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, for instance Y(t) - Y(0) and Y(t + h) - Y(t).

The expected value of the product of two non-overlapping increments 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.

Almost always, the graph of index H fBm has box-counting dimension 2 - H.

Return to fractional Brownian motion.