restatement of main research issues in volatility storm modeling
June 24, 2015 by zulfahmed
We have the empirical data and we want to carefully evaluate whether on daily data TURBULENCE and PHASE TRANSITION between LAMINAR and TURBULENT phases can be seen from the underlying model of volatility. DIFFUSION models versus HYDRODYNAMIC models differ only in whether we:
(a) Look at SAMPLE STOCHASTIC PATHS of Diffusion as volatility such as Brownian or fractional Brownian motion and their variants
(b) Look at volatility as a deterministic quantity governed by some FPDE and also whether that FPDE has the extra F(u) term.
Oh come on, I’m not a maniac. I’m excited about something important.
Sir, I think we can see clearly using the fact that subordination affects anomalous diffusion but that anomalous diffusion does not necessarily have the nonlinearity of Navier-Stokes but the global volatility is not yet a resolved question and we can check on data obtained quite easily by considering removing a diffusion term with what I had called the excess function defined as
Excess(u) = ((d/dt)^alpha + (-Laplacian)^beta)u
which is zero when u satisfies a diffusion equation but is F(u) otherwise. If for all alpha,beta of interest this excess is nontrivial, then we could consider the diffusion equations to be misspecified models. This is a simplistic test to see if Hydrodynamic models (which can have phase transition to turbulence while diffusions cannot possibly) are better descriptions of the univariate daily time series.