pseudodifferential equation context available
June 9, 2015 by zulfahmed
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>






Sir,
Although my idea of jumping to pseudodifferential equations has been anticipated already in 2010, I believe that an exact pseudodifferential equation for our volatility science is a tractable exercise. This is a big problem, very big in the sense that OUR exact volatility solution fitting empirical EN(t) gives the EMPIRICAL side of the story better than some of the physical applications because we already have data and empirical curve for an exact correction for financial volatility and most likely ANOMALOUS DIFFUSION problems also do not have exact power law in wating times. This is as decent a step in science as when in 1905 Einstein unified micro and macro diffusion theory. The Einsteinian solution was to seek parsimony and this field is lacking parsimony and communication. Finance as an exact science is very important for quelling volatility storms and making this miserable HellonEarth a bit better for the seven billion who have other problems besides being victims to volatility in prices.
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


1:38 PM (0 minutes ago)




The attached paper generalizes the KolmogorovFokkerPlanck parabolic equations to timefractional pseudodifferential equations. This focuses on stochastic differential equations driven by Levy (Markov) processes.
Formally, the fractional integral is often defined by integrals which multiply by a translate of the integrating variable u and taking a fractional power. I want to consider just to have a simplistic conceptual understanding the same replaced by a family of modifications (ua)^{beta1} exp^{alpha*u^gamma). The considerations of powers replaced by these products in the fields where ‘fractional’ appears would be justified by our residual pattern for me. GradshteynRizhik treats some of the exact values for integrals of this type (attached).
If we replace all the power laws with these combination laws, we might capture what in these phenomena we’re getting wrong with ‘fractional’. There seems to be a conceptual chaos when I look at the literature since the path from differential and general pseudodifferential equations traversed through the first order correction to ‘fractional’ and now we have the second order correction perhaps on actual empirical data for an exact fit to an important problem.
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