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I have been playing with some Brownian motion simulations (Euler scheme), when I discovered some strange behavior. When simulation 2 classical Brownian motions, the (Pearson) correlation between them is almost always quite significant (be it the absolute value and the p-value). Moreover, when increasing the number of increments, the correlation increases as well.

Here is my sample code:

import numpy as np
from scipy.stats import pearsonr

np.random.seed(12)


def BMpath(N,T,d=2):
    dt = T/N
    B = np.zeros((d,N+1))
    for j in range(N):
        dZ = np.random.normal(0,1,d)
        dB = dZ*np.sqrt(dt)
        B[:,j+1] = B[:,j] + dB
    return B


b = BMpath(1000,100,2)
pearsonr(b[0,:],b[1,:])

When the correlation for this seed is (0.6067007021721034, 1.1145014850169111e-101) (the latter is the p-value). Is there an intuitive explanation why is that? My intuition says, that the correlation should be close to 0, as the increments are from i.i.d. $N(0,1)\sqrt{1/N}$.

PK1998
  • 157

1 Answers1

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If you run several simulations you will find the correlations are widely distributed and are as likely to be negative as positive.

The cause of the apparently high magnitudes for the correlations is that Brownian motion is not a stationary process (among other properties, the variances increases over time). Their increments are stationary, by construction, and if you considered correlation between the increments of the two series then you would get the sort of low-magnitude distribution you seem to expect. When considering correlation between time-series, you need stationarity to get meaningful results.

This sort of spurious correlation was the topic of Yule's presidential address to the Royal Statistical Society in 1925-26, and has been investigated in depth ever since. As a real world example of high correlation between time-series and low correlation between increments (though also affected by drift), see an answer on economics.stackexchange.

In summary, each Brownian motion is autocorrelated, and this is likely to lead to apparent but misleading relationships appearing between them.

Henry
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