Basic measures used by people studying ODE solutions to measure senstivity of an individual trajectory of a solution to move exponentially by iteration. Precise definition in snapshot. Intuition from iterative mappings: lim_{N tends to infinity}  f^N(x0+h)f^N(x0) = O( f(x0) exp( lamda * N) ) for trajectories this classifies by sign: lambda0 means it’s diverging. in R an example from our raw undenoised volatility data is: library(fractal) lamda<lyapunov(h.noisy[,1]) The package fNonlinear also has some methods. We’re interested in dynamical laws of volatility. The goal is to find the dynamical laws as DDE equivalent to NavierStokes type PDE on a graph and then analyze the PHASE TRANSITION to turbulence theoretically with the expectation that there do exist some parameters which separate ‘laminar’ from ‘turbulent’ flow for the dynamical system. The problem of estimating Lyapunov series is a long solved problem so this is just checking software. Attached paper shows the solution, and a scan is enough to interpret the function in R above. So we have computational infrastructure for calculating these. Now we return to the problem of theory/stats fitting for dynamical laws. We know that we want to study the system in terms of a parameter. An easy solution is simply to pick parameters using some