stochastic calculus for f Bm and their relations. We also give
some applications, including discussions of the (sometimes controversial) use
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VIII
Preface
of f Bm in finance, stochastic partial differential equations, stochastic optimal,
control and local time for f Bm.
As shown by the reference section, there is a large literature concerning
stochastic calculus for f Bm and its applications. We have tried to cite rigorously every paper, preprint, or book we were aware of, and we apologize if we
accidentally overlooked some works.
We want to thank Birgit Beck, Christian Bender, Catriona M. Byrne,
Alessandra Cretarola, Robert Elliott, Nils Christian Framstad, Serena Fuschini,
Thilo Meyer-Brandis, Kirsten Minkos, Sebastian Queißer, Donna Mary Salopek,
Agn`es Sulem, Esko Valkeila, John van der Hoek, and three anonymous referees
for many valuable communications and comments. Yaozhong Hu acknowledges
the support of the National Science Foundation under Grant No. DMS0204613
and DMS0504783. We are also very grateful to our editors Karen Borthwick,
Helen Desmond and Stephanie Harding for their patience and support.
Any remaining errors are ours.
Francesca Biagini, Yaozhong Hu, Bernt Øksendal and Tusheng Zhang,
Munich, Lawrence, Oslo, and Manchester, November 2006.
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Part I Fractional Brownian motion
1
Intrinsic properties of the fractional Brownian motion . . . . .
1.1 Fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Stochastic integral representation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Correlation between two increments . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Long-range dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Hă
older continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Path differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 The fBm is not a semimartingale for H = 1/2 . . . . . . . . . . . . . . .
1.9 Invariance principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
8
9
10
11
11
12
14
Part II Stochastic calculus
2
Wiener and divergence-type integrals for fractional
Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Wiener integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Wiener integrals for H > 1/2 . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Wiener integrals for H < 1/2 . . . . . . . . . . . . . . . . . . . . . . .
2.2 Divergence-type integrals for f Bm . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Divergence-type integral for H > 1/2 . . . . . . . . . . . . . . . .
2.2.2 Divergence-type integral for H < 1/2 . . . . . . . . . . . . . . . .
23
23
27
34
37
39
41
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Fractional Wick Itˆ
o Skorohod (fWIS) integrals for fBm of
Hurst index H > 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Fractional white noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Fractional Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Fractional stochastic gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Fractional Wick Itˆ
o Skorohod integral . . . . . . . . . . . . . . . . . . . . . .
3.5 The φ-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Fractional Wick Itˆ
o Skorohod integrals in L2 . . . . . . . . . . . . . . . .
3.7 An Itˆ
o formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Lp estimate for the fWIS integral . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Iterated integrals and chaos expansion . . . . . . . . . . . . . . . . . . . . .
3.10 Fractional Clark Hausmann Ocone theorem . . . . . . . . . . . . . . . . .
3.11 Multidimensional fWIS integral . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Relation between the fWIS integral and the divergence-type
integral for H > 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
47
59
62
64
65
68
71
75
78
83
87
96
4
Wick Itˆ
o Skorohod (WIS) integrals for fractional
Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1 The M operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 The Wick Itˆ
o Skorohod (WIS) integral . . . . . . . . . . . . . . . . . . . . . 103
4.3 Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.5 Relation with the standard Malliavin calculus . . . . . . . . . . . . . . . 115
4.6 The multidimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5
Pathwise integrals for fractional Brownian motion . . . . . . . . . 123
5.1 Symmetric, forward and backward integrals for fBm . . . . . . . . . 123
5.2 On the link between fractional and stochastic calculus . . . . . . . . 125
5.3 The case H < 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 Relation with the divergence integral . . . . . . . . . . . . . . . . . . . . . . . 130
5.5 Relation with the fWIS integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.6 Relation with the WIS integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6
A useful summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.1 Integrals with respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.1.1 Wiener integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.1.2 Divergence-type integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.1.3 fWIS integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1.4 WIS integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.1.5 Pathwise integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 Relations among the different definitions of stochastic integral . 155
6.2.1 Relation between Wiener integrals and the divergence . . 156
6.2.2 Relation between the divergence and the fWIS integral . 156
6.2.3 Relation between the fWIS and the WIS integrals . . . . . 157
6.2.4 Relations with the pathwise integrals . . . . . . . . . . . . . . . . 158
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XI
6.3 Itˆ
o formulas with respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Part III Applications of stochastic calculus
7
Fractional Brownian motion in finance . . . . . . . . . . . . . . . . . . . . . 169
7.1 The pathwise integration model (1/2 < H < 1) . . . . . . . . . . . . . . 170
