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(a) A deterministic fractional differential equation that is perhaps the solution of a fractional partial differential equation on a graph:
(d/dt^alpha + (Laplacian_G)) u = F(u)
(i) Volatility is a global object and therefore could be considered a macroscopic variable like temperature and pressure whose underlying statistical mechanics could be considered not known as in thermodynamics and hydrodynamics. This would suggest that exact deterministic laws could be the basis of volatility
(ii) In the case that volatility does have deterministic dynamics the size of the excess terms would be a reasonable candidate for source of turbulence in volatility dynamics as this is not just the case for classical NavierStokes equation where the term is kappa*<u,grad u> where kappa is the Reynold’s number but this phenomenon has been considered in the monumental Russian project of tabulating solutions (when space is R^d) of semilinear parabolic equations with a nonlinear ‘convection’ terms where the intuition is that diffusion term smoothens and convection term leads to turbulence. We have a natural space variable so this is a natural question of exploration.
The code is still under development and it is written using scipy facilities from scratch.
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