Fractional Brownian motion
In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process "B"("t") on [0, "T"], that starts at zero, has expectation zero for all "t" in [0, "T"], and has the following covariance function: where "H" is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value  leading to a smoother motion. It was introduced by . The value of "H" determines what kind of process the "fBm" is: The increment process, "X"("t") = "B"("t"+1) − "B"("t"), is known as fractional Gaussian noise. There is also a generalization of fractional Brownian motion: "n"-th order fractional Brownian motion, abbreviated as n-fBm. n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order "n" are stationary. For "n" = 1, n-fBm is classical fBm. Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown; fractional Gaussian noise  is named after mathematician Carl Friedrich Gauss. Background and definition. Prior to the introduction of the fractional Brownian motion, used the Riemann–Liouville fractional integral to define the process where integration is with respect to the white noise measure "dB"("s"). This integral turns out to be ill-suited to applications of fractional Brownian motion because of its over-emphasis of the origin . The idea instead is to use a different fractional integral of white noise to define the process: the Weyl integral for "t" > 0 (and similarly for "t" < 0). The main difference between fractional Brownian motion and  regular Brownian motion is that while the increments in Brownian Motion are independent, increments for fractional Brownian motion are not. If H > 1/2, then there is positive autocorrelation: if there is an increasing pattern in the previous steps, then it is likely that the current step will be increasing as well. If H < 1/2, the autocorrelation is negative. Properties. Self-similarity. The process is self-similar, since in terms of probability distributions: This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a fractal property. FBm can  also be defined as the unique mean-zero Gaussian process, null at the origin, with stationary and self-similar increments. Stationary increments. It has stationary increments: Long-range dependence. For "H" > ½ the process exhibits long-range dependence, Regularity. Sample-paths are almost nowhere differentiable. However, almost-all trajectories are locally Hölder continuous of any order strictly less than "H": for each such trajectory, for every "T" > 0 and for every "ε" > 0 there exists a (random) constant "c" such that for 0 < "s","t" < "T". Dimension. With probability 1, the graph of "B"("t") has both  Hausdorff dimension and box dimension of 2−"H". Integration. As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional stochastic integrals". In general though, unlike integrals with respect to regular Brownian motion, fractional stochastic integrals are not semimartingales. Frequency-domain interpretation. Just as Brownian motion can be viewed as white noise filtered by formula_8 (i.e. integrated), fractional Brownian motion is white noise filtered by formula_9 (corresponding to fractional integration). Sample paths. Practical computer realisations of an "fBm" can be generated, although they are only a finite approximation. The sample paths chosen can  be thought of as showing discrete sampled points on an "fBm" process. Three realizations are shown below, each with 1000 points of an "fBm" with Hurst parameter 0.75. Realizations of three different types of "fBm" are shown below, each showing 1000 points, the first with Hurst parameter 0.15, the second with Hurst parameter 0.55, and the third with Hurst parameter 0.95. The higher the Hurst parameter is, the smoother the curve will be. Method 1 of simulation. One can simulate sample-paths of an "fBm" using methods for generating stationary Gaussian processes with known covariance function. The simplest method relies  on the Cholesky decomposition method of the covariance matrix (explained below), which on a grid of size formula_10 has complexity of order formula_11. A more complex, but computationally faster method is the circulant embedding method of . Suppose we want to simulate the values of the "fBM" at times formula_12 using the Cholesky decomposition method. In order to compute formula_15, we can use for instance the Cholesky decomposition method. An alternative method uses the eigenvalues of formula_16: Note that the result is real-valued because formula_36. Note that since the eigenvectors are linearly independent, the matrix formula_39 is invertible. Method 2  of simulation. It is also known that where "B" is a standard Brownian motion and Where formula_47 is the Euler hypergeometric integral. Say we want to simulate an "fBm" at points formula_48. The integral may be efficiently computed by Gaussian quadrature.