7.2 The WIS integration model (0 < H < 1) . . . . . . . . . . . . . . . . . . . . 172
7.3 A connection between the pathwise and the WIS model . . . . . . 179
7.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8
Stochastic partial differential equations driven by
fractional Brownian fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.1 Fractional Brownian fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.2 Multiparameter fractional white noise calculus . . . . . . . . . . . . . . 185
8.3 The stochastic Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.4 The linear heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.5 The quasi-linear stochastic fractional heat equation . . . . . . . . . . 198
9
Stochastic optimal control and applications . . . . . . . . . . . . . . . . 207
9.1 Fractional backward stochastic differential equations . . . . . . . . . 207
9.2 A stochastic maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
9.3 Linear quadratic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
9.4 A minimal variance hedging problem . . . . . . . . . . . . . . . . . . . . . . . 218
9.5 Optimal consumption and portfolio in a fractional Black and
Scholes market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.6 Optimal consumption and portfolio in presence of stochastic
volatility driven by fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10 Local time for fractional Brownian motion . . . . . . . . . . . . . . . . . 239
10.1 Local time for fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.2 The chaos expansion of local time for fBm . . . . . . . . . . . . . . . . . . 245
10.3 Weighted local time for fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.4 A Meyer Tanaka formula for fBm . . . . . . . . . . . . . . . . . . . . . . . . . . 253
10.5 A Meyer Tanaka formula for geometric fBm . . . . . . . . . . . . . . . . 255
10.6 Renormalized self-intersection local time for fBm . . . . . . . . . . . . 258
10.7 Application to finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Part IV Appendixes
A
Classical Malliavin calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
A.1 Classical white noise theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
A.2 Stochastic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
A.3 Malliavin derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
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B
Notions from fractional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
B.1 Fractional calculus on an interval . . . . . . . . . . . . . . . . . . . . . . . . . . 285
B.2 Fractional calculus on the whole real line . . . . . . . . . . . . . . . . . . . 288
C
Estimation of Hurst parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
C.1 Absolute value method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
C.2 Variance Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
C.3 Variance residuals methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
C.4 Hurst’s rescaled range (R/S ) analysis . . . . . . . . . . . . . . . . . . . . . . 291
C.5 Periodogram method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
C.6 Discrete variation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
C.7 Whittle method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
C.8 Maximum likelihood estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
C.9 Quasi maximum likelihood estimator . . . . . . . . . . . . . . . . . . . . . . . 294
D
Stochastic differential equations for fractional Brownian
motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
D.1 Stochastic differential equations with Wiener integrals . . . . . . . . 297
D.2 Stochastic differential equations with pathwise integrals . . . . . . 300
D.3 Stochastic differential equations via rough path analysis . . . . . . 305
D.3.1 Rough path analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
D.3.2 Stochastic calculus with rough path analysis . . . . . . . . . . 306
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Index of symbols and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
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Introduction
This book originates from the need of a comprehensive account of the stochastic integration theory of the fractional Brownian motion (f Bm). However
there are many important aspects of f Bm that are not discussed here. For
example, for an analysis of the theory and the applications of long-range dependence from a more statistical point of view, we refer to [81]. Our selection
of topics is based mainly on our main interests and background in corresponding research papers. However besides our (fractional and standard) white noise
approach, we have tried to provide an overview of some of the most important
methods of introducing a stochastic integral for f Bm.
After reviewing in Chapter 1 the properties of the fractional Brownian
motion, in Chapter 2 we start our tour through several definitions of stochastic
integral for f Bm by the Wiener integral since it deals with the simplest case
of deterministic integrands. We proceed then to introduce the divergence type
integral seen as adjoint operator of the stochastic derivative.
In Chapters 3 and 4 we present a stochastic integration based on the white
noise theory. In Chapter 3 the stochastic integral is introduced as an element
of fractional Hida distribution space for Hurst index 1/2 < H < 1 and then
conditions are clarified that guarantee the existence of this type of integral in
L2 . In Chapter 4 the integral is defined as an element in the classical Hida
distribution space by using the white noise theory and Malliavin calculus for
standard Brownian motion introduced in Appendix A. The main advantage
of this method with respect to the one presented in Chapter 3 is that it
permits us to define the stochastic integral for any H ∈ (0, 1). In addition it
doesn’t require the introduction of the fractional white noise theory since it
uses the well-established theory for the standard case. However, the approach
of Chapter 3 can be seen as more intrinsic.
