Fractional Brownian motion stationary and self-similar increments. It has stationary increments: For "H" > ½ the process exhibits long-range dependence, 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". With probability 1, the graph of "B"("t") has both Hausdorff dimension and box dimension of 2−"H". As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional
Geometric Brownian motion above: formula_27. import numpy as np import pandas as pd import matplotlib.pyplot as plt mu=1 n=50 dt=0.1 x0=100 x=pd.DataFrame() np.random.seed(1) for sigma in np.arange(0.8,2,0.2): x.columns=np.arange(0.8,2,0.2) plt.plot(x) plt.legend(x.columns) plt.xlabel('t') plt.ylabel('X') plt.title('Realizations of Geometric Brownian Motion with different variances\n mu=1') plt.show() GBM can be extended to the case where there are multiple correlated price paths. Each price path follows the underlying process where the Wiener processes are correlated such that formula_31 where formula_32. For the multivariate case, this implies that Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock