## VOLATILITY STORM PROBLEM AS A HIGH DIMENSIONAL FUNCTION

August 14, 2015 by zulfahmed

## standard nonlinear reaction-diffusion theory

Dear Professor Donoho,

Ignoring the fractional portion of ((d/dt)^alpha + (-Laplacian)^beta )u= F(u) i.e. alpha=1, beta=1, I learned it is well-known that the way to study stability is to look at the Jacobian matrix JF(u0) of F at u0 with conditions trace(JF(u0))<0 and det(JF(u0)) > 0 for space variables of dimensions > 1. Maybe this is the easiest way to understand the stability of global financial volatility quantatively:

http://pauli.uni-muenster.de/tp/fileadmin/lehre/NumMethoden/SS14/RD.pdf

We consider the best parameters (alpha,beta) that statistically describe a linear fractional diffusion system for 1900 stocks for which I had collected data several months ago, let F be the residual of the fit, and then numerically consider the behavior of the Jacobian of this F and check it’s trace and determinant around a solution u0? If the trace and determinant conditions for the empirical JF is violated then we could consider ‘calm water diffusion models’ to be inappropriate quantitatively to modeling global volatility and consider ‘volatility storms’ rigorously via the resulting F and JF?

I apologize for these issues but it has been 15 years since I worked on analytical probability and am currently fairly isolated from technicians.

General F(z) are likely to produce instabilities anyway and it seems from the work on cubic instabilities that the interest at least in hard science applications have been restricted to fairly concrete analytical expressions for F. In our situation, we do not expect any analytical expression of F to exist necessarily that is closed form, but assuming that one existed hypothetically — intuitively one might want to consider a d1 dimensional latent variable system where a nonlinear fractional diffusion informs the entire financial market representing ‘behavioral factors’ and a mapping of its dynamics to all nodes of the market assuming each node is a function of this latent system, then one would like to understand if the underlying F has Jacobian with negative trace and positive determinant.

A small step toward this problem is determining the parameters of a univariate function of the one-dimensional brownian motion whose solution is here:

https://zulfahmed.wordpress.com/2015/08/14/observing-functions-of-brownian-motion-hermite-expansion/

There exists a multidimensional Ito formula which could be used for higher dimensions which is more complicated only for the algebra. This approach uses the special feature of the brownian motion to Hermite polynomials. More generally we would like to understand how to take observed vector time series Y_t = (Y_t^1, …, Y_t^N) with say N=1900 and consider it a functional of a more general process which is Y_t = Phi(Z_t) whose probability density satisfies some nonlinear fractional reaction-diffusion equation and estimate the nonlinear term which is a much more difficult problem.

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