In [1]:
import numpy as np
import matplotlib.pyplot as plt
In [2]:
plt.rcParams['figure.figsize'] = [16, 6]
Another random walk¶
We know if $B(t)$ is standard Brownian motion, with $B(0) = 0$, then $\text{cov}[B(s), B(t)] = \min(s,t)$. So, for $t_0 < t_1 < t_2 < \cdots < t_n$, the covariance matrix for $(B(t_0), \ldots, B(t_n))$ is $$ \Sigma_{ij} = \min(t_i, t_j) . $$
This means that we can sample locations along the path of Brownian motion by sampling from a multivariate Normal.
Note: this is not the most straightforward way to do this (better would be to simulate independent increments) but this way we get to practice simulating correlated Gaussians.
In [37]:
def B(t):
starts_at_zero = (t[0] == 0)
if starts_at_zero:
tt = t[1:]
else:
tt = t
n = len(tt)
sigma = np.zeros((n,n))
for i in range(n):
for j in range(n):
sigma[i,j] = min(tt[i], tt[j])
assert(np.all(sigma == sigma.T))
chol_sigma = np.linalg.cholesky(sigma)
z = np.random.normal(size=n)
b = np.matmul(chol_sigma, z)
if starts_at_zero:
b = np.concatenate([[0.0], b])
return b
In [39]:
t = np.linspace(0, 2, 1000)
Bt = B(t)
fig, ax = plt.subplots()
ax.plot(t, Bt)
Out[39]:
