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## a mathematics muse?

### Zulfikar Ahmed<zulfikar.moinuddin.ahmed@gmail.com>

11:21 AM (1 hour ago)

 to aimee
So I am taking up a purely mathematical question in order to solve the volatility storms question — which is to produce a stochastic understanding of the nonliner diffusion models which are traditionally (since McKean 1975) tied to branching or chain reaction or cascading processes.  The question is whether one can prove mathematically using probabilistic methods that the KPP equation has a phase transition to turbulence, showing that volatility in global finance is predicated on these sorts of chain reactions like bombs dropped on the financial markets.  So I expect this is true when the nonlinearity of the fractional diffusion is F(u) = u^2-u but have no sense of why this should hold for other sorts of nonlinearities. General F does not have such a neat branching chain reaction interpretation of course so the analysis problem is open.

My uncle suggests that instead of trying to save the world I use this research to make mobile apps because that may be more useful to many people.  Both are possible.
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### Zulfikar Ahmed<zulfikar.moinuddin.ahmed@gmail.com>

12:31 PM (21 minutes ago)

 to aimee
I love this guy Mainardi …

### Zulfikar Ahmed<zulfikar.moinuddin.ahmed@gmail.com>

12:48 PM (4 minutes ago)

 to aimee
Mittag-Leffler use in fractional calculus has a couple of approximations from 2014 which are pretty useful by Mainardi.  The function E_alpha(-t^alpha) is the survival probability of a fractional process of a particle up to time t when it dies and two particles are born, in other words the chain reaction probability in the branching process.  When the nonlinearity F(u) = u^2-u, which is a simple intuitive model foe what one might expect in financial volatility time series, we should expect a phase transtion to turbulence for these sorts of fractional nonlinear diffusions.  It’s unclear still whether such an F makes sense generally — this is the question of the science of finance to be checked on empirical data.  There is deep knowledge here hidden somewhere regarding natural science of markets without ‘irrational exhuberance’ and whatever.

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### Zulfikar Ahmed<zulfikar.moinuddin.ahmed@gmail.com>

11:41 AM (1 hour ago)

 to aimee
And the analysis of these fractional diffusions are complicated by the fact that these Mittag-Leffler functions are not monotonic but have infinitely many values for zeros, etc. and of course these are involved in the probability density and survival probability for the branching process even for the KPP fractional diffusion.  Ah I wish I were back at Princeton when my mind worked much better on these things.

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