Fractional Levy stable motion

The last post in the ARFIMA series was not the last stop in understanding the model we have studied in [1]. In the paper we have looked at ARFIMA as a model for fractional Levy stable motion (abbr. FLSM), which is a generalization of Brownian motion in two regards: fractionality and noise distribution.

Typical Brownian motion, whether fractional or not, uses normal distribution to describe noise. FLSM uses \( \alpha \)-stable distribution instead. This generalization allows to take into account systems in which large jumps are common (distribution has heavy tails).

Now FLSM has two parameters: differentiation order \( d \) and index of stable distribution \( \alpha \). If \( \alpha = 2 \) then stable distribution is equivalent to normal distribution and FLSM is equivalent to fBm. Otherwise we will observe deviations from the usual shape of Brownian motion. Note that \( \alpha = 1 \) is also a special case. In this case the distribution corresponds to Cauchy distribution.

Interestingly, having heavy tailed noise distribution also affects Hurst index of the process. In general case it is given by:

\begin{equation} H = d + \frac{1}{\alpha} . \end{equation}

Note that the app also allows you to generate fractional Levy stable noise (abbr. FLSN), too. It is simply but a straightforward derivative of FLSM.