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We believe that the residual from our initial powerlaw fit for the expected integrated shock distribution showed systematic deviation rather than any noise but the linear fit is statistically good in the sense that R2=0.99+ and pvalue is much less than 0.01. The residual graph is attached to a power law fit.
Now power law fit implies fractional Poisson distribution worked out in 2003 by Nick Laskin, and this in turn implies the use of the MittagLeffler function probability calculations thought of as a ‘fractional exponential’. Attached you can see some work on these functions especially how to compute them algorithmically as well as some theory.
The fact that this MittagLeffler function is thought to be a fractional version of exponential function is very interesting. Now these functions are defined modulo constants by a power series with Gamma(1+k*mu) in the denominator where the numerator has z^m. Although I proposed an easy class of powers and exponentials as the panacea excitedly in the past couple of days, I am known to make errors in excitement which is rash, childish, and fun, but it’s time to be serious which implies that we have to look at this mathematics of MittagLeffer and ‘fractional exponential’ through new eyes. One generalization of MittagLeffler addressed here that looks exceedingly interesting for possible exact theories for volatility is that the denominator changes to Gamma(alpha+k*beta) replacing the single parameter mu. This needs further investigation on data as the possible second order residual wrongness discovered by us for power law in waiting time distributions in empirical data for 1900 stock stochastic volatility (which correspond to the singleparameter MittagLeffler function).
Preview attachment Screenshot from 20150610 16:07:26.png
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Preview attachment computationmittaglefflerfunctions.PDF
Preview attachment fractionalwaitingtimes199551R848a.pdf
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