Fractional Brownian Motion Approximations and Projections

This monograph studies the relationships between fractional Brownian motion (fBm) and other processes of more simple form. In particular, this book solves the problem of the projection of fBm onto the space of Gaussian martingales that can be represented as Wiener integrals with respect to a Wiener process. It is proved that there exists a unique martingale closest to fBm in the uniform integral norm. Numerical results concerning the approximation problem are given. The upper bounds of distances from fBm to the different subspaces of Gaussian martingales are evaluated and the numerical calculations are involved. The approximations of fBm by a uniformly convergent series of Lebesgue integrals, semimartingales and absolutely continuous processes are presented.

As auxiliary but interesting results, the bounds from below and from above for the coefficient appearing in the representation of fBm via the Wiener process are established and some new inequalities for Gamma functions, and even for trigonometric functions, are obtained.

Oksana Banna is Assistant Professor at the Department of Economic Cybernetics at Taras Shevchenko National University of Kyiv (KNU) in Ukraine.

Yuliya Mishura is Full Professor and Head of the Department of Probability, Statistics and Actuarial Mathematics at KNU.

Kostiantyn Ralchenko is Associate Professor at the Department of Probability, Statistics and Actuarial Mathematics at KNU.

Sergiy Shklyar is Senior Researcher at the Department of Probability, Statistics and Actuarial Mathematics at KNU.

Notations ix

Introduction xiii

Chapter 1. Projection of fBm on the Space of Martingales 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 function for functional with the Molchan kernel and H ∈ (1/2,1) 17

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

Chapter 2. Distance Between fBm and Subclasses of Gaussian Martingales 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) = a₀s^α + a₁s^α+1 80

2.1.6. fBm and Wiener integrals involving integrands k₁ + k₂s^α 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 c_H 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 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

Chapter 3. Approximation of fBm by Various Classes of Stochastic Processes 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

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

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