Finally, in Chapter 5 we investigate the definition and the properties of the
pathwise integrals, respectively, symmetric, forward, and backward integrals.
All through this part we underline and investigate the relations between
the different approaches and in Chapter 6 we provide what in our eyes is
a useful summary. Here we present a synthesis of all the definitions and
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2
Introduction
relations of the several kinds of stochastic integration for f Bm together with
an overview of the Itˆ
o formula relative to each approach, trying to emphasize
how they can derive from each other by using the connections among the
different stochastic integrals.
In the second part we illustrate some application to finance, stochastic partial differential equations, stochastic optimal control, and local time for f Bm.
In the appendixes we gather the main results concerning the standard white
noise theory and Malliavin calculus for Brownian motion and fractional calculus. Without aiming at completeness, for the reader’s convenience we also
provide a short summary of the main methods used to estimate the Hurst
parameter from sequences of data and some results concerning stochastic differential equations for f Bm.
In spite of its high level of technicality, we hope that this book will provide
a reference text for further development of the theory and the applications of
f Bm.
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1
Intrinsic properties of the fractional Brownian
motion
The aim of this book is to provide a comprehensive overview and systematization of stochastic calculus with respect to fractional Brownian motion.
However, for the reader’s convenience, in this chapter we review the main
properties that make fractional Brownian motion interesting for many applications in different fields.
The main references for this chapter are [76], [156], [177], [195], [209], [215].
For further details concerning the theory and the applications of long-range
dependence from a more statistical point of view, we also refer to [81].
1.1 Fractional Brownian motion
The fractional Brownian motion was first introduced within a Hilbert space
framework by Kolmogorov in 1940 in [141], where it was called Wiener Helix.
It was further studied by Yaglom in [230]. The name fractional Brownian
motion is due to Mandelbrot and Van Ness, who in 1968 provided in [156]
a stochastic integral representation of this process in terms of a standard
Brownian motion.
Definition 1.1.1. Let H be a constant belonging to (0, 1). A fractional Brownian motion ( fBm) (B (H) (t))t≥0 of Hurst index H is a continuous and centered
Gaussian process with covariance function
E B (H) (t)B (H) (s) = 1/2(t2H + s2H − |t − s|2H ).
For H = 1/2, the fBm is then a standard Brownian motion. By Definition
1.1.1 we obtain that a standard fBm B (H) has the following properties:
1. B (H) (0) = 0 and E B (H) (t) = 0 for all t ≥ 0.
2. B (H) has homogeneous increments, i.e., B (H) (t + s) − B (H) (s) has the
same law of B (H) (t) for s, t ≥ 0.
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6
1 Intrinsic properties of the fractional Brownian motion
3. B (H) is a Gaussian process and E B (H) (t)2 = t2H , t ≥ 0, for all H ∈
(0, 1).
4. B (H) has continuous trajectories.
The existence of the fBm follows from the general existence theorem of centered Gaussian processes with given covariance functions (see [196]). We will
also give some constructions of the fBm through the white noise theory for
our special purposes in later chapters. The fBm is divided into three very
different families corresponding to 0 < H < 1/2, H = 1/2, and 1/2 < H < 1,
respectively, as we will see in the sequel. It was Mandelbrot that named the
parameter H of B (H) after the name of the hydrologist Hurst, who made
a statistical study of yearly water run-offs of the Nile river (see [129]). He
considered the values δ1 , . . . , δn of n successive yearly run-offs and their corn
responding cumulative value ∆n =
i=1 δi over the period from the year
662 until 1469. He discovered that the behavior of the normalized values of
the amplitude of the deviation from the empirical mean was approximately
n
cnH , where H = 0.7. Moreover, the distribution of ∆n = i=1 δi was approximately the same as nH δ1 , with H > 1/2. Hence, this phenomenon could
not be modeled by using a process with independent increments, but rather
the δi could be thought as the increments of a fBm. Because of this study,
Mandelbrot introduced the name Hurst index.
