
1:12 PM (0 minutes ago)



Dear Professor Donoho,
MittagLeffler functions I finally understand now as a construction for solutions of
y^{alpha}(t) = lamda y(t)
y(0) = 1
so that the solutions are precisely given by MittagLeffler functions of a single parameter. We are clearly dealing with situations where an exact model is a perturbation to some fractional equation of this type. We are detecting the exact difference in terms of the Ito density of a process’ Ito formula. Metrically, the perturbation is tiny compared to the fractional differential equation approximation. So we want to perhaps return to the question of volatility control now in terms of KAM theory of stability for FRACTIONAL DIFFERENTIAL EQUATIONS. So that’s what we were missing — we need a chaos theory for fractional equations which will then be able to describe the volatility dynamics and tell us something about phases for these. Power laws are then useful as an approximation and perhaps less useful as a principle defining nature. Mandelbrotian paradigm would suggest: Nature does not work as Newton said, by differential equations; rather it prefers these perturbations of fractional differential equations for its dynamics. Now this is perfectly sensible for a statistical mechanical description of nature because our original stab at heat equation from Joseph Fourier was based on the Newtonian paradigm that was then adapted to produce quantum theory’s Schroedinger equation which is probably not quite right either beyond Dirac’s correction in essentially forcing linearity on nature.
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numericalsolutionsfractionaldifferentialequations.pdf
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