Fractional Brownian Motion

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)s^1/2−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 c_H for an fBm. The new upper bound for is provided. Comparison is illustrated ...