A New Approach to Fractional Brownian Motion of Order n Via Random Walk in the Complex Plane

By using a very simple model of random walk defined on the roots of the unity in the complex plane, one can obtain the model of fractional brownian motion of order n which has been previously introduced in the form of rotating Gaussian white noise. This definition of fractional Brownian motion of order n as the limit of complex random walk, provides new insights in its genuine practical meaning, and in the derivation of most of the related theoretical results. Itôs stochastic calculus can be extended in a straightforward manner to the path integral so generated in the complex plane. The corresponding probability distribution is stable in Levys sense, a Lindebergs like central limit theorem is stated, together with a Feyman–Kacs formula and a Dinkins formula. Then one exhibits the relation between the Hausdorffs dimension and the pattern entropy of the process. The probabilistic approach here is different from Hochbergs and Mandelbrots. Like Saintys, it uses the complex roots of the unity, but it is much more straightforward and simple, and it is the only one which provides results which are fully consistent with the so-called Kramers–Moyal expansion.