1.2 Stochastic integral representation
Here we discuss some of the integral representations for the fBm. In [156], it
is proved that the process
Z(t) =
=
1
Γ (H + 1/2)
1
Γ (H + 1/2)
H−1/2
R
(t − s)+
H−1/2
− (−s)+
dB(s)
0
−∞
(t − s)H−1/2 − (−s)H−1/2 dB(s)
t
(t − s)H−1/2 dB(s)
+
(1.1)
0
where B(t) is a standard Brownian motion and Γ represents the gamma
function, is a fBm with Hurst index H ∈ (0, 1). If B(t) is replaced by a
complex-valued Brownian motion, the integral (1.1) gives the complex fBm.
By following [177] we sketch a proof for the representation (1.1). For further
detail we refer also to [207]. First we notice that Z(t) is a continuous centered
Gaussian process. Hence, we need only to compute the covariance functions.
In the following computations we drop the constant 1/Γ (H +1/2) for the sake
of simplicity. We obtain
H−1/2
E Z 2 (t) =
R
(t − s)+
H−1/2
− (−s)+
2
ds
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1.2 Stochastic integral representation
H−1/2
= t2H
R
(1 − u)+
7
2
H−1/2
− (−u)+
du
= C(H)t2H ,
where we have used the change of variable s = tu. Analogously, we have that
E |Z(t) − Z(s)|2 =
H−1/2
R
(t − u)+
H−1/2
− (s − u)+
H−1/2
R
ds
H−1/2
(t − s − u)+
= t2H
2
− (−u)+
2
du
= C(H)|t − s|2H .
Now
E [Z(t)Z(s)] = −
=
1
E |Z(t) − Z(s)|2 − E Z(t)2 − E Z(s)2
2
1 2H
(t + s2H − |t − s|2H ).
2
Hence we can conclude that Z(t) is a fBm of Hurst index H.
Several other stochastic integral representations have been developed in
the literature. By [207], we get the following spectral representation of fBm
B (H) (t) :=
1
C2 (H)
R
eits − 1 1/2−H ˜
|s|
dB(s),
is
˜
where B(s)
= B 1 + iB 2 is a complex Brownian measure on R such that
1
1
B (A) = B (−A), B 2 (A) = −B 2 (−A), and E B 1 (A)2 = E B 2 (A)2 =
|A|/2 for every A ∈ B(R), and
π
HΓ (2H) sin Hπ
C2 (H) =
1/2
,
where B i (A) := A dB i (t). Equation (1.1) provides an integral representation
for fBm over the whole real line. By following the approach of [172], we can
also represent the fBm over a finite interval, i.e.,
t
B (H) (t) :=
KH (t, s) dB(s),
t ≥ 0,
0
where
1. For H > 1/2,
t
|u − s|H−3/2 uH−1/2 du,
KH (t, s) = cH s1/2−H
s
where cH = [H(2H − 1)/β(2 − 2H, H − 1/2)]
1/2
and t > s.
(1.2)
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8
1 Intrinsic properties of the fractional Brownian motion
2. For H < 1/2,
t
s
KH (t, s) = bH
H−1/2
(t − s)H−1/2
1
− H−
2
(1.3)
t
(u − s)
1/2−H
H−1/2 H−3/2
s
u
du
s
with bH = [2H/((1 − 2H)β(1 − 2H, H + 1/2))]
1/2
and t > s.
For the proof, we refer to [114], [172], and [177]. Note that this representation
is canonical in the sense that the filtrations generated by B (H) and B coincide.
In Chapter 2 a definition of stochastic integral with respect to fBm will
be introduced by exploiting the stochastic integral representation of B (H) in
terms of (1.2) and of (1.3).
Remark 1.2.1. Integral representations that change fBm of arbitrary Hurst
index K into fBm of index H have been studied in [191], [133] and [134]. In
Theorem 1.1 of [191] it is shown that for any K ∈ (0, 1), there exists a unique
˜ (K) such that for all t ∈ R there holds
K-fBm B
B (H) (t) = c˜H,K
R
˜ (K) (s),
(t − s)H−K
dB
− (−s)H−K
+
+
a.s.,
(1.4)
1/2
with c˜H,K = 1/Γ (H − K + 1) (Γ (2K + 1) sin (πK)/Γ (2H + 1) sin (πH)) .
