conceptual analysis of generalizations of Poisson process
June 9, 2015 by zulfahmed
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>






Dear Professor Donoho,
The physicists had generalized a very large part of their infrastructure to analyze many classical phenomena such as diffusion — first written down in 1855 and then built into a giant machine which seems to have built the powerlaw tail for digging into phenomena that ‘exhibits power law tails’. This is the physicists’ way of moving forward, from their arrogant empiricism that does not fully grasp the fact that if nature is described by mathematics, then mathematical guidance cannot simply be considered secondary “help for theory”. Empirical fit are certainly fundamental to scientific development and depth, as the example of string theory shows which is pure nonsense outofcontrol theory which have nothing to do with nature and its workings. This ‘power law’ phenomena is more serious and our reaction to it as mathematicians in heart and early training in my case and in great accomplishments in yours to note that mathematics never developed depth without physics either. Therefore ‘physicists’ in my reference is not targeted to any individual but the collective fate of science since the atomic bomb became the fundamental achievement on which physics would pride itself. In contrast, if one looks at truly beautiful development in mathematicsphysics interplay it is in the nonsuperstring theory purely mathematical developments in the theory of fourmanifolds where string theorist Witten had genuine geometric and topological insights etc. Taking this analogy and mapping to the problem of volatility in finance, we see that physicists thinking in terms of ‘power law’ as what is being observed is obscuring the mathematical structure of the renewal process that is probably behind shock distribution in the market. I propose the claseses of distributions f(t) ~ a t^p exp( b t^q) as distributions of interest for volatility modeling because:
(a) the derivatives stay in the same class with superpositions and at the same time extends powers and exponentials suggesting deep discoveries in this unification of Gauss and powers. I’m not sure if is already worked out but I expect that this volatility problem is clarifying something about anomalous diffusion which is hidden by the usual R^2 and pvalue metrics of goodness of fit.
(b) more importantly EMPIRICAL DATA shows that in the power law tail for our data convexity which implies that the residual has a tail that behaves as a nonlinear function of log(t).
This is a happy insight which will require a great deal more study and work to work out but this is in my view the first lesson I learn from volatility problem. Volatility is a hard problem because it’s hard to believe in our current collective frame of mind that it can have laws MORE ACCURATE then physical laws but this is a false paradigm. To the extent human beings behave like particles, and this is a very physics manner of looking at social systems, and their convictions, to give due credit, had not failed them for volatility. MATHEMATICAL insight would be to look at their bad residual and notice that we need fractional Poisson processes generalized to a new function class uniting Gaussian and powers and ensure that formulae from renewal theory are worked out and we have a fundamental advance in a bunch of fields including anomalous diffusions and other scientifically active fields.
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