Table Of ContentFractional 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.
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Contents
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter1.ProjectionoffBmontheSpaceofMartingales . . . . . . 1
1.1. fBm and its integral representations . . . . . . . . . . . . . . . . . 2
1.2. Formulation of the main problem . . . . . . . . . . . . . . . . . . 5
1.3. The lower bound for the distance between fBm
and Gaussian martingales . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4. The existence of minimizing function for the
principal functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5. An example of the principal functional with infinite
set of minimizing functions . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6. Uniqueness of the minimizing(cid:2)funct(cid:3)ion for functional
with the Molchan kernel and H ∈ 1,1 . . . . . . . . . . . . . . . . . 17
2
1.7. Representation of the minimizing function . . . . . . . . . . . . . 21
1.7.1. Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7.2. Main properties of the minimizing function . . . . . . . . . . 28
1.8. Approximation of a discrete-time fBm by martingales . . . . . . 31
1.8.1. Description of the discrete-time model . . . . . . . . . . . . . 31
1.8.2. Iterative minimization of the squared distance
using alternating minimization method. . . . . . . . . . . . . . . . . 33
1.8.3. Implementation of the alternating minimization
algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.8.4. Computation of the minimizing function . . . . . . . . . . . . 44
1.9. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
vi FractionalBrownianMotion
Chapter2.DistanceBetweenfBmandSubclassesof
GaussianMartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1. fBm and Wiener integrals with power functions . . . . . . . . . . 54
2.1.1. fBm and Wiener integrals with constant integrands . . . . . 54
2.1.2. fBm and Wiener integrals involving power integrands
with a positive exponent . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.1.3. fBm and integrands a(s) with a(s)s−α
non-decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1.4. fBm and Gaussian martingales involving power
integrands with a negative exponent . . . . . . . . . . . . . . . . . . 67
2.1.5. fBm and the integrands a(s)=a0sα+a1sα+1 . . . . . . . . . 80
2.1.6. fBm and Wiener integrals involving
integrands k1+k2sα . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.2. The comparison of distances between fBm and subspaces
of Gaussian martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.2.1. Summary of the results concerning the values
of the distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.2.2. The comparison of distances . . . . . . . . . . . . . . . . . . . 112
2.2.3. The comparison of upper and lower bounds for
the constant cH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.3. Distance between fBm and class of “similar” functions . . . . . . 118
2.3.1. Lower bounds for the distance . . . . . . . . . . . . . . . . . . 121
2.3.2. Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.4. Distance between fBm and Gaussian martingales
in the integral norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.5. Distance between fBm with Mandelbrot–Van Ness
kernel and Gaussian martingales . . . . . . . . . . . . . . . . . . . . . 129
2.5.1. Constant function as an integrand . . . . . . . . . . . . . . . . 129
2.5.2. Power function as an integrand. . . . . . . . . . . . . . . . . . 131
2.5.3. Comparison of Molchan and Mandelbrot–Van
Ness kernels . . . . . . . . . . . . . . . . . .(cid:2). . .(cid:3). . . . . . . . . . . . 132
2.6. fBm with the Molchan kernel and H ∈ 0, 1 ,
2
in relation to Gaussian martingales . . . . . . . . . . . . . . . . . . . . 133
2.7. Distance between the Wiener process and integrals
with respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.7.1. Wiener integration with respect to fBm . . . . . . . . . . . . 138
2.7.2. Wiener process and integrals of power functions
with respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
2.8. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Contents vii
Chapter3.ApproximationoffBmbyVariousClasses
ofStochasticProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.1. Approximation of fBm by uniformly convergent series of
Lebesgue integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.2. Approximation of fBm by semimartingales . . . . . . . . . . . . . 157
3.2.1. Construction and convergence of approximations . . . . . . . 157
3.2.2. Approximation of an integral with respect to fBm
by integrals with respect to semimartingales . . . . . . . . . . . . . 159
3.3. Approximation of fBm by absolutely continuous
processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.4. Approximation of multifractional Brownian motion by
absolutely continuous processes . . . . . . . . . . . . . . . . . . . . . . 171
3.4.1. Definition and examples . . . . . . . . . . . . . . . . . . . . . . 171
3.4.2. Hölder continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.4.3. Construction and convergence of approximations . . . . . . . 175
3.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Appendix1.AuxiliaryResultsfromMathematical,Functional
andStochasticAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Appendix2.EvaluationoftheChebyshevCenterofaSet
ofPointsintheEuclideanSpace . . . . . . . . . . . . . . . . . . . . . . 205
Appendix3.SimulationoffBm . . . . . . . . . . . . . . . . . . . . . . . 239
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