In Theorem 5.1 of [133] integration is carried out on [0, t] and showed that for
given K ∈ (0, 1), there exists a unique K-fBm B (K) (t), t ≥ 0, such that for
all t ≥ 0 we have a.s. that
t
(t − s)H−K
B (H) (t) = cH,K
0
·F
1 − K − H, H − K, 1 + H − K,
s−t
s
dB (K) (s),
(1.5)
where F is Gauss hypergeometric function and
cH,K =
1
Γ (H − K + 1)
2HΓ (H + 1/2)Γ (3/2 − H)Γ (2 − 2K)
2KΓ (K + 1/2)Γ (3/2 − K)Γ (2 − 2H)
1/2
.
In [134] an analytical connection between (1.4) and (1.5) is proved.
1.3 Correlation between two increments
For H = 1/2, B (H) is a standard Brownian motion; hence, in this case the
increments of the process are independent. On the contrary, for H = 1/2 the
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1.4 Long-range dependence
9
increments are not independent. More precisely, by Definition 1.1.1 we know
that the covariance between B (H) (t + h) − B (H) (t) and B (H) (s + h) − B (H) (s)
with s + h ≤ t and t − s = nh is
ρH (n) =
1 2H
h [(n + 1)2H + (n − 1)2H − 2n2H ].
2
In particular, we obtain that two increments of the form B (H) (t+h)−B (H) (t)
and B (H) (t + 2h) − B (H) (t + h) are positively correlated for H > 1/2, while
they are negatively correlated for H < 1/2. In the first case the process
presents an aggregation behavior and this property can be used in order to
describe “cluster” phenomena (systems with memory and persistence). In the
second case it can be used to model sequences with intermittency and antipersistence.
1.4 Long-range dependence
Definition 1.4.1. A stationary sequence (Xn )n∈N exhibits long-range dependence if the autocovariance functions ρ(n) := cov(Xk , Xk+n ) satisfy
lim
n→∞
ρ(n)
=1
cn−α
for some constant c and α ∈ (0, 1). In this case, the dependence between Xk
and Xk+n decays slowly as n tends to infinity and
∞
ρ(n) = ∞.
n=1
Hence, we obtain immediately that the increments Xk := B (H) (k) − B (H) (k −
1) of B (H) and Xk+n := B (H) (k + n) − B (H) (k + n − 1) of B (H) have the
long-range dependence property for H > 1/2 since
ρH (n) =
1
[(n + 1)2H + (n − 1)2H − 2n2H ] ∼ H(2H − 1)n2H−2
2
as n goes to infinity. In particular,
lim
n→∞
ρH (n)
= 1.
H(2H − 1)n2H−2
Summarizing, we obtain
1. For H > 1/2,
2. For H < 1/2,
∞
n=1
∞
n=1
ρH (n) = ∞.
|ρH (n)| < ∞.
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10
1 Intrinsic properties of the fractional Brownian motion
There are alternative definitions of long-range dependence. We recall that a
function L is slowly varying at zero (respectively, at infinity) if it is bounded
on a finite interval and if, for all a > 0, L(ax)/L(x) tends to 1 as x tends to
zero (respectively, to infinity).
We introduce now the spectral density of the autocovariance functions ρ(k)
f (λ) :=
1
2π
∞
e−iλk ρ(k)
k=−∞
for λ ∈ [−π, π].
Definition 1.4.2. For stationary sequences (Xn )n∈N with finite variance, we
say that (Xn )n∈N exhibits long-range dependence if one of the following holds:
n
1. limn→∞ ( k=−n ρ(k))/(cnβ L1 (n)) = 1 for some constant c and β ∈ (0, 1).
2. limk→∞ ρ(k)/ck −γ L2 (k) = 1 for some constant c and γ ∈ (0, 1).
3. limλ→0 f (λ)/c|λ|−δ L3 (|λ|) = 1 for some constant c and δ ∈ (0, 1).
Here L1 , L2 are slowly varying functions at infinity, while L3 is slowly varying
at zero.
Lemma 1.4.3. For fBm B (H) of Hurst index H ∈ (1/2, 1), the three definitions of long-range dependence of Definition 1.4.2 are equivalent. They hold
with the following choice of parameters and slowly varying functions:
1. β = 2H − 1, L1 (x) = 2H.
2. γ = 2 − 2H, L2 (x) = H(2H − 1).
3. δ = 2H − 1, L3 (x) = π −1 HΓ (2H) sin πH.
Proof. For the proof, we refer to Section 4 in [221].
For a survey on theory and applications of long-range dependence, see also
[81].
