Fractional Brownian motion

The fractional Brownian motion or fractional Brownian motion is a class of centered Gaussian processes, which are characterized by the following covariance:

Where H is a real number (0, 1). H is often called the Hurst parameter. H = 1 /2 fractional Brownian motion is a one-dimensional Brownian movement.

Properties

Self-similarity

Is self-similar. More precisely, that have the processes and c for each fixed > 0 has the same distribution.

Stationary increments

From the representation of the covariance function, the relationship is directly followed

In particular, the increments are therefore stationary. In addition:

• If H = 1/2, the process is independent increments;
• If H> 1/2, then the increment is positively correlated;
• If h < 1/2, so that the increments are negatively correlated.

Path Properties

The paths of the fractional Brownian motion with Hurst parameter H are Hölder continuous with index for each.

Stochastic integration

It is possible to define stochastic integrals with respect to the broken Brownian motion.

See also

• Anomalous diffusion