Brownian web is what we now call the Brownian web. Formally speaking, it is a collection of one-dimensional coalescing Brownian motions starting from every space-time point in formula_1. The fact that the Brownian web consists of an "uncountable" number of Brownian motions is what makes the construction highly non-trivial. Arratia gave a construction but was unable to prove convergence of coalescing random walks to a limiting object and characterize such a limiting object. Tóth and Werner in their study of the true self-repelling motion obtained many detailed properties of this limiting object and its dual but did not prove convergence of coalescing
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