1.5 Self-similarity
By following [209], we introduce the following:
Definition 1.5.1. We say that an Rd -valued random process X = (Xt )t≥0 is
self-similar or satisfies the property of self-similarity if for every a > 0 there
exists b > 0 such that
Law(Xat , t ≥ 0) = Law(bXt , t ≥ 0).
(1.6)
Note that (1.6) means that the two processes Xat and bXt have the same
finite-dimensional distribution functions, i.e., for every choice t0 , . . . , tn in R,
P (Xat0 ≤ x0 , . . . , Xatn ≤ xn ) = P (bXt0 ≤ x0 , . . . , bXtn ≤ xn )
for every x0 , . . . , xn in R.
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1.7 Path differentiability
11
Definition 1.5.2. If b = a−H in Definition 1.5.1, then we say that X =
(Xt )t≥0 is a self-similar process with Hurst index H or that it satisfies the
property of (statistical) self-similarity with Hurst index H. The quantity D =
1/H is called the statistical fractal dimension of X.
Since the covariance function of the fBm is homogeneous of order 2H, we
obtain that B (H) is a self-similar process with Hurst index H, i.e., for any
constant a > 0 the processes B (H) (at) and a−H B (H) (t) have the same distribution law.
1.6 Hă
older continuity
We recall that according to the Kolmogorov criterion (see [228]), a process
X = (Xt )t∈R admits a continuous modification if there exist constants α ≥ 1,
β > 0, and k > 0 such that
E [|X(t) − X(s)|α ] ≤ k|t − s|1+β
for all s, t ∈ R.
Theorem 1.6.1. Let H ∈ (0, 1). The fBm B (H) admits a version whose sample paths are almost surely Hă
older continuous of order strictly less than H.
Proof. We recall that a function f : R → R is Hă
older continuous of order ,
0 < 1, and write f ∈ C α (R), if there exists M > 0 such that
|f (t) − f (s)| ≤ M |t − s|α ,
for every s, t ∈ R. For any α > 0 we have
E |B (H) (t) − B (H) (s)|α = E |B (H) (1)|α |t − s|αH ;
hence, by the Kolmogorov criterion we get that the sample paths of B (H) are
almost everywhere Hăolder continuous of order strictly less than H. Moreover,
by [9] we have
|B (H) (t)|
= cH
lim sup
log log t−1
t→0+ tH
with probability one, where cH is a suitable constant. Hence B (H) cannot have
sample paths with Hă
older continuitys order greater than H.
1.7 Path dierentiability
By [156] we also obtain that the process B (H) is not mean square differentiable
and it does not have differentiable sample paths.
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12
1 Intrinsic properties of the fractional Brownian motion
Proposition 1.7.1. Let H ∈ (0, 1). The fBm sample path B (H) (.) is not
differentiable.
In fact, for every t0 ∈ [0, ∞)
lim sup |
t→t0
B (H) (t) − B (H) (t0 )
|=∞
t − t0
with probability one.
Proof. Here we recall the proof of [156]. Note that we assume B (H) (0) = 0.
The result is proved by exploiting the self-similarity of B (H) . Consider the
random variable
B (H) (t) − B (H) (t0 )
Rt,t0 :=
t − t0
that represents the incremental ratio of B (H) . Since B (H) is self-similar, we
have that the law of Rt,t0 is the same of (t − t0 )H−1 B (H) (1). If one considers
the event
B (H) (s)
>d ,
A(t, ω) := sup
s
0≤s≤t
then for any sequence (tn )n∈N decreasing to 0, we have
A(tn , ω) ⊇ A(tn+1 , ω),
and
A(tn , ω) ⊇ (|
B (H) (tn )
| > d) = (|B (H) (1)| > t1−H
d).
n
tn
The thesis follows since the probability of the last term tends to 1 as n → ∞.
1.8 The fBm is not a semimartingale for H = 1/2
The fact that the fBm is not a semimartingale for H = 1/2 has been proved
by several authors. For example, for H > 1/2 we refer to [82], [150], [152].
Here we recall the proof of [195] that is valid for every H = 1/2. In order
to verify that B (H) is not a semimartingale for H = 1/2, it is sufficient to
compute the p-variation of B (H) .
