Keywords: forward and backward fractional derivatives, generalised Cauchy derivative, Liouville derivative, differintegration, central fractional derivatives, fractional stochastic process, fractional Brownian motion.

It is no use to refer the importance of stochastic processes with fractional characteristics. In fact, they are very frequent in nature and in daily applications. Fractional Brownian motion (fBm) and noises are well known designations for some of these kinds of signals [1,2,3,4]. In parallel, self-similarity and long range dependence are interconnected notions and appear in a variety of contexts [1,2,3,4]. However, it is not clear how we can establish a bridge between these notions and the current definitions of fractional derivative. Here we will try to do a new step into that goal by using two sets of derivatives [5,6,7]: a) the casual and b) the central. For both classes, summation and integral formulae are presented.

Starting from those fractional derivative definitions, we apply them to stationary stochastic processes. The computation of the autocorrelations of the derivative processes shows equivalence among all the derivatives. This is interesting and possibly expected, since the time arrow does not have any importance for stationary stochastic processes. However the derivative processes are not stationary in general.

The above considerations are applied to the particular case of the white noise and use the results to define a fractional Brownian motion. This is a model for non stationary signals, but with stationary increments, that are useful in understanding phenomena with long range dependence and with a frequency dependence of the form , with non integer. A mathematical representation for this kind of process was proposed by Mandelbrot and Van Ness [1]. This definition has some inconvenient features that we try to overcome with a different definition [8]. This one is a generalization of the usual Brownian motion definition. The properties of the signal obtained from the application of this definition are studied, mainly:

It is a non stationary process with stationary increments, The process is self similar; The incremental process has a spectrum

From this last property, we conclude also that, if H is the Hurst parameter [1,2]:

the spectrum is parabolic, if , and corresponds to an antipersistent fBm, because the increments tend to have opposite signs; the spectrum has a hyperbolic character, if , and corresponds to a persistent fBm, because the increments tend to have the same sign.

We show that the formulation of Mandelbrot and Van Ness can be deduced from ours. The formulation we propose here is more general in the sense of giving the possibility of using other derivatives, especially the Grünwald-Letnikov derivative that can be useful in a discrete-time implementation.

1

Mandelbrot, B.B. and Van Ness, J.W., "The Fractional Brownian Motions, Fractional Noises and Applications", SIAM Review, Vol. 10, No. 4, Oct. 1968, pp. 422-437. doi:10.1137/1010093

2

Mandelbrot, B.B., "The Fractal Geometry of Nature", W.H.Freeman and Company, New York, 1983.

3

Keshner, M.S. "1/f Noise", Proceedings of IEEE, Vol. 70, Mar. 1982, pp. 212-218. doi:10.1109/PROC.1982.12282

4

Reed, I.S., Lee, P.C. and Truong, T.K., "Spectral Representation of Fractional Brownian Motion in n Dimensions and its Properties", IEEE Trans. on Information Theory, Vol. 41, No. 5,September 1995, pp. 1439-1451. doi:10.1109/18.412687

5

Ortigueira, M.D., "A coherent approach to non integer order derivatives", accepted for publication in Signal Processing. doi:10.1016/j.sigpro.2006.02.002

6

Ortigueira, M.D., "Fractional Centred Differences and Derivatives", to be presented at the 2nd IFAC Workshop on Fractional Differentiation and its Applications. >

7

Ortigueira, M.D., "Riesz Potentials and their inverses via Centred Derivatives", accepted for publication in the International Journal of Mathematics and Mathematical Sciences.

8

Ortigueira, M.D. e Batista, A.G. "A Fractional Linear System View of the Fractional Brownian Motion", Nonlinear Dynamics, 38: 295-303, 2004. doi:10.1007/s11071-004-3762-8

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