Fractional Brownian Motion

To understand the role of the index H in fBm, we recall the expected value, E(Y), of a random process Y(t).

One way to measure the correlation of a random process Y(t) is to compute the expected value of the product of non-overlapping increments.

With some work, it can be shown that for index H fBm,

E((Y(t) - Y(0))⋅(Y(t + h) - Y(t))) = ((t + h)2H - t2H - h2H)/2

Though some more work is required to see this, this expectation is

positive for H > 1/2 0 for H = 1/2 (This one is easy.) negative for H < 1/2

So we shall consider three examples.

H > 1/2 gives persistent fBm H = 1/2 gives standard Brownian motion H < 1/2 gives anti-persistent fBm

Note as H decreases, the graph of index H fBm appears to get rougher.

For this reason, H is called the roughness exponent.

It is also called the Holder or Hurst exponent.

With probability one, the graph of index H fBm has box-counting dimension 2 - H.

Return to fractional Brownian motion.