Paths of Brownian motion

Question

Suppose $B_t$ is a Brownian motion. A path of Brownian motion is a function $t \rightarrow B_t(\omega)$. Now, I want to plot two paths of Brownian motion for fixed $\omega_1$ and $\omega_2$. Isn't the same as plotting arbitrarily two Brownian motions, or how should fixed $\omega$ be understood here?

Answer 1

For all omegas (or almost surely every) it holds that the path is continous, undifferentiable... and yes, one particular omega corresponds to one particular path. So if you generate BM in R, one generation corresponds in this case to one omega.

E.g. this code gives 10 different paths, i.e. ten different $B_t(\omega_i), i=1, ..., 10$.

\begin{verbatim}

n = 1000

t = 100

No.Ex = 10

steps = seq(0,t,length=n+1)

A = replicate(No.Ex, {

bm <- c(0, cumsum(rnorm(n,0,sqrt(t/n))))

})

cols = rainbow(No.Ex)

matplot(A, type = "l", col = cols, lty = 1)

\end{verbatim}