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## fractional differentiation of the Mittag-Leffler function

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

8:56 PM (0 minutes ago)

 to David, bcc: aimee
Dear Professor Donoho,

I’m looking closely at the nature of the two-parameter Mittag-Leffler under fractional differentiation for some understanding.  We can note that this relation, in snapshot of attached paper, shows an important thing regards the effect on the Mittag-Leffler parameters a,b of fractional differentiation by D_t^alpha:

D_t^alpha:   t^A E_{a,b}(lambda* t^delta) = t^{A-alpha} E_{a,b-alpha}(lambda*t^delta)

In other words, the power of t changes and the Mittag-Leffler second parameter changes.  To me this suggests that rather than power law, the Mittag-Leffler function mixed with powers is the key function; but there is also the Laplace transform that turns fractional derivatives to multiplication by fractional powers in the transformed-space:

Laplace{( D_t^alpha – beta ) u}(p) = p^alpha Laplace{u} – beta*Laplace{u}

This is helpful since then we can just do a fit to data by using the Laplace transform of data as follows assuming v is the signal

n<-length(v)
t<-seq(1,n)**alpha

x<-laplace(v)
So if we assume Laplace{(-Delta)^{2*beta} u) = s^{2*beta} Laplace(u), then
we’d have to look for solutions for our intended FDE with alpha time and beta (-Laplacian) power equalling F(u):
s^alpha Laplace(u) + s^{2*beta} *Laplace(u) = Laplace{F(u)}
This then can be transformed by manipulation to just a Laplacian power evolution (power=2*beta-alpha) calculated easily and the right hand side will be adjusted by s^{-alpha} Laplace{F(u)}.  Or we can just take the log of this equation and obtain a linear model without having to calculate the fractional derivative at all.

Ok this looks like a more promising manner in which to fit data!!

Sir, since 1995 when I held my first job at Lehman Brothers Fixed Income Research with a great deal of quant work done, this is the first time in this time that I think there is a real science in finance that is deep and it’s in global financial volatility and we’re close to making a breakthrough in this and related areas if we can prove that volatility can be modeled as a perturbation of deterministic FDE.

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