Fractional Brownian Motion

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 ixIntroduction xiiiChapter 1. Projection of fBm on the Space of Martingales 11.1. fBm and its integral representations 21.2. Formulation of the main problem 51.3. The lower bound for the distance between fBm and Gaussian martingales 81.4. The existence of minimizing function for the principal functional 101.5. An example of the principal functional with infinite set of minimizing functions 121.6. Uniqueness of the minimizing function for functional with the Molchan kernel and H ∈ (1/2,1) 171.7. Representation of the minimizing function 211.7.1. Auxiliary results 211.7.2. Main properties of the minimizing function 281.8. Approximation of a discrete-time fBm by martingales 311.8.1. Description of the discrete-time model 311.8.2. Iterative minimization of the squared distance using alternating minimization method 331.8.3. Implementation of the alternating minimization algorithm 431.8.4. Computation of the minimizing function 441.9. Exercises 50Chapter 2. Distance Between fBm and Subclasses of Gaussian Martingales 532.1. fBm and Wiener integrals with power functions 542.1.1. fBm and Wiener integrals with constant integrands 542.1.2. fBm and Wiener integrals involving power integrands with a positive exponent 572.1.3. fBm and integrands a(s) with a(s)s−α non-decreasing 652.1.4. fBm and Gaussian martingales involving power integrands with a negative exponent 672.1.5. fBm and the integrands a(s) = a0sα + a1sα+1 802.1.6. fBm and Wiener integrals involving integrands k1 + k2sα 842.2. The comparison of distances between fBm and subspaces of Gaussian martingales 1092.2.1. Summary of the results concerning the values of the distances 1092.2.2. The comparison of distances 1122.2.3. The comparison of upper and lower bounds for the constant cH 1132.3. Distance between fBm and class of “similar” functions 1182.3.1. Lower bounds for the distance 1212.3.2. Evaluation 1242.4. Distance between fBm and Gaussian martingales in the integral norm 1272.5. Distance between fBm with Mandelbrot–Van Ness kernel and Gaussian martingales 1292.5.1. Constant function as an integrand 1292.5.2. Power function as an integrand 1312.5.3. Comparison of Molchan and Mandelbrot–Van Ness kernels 1322.6. fBm with the Molchan kernel and H ∈ (0,1/2), in relation to Gaussian martingales 1332.7. Distance between the Wiener process and integrals with respect to fBm 1382.7.1. Wiener integration with respect to fBm 1382.7.2. Wiener process and integrals of power functions with respect to fBm 1402.8. Exercises 150Chapter 3. Approximation of fBm by Various Classes of Stochastic Processes 1533.1. Approximation of fBm by uniformly convergent series of Lebesgue integrals 1533.2. Approximation of fBm by semimartingales 1573.2.1. Construction and convergence of approximations 1573.2.2. Approximation of an integral with respect to fBm by integrals with respect to semimartingales 1593.3. Approximation of fBm by absolutely continuous processes 1643.4. Approximation of multifractional Brownian motion by absolutely continuous processes 1713.4.1. Definition and examples 1713.4.2. Hölder continuity 1733.4.3. Construction and convergence of approximations 1753.5. Exercises 180Appendix 1. Auxiliary Results from Mathematical, Functional and Stochastic Analysis 181Appendix 2. Evaluation of the Chebyshev Center of a Set of Points in the Euclidean Space 205Appendix 3. Simulation of fBm 239Solutions 251References 257Index 265