Definition 1.8.1. Let (X(t))t∈[0,T ] be a stochastic process and consider a partition π = {0 = t0 < t1 < . . . < tn = T }. Put
n
p
Sp (X, π) :=
|X(tk ) − X(tk−1 )| .
i=1
The p-variation of X over the interval [0, T ] is defined as
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1.8 The fBm is not a semimartingale for H = 1/2
13
Vp (X, [0, T ]) := sup Sp (X, π),
π
where π is a finite partition of [0, T ]. The index of p-variation of a process is
defined as
I(X, [0, T ]) := inf {p > 0; Vp (X, [0, T ]) < ∞} .
We claim that
I(B (H) , [0, T ]) =
1
.
H
In fact, consider for p > 0,
n
Yn,p = npH−1
B (H)
i=1
i
n
− B (H)
i−1
n
p
.
Since B (H) has the self-similarity property, the sequence (Yn,p )n∈N has the
same distribution as
n
Y˜n,p = n−1
|B (H) (i) − B (H) (i − 1)|p .
i=1
By the Ergodic theorem (see, for example, [69]) the sequence Y˜n,p converges
almost surely and in L1 to E |B (H) (1)|p as n tends to infinity; hence, it
converges also in probability to E |B (H) (1)|p . It follows that
n
B (H)
Vn,p =
i=1
i
n
− B (H)
i−1
n
p
converges in probability respectively to 0 if pH > 1 and to infinity if pH < 1
as n tends to infinity. Thus we can conclude that I(B (H) , [0, T ]) = 1/H. Since
for every semimartingale X, the index I(X, [0, T ]) must belong to [0, 1] ∪ {2},
the fBm B (H) cannot be a semimartingale unless H = 1/2.
As a direct consequence of this fact, one cannot use the Itˆ
o stochastic calculus developed for semimartingales in order to define the stochastic integral
with respect to B (H) . In the following chapters we will summarize the different
approaches developed in the literature in order to overcome this problem.
In [53] it has been introduced the new notion of weak semimartingale and
shown that B (H) is not even a weak semimartingale if H = 1/2. A stochastic
process (X(t))t≥0 is said to be a weak semimartingale if for every T > 0 the
family of random variables
n
ai [X(ti ) − X(ti−1 )], n ≥ 1, 0 = t0 < . . . < tn = T, |ai | < 1, ai ∈ FtXi−1
i=1
is bounded in L0 . Here FX represents the natural filtration associated to the
process X. Moreover, in [53] it is shown that if B(t) is a standard Brownian
motion independent of B (H) , then the process
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14
1 Intrinsic properties of the fractional Brownian motion
M H (t) := B (H) (t) + B(t)
is not a weak semimartingale if H ∈ (0, 1/2) ∪ (1/2, 3/4), while it is a
semimartingale equivalent in law to B on any finite time interval [0, T ] if
H ∈ (3/4, 1). We refer to [53] for further details.
1.9 Invariance principle
Here we present an invariance principle for fBms due to [36].
Assume that {Xn , n = 1, 2, ...} is a stationary Gaussian sequence with
E [Xi ] = 0 and E[Xi2 ] = 1. Define
1
Zn (t) = H
n
[nt]−1
Xk ,
0 ≤ t ≤ 1,
k=0
where [·] stands for the integer part. We will show that if the covariance of
n
X is proportional to Cn2H for large n, Zn (t), t ≥ 0 converges weakly
k=0
√ k(H)
to CBt
in a suitable metric space. Let us first introduce the the metric
space. Let I = [0, 1] and denote by Lp (I) the space of Lebesgue integrable
functions with exponent p. For f ∈ Lp (I), t ∈ I, define
1/p
|f (x + h) − f (x)|p dx
ωp (f, t) = sup
|h|≤t
,
Ih
where Ih = {x ∈ I, x + h ∈ I}. For 0 < α < 1 and β > 0, consider the
real-valued function ωβα (·) defined by
ωβα (t) = tα 1 + log
and we let
f
α
ωβ
p
1
t
β
,
t > 0,
ωp (f, t)
.
α
0
= ||f ||Lp (I) + sup
The Besov space Lipp (α, β) is the class of functions f in Lp (I) such that
ωα
ωα
f p β < ∞. Lipp (α, β) endowed with the norm || · ||p β is a nonseparable
Banach space. Let Bpα,β denote the separable subspace of Lipp (α, β) formed
by functions f ∈ Lipp (α, β) satisfying ωp (f, t) = o(ωβα (t)) as t → 0. For
a continuous function f , denote by {Cn (f ), n ≥ 0} the coefficients of the
decomposition of f in the Schauder basis given by
C0 (f ) = f (0), C1 (f ) = f (1) − f (0),
and for n = 2j + k, j ≥ 0, and k = 0, ..., 2j − 1,
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1.9 Invariance principle
2k − 1
2j+1
Cn (f ) = 2 · 2j/2 f
−
2k
2j+1
1
f
2
+f
2k − 2
2j+1
15
.
