Running maximum and minimum
How high does a Brownian path climb by a given time, and how is that peak distributed?
Goal: follow the running maximum and minimum through time and compare the law of M(t) with |W(t)| and the density 2φ(m/√t)/√t.
t = 1.00
The running maximum climbs in steps: it only increases when the path sets a new high. The faint cloud shows the other simulated paths up to the same time.
M(t) empirical|W(t)| empirical2φ(m/√t)/√t
The empirical densities of M(t) and |W(t)| both match the theoretical 2φ(m/√t)/√t — two different random variables with the same distribution.