np.random.seed(1234)

Nsteps = 500

t = np.arange(Nsteps)

mu = 0.002

sigma = 0.01

# the steps and position

S = mu + sigma*np.random.randn(Nsteps)

X = S.cumsum()

# the 1 sigma upper and lower analytic population bounds

lower_bound = mu*t - sigma*np.sqrt(t)

upper_bound = mu*t + sigma*np.sqrt(t)

fig, ax = plt.subplots(1)

ax.plot(t, X, lw=2, label='walker position', color='blue')

ax.plot(t, mu*t, lw=1, label='population mean', color='black', ls='--')

ax.fill_between(t, lower_bound, upper_bound, facecolor='yellow', alpha=0.5,

label='1 sigma range')

ax.legend(loc='upper left')

# here we use the where argument to only fill the region where the

# walker is above the population 1 sigma boundary

ax.fill_between(t, upper_bound, X, where=X>upper_bound, facecolor='blue', alpha=0.5)

ax.set_xlabel('num steps')

ax.set_ylabel('position')

ax.grid()