The following characterization theorem proved in [55] will be used.
Theorem 1.9.1. 1. If α > 1/p, then Lipp (α, β) is the space of continuous
functions with the following equivalence of norms:
ωα
||f ||p β ∼ max |C0 (f )|, |C1 (f )|,
sup
j≥0
2−j(1/2−α+1/p)
(1 + j)β
2j+1
1/p
|Cn (f )|p
.
n=2j +1
2. f belongs to Bpα,β if and only if
2−j(1/2−α+1/p)
lim
j→∞
(1 + j)β
2j+1
1/p
|Cn (f )|p
= 0.
n=2j +1
Lemma 1.9.2. Let 1 ≤ p < ∞, 1/p < α < 1, and β > 0. A set F of
measurable functions f : I → R is relatively compact in Bpα,β if
ωα
1. supf ∈F ||f ||p β < ∞,
2. lim supδ→0 supf ∈F Kδ (f, α, β, p) = 0, where
Kδ (f, α, β, p) = sup
0
ωp (f, t)
.
ωβα (t)
Proof. It is a consequence of the Frechet–Kolmogorov theorem: a subset K ⊂
Lp (I) is relatively compact if and only if
|f (s)|p ds
sup
f ∈K
(|f (s + t) − f (s)|p ) ds = 0.
lim sup
t→0 f ∈K
< ∞,
I
I
Now assume (1) and (2) hold for a set F . To prove that F is relatively compact,
we need to show that any sequence {fn , n ≥ 1} ⊂ F admits a convergent
subsequence. Pick a sequence {fn , n ≥ 1} from F . By the Frechet–Kolmogorov
theorem, {fn , n ≥ 1} has a convergent subsequence in Lp (I). Without loss of
generality, we assume that fn → f in Lp (I). First we show that f ∈ Bpα,β . By
the Fatou lemma,
1/p
|f (x + h) − f (x)|p dx
ωp (f, t) = sup
|h|≤t
Ih
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16
1 Intrinsic properties of the fractional Brownian motion
1/p
≤ sup lim inf
|fn (x + h) − fn (x)|p dx
|h|≤t n→∞
Ih
1/p
≤ supn sup
|fn (x + h) − fn (x)|p dx
|h|≤t
.
Ih
This together with assumption (2) implies ωp (f, t) = o(ωβα (t)) as t → 0. Hence,
f ∈ Bpα,β . We will finish the proof by showing fn → f also in Bpα,β . From the
definition of the norm in Bpα,β , it is sufficient to show
lim sup
n→∞ 0
ωp (fn − f, t)
= 0.
ωβα (t)
For any 0 < δ < 1, we have
ωp (fn − f, t)
ωp (fn − f, t)
ωp (fn − f, t)
≤ sup
+ sup
.
α
α
ωβ (t)
ωβ (t)
ωβα (t)
0
0
δ
sup
Let ε > 0. By assumption (2) we can find δ > 0 such that
sup
0
ε
ωp (fn − f, t)
≤ ,
ωβα (t)
2
for all n ≥ 1. On the other hand,
ωp (fn − f, t)
≤ cδ sup ωp (fn − f, t) ≤ 2cδ ||fn − f ||Lp (I) .
ωβα (t)
δ
δ
sup
Thus, there exists N > 0 such that for n ≥ N ,
ωp (fn − f, t)
ε
≤ .
α (t)
ω
2
δ
β
sup
Combining the above arguments, we arrive at
lim sup
n→∞ 0
ωp (fn − f, t)
= 0.
ωβα (t)
Lemma 1.9.3. Let α > 1/p and 0 < β < β . The space Lipp (α, β) is compactly embedded in Bpα,β .
Proof. Let B = {f ∈ Lipp (α, β);
ωα
||f ||p β ≤ M } be a bounded subset
ωα
of Lipp (α, β). It is clear that if β < β , then ||f ||p β
ωα
supf ∈B ||f ||p β < ∞. On the other hand,
ωα
≤ ||f ||p β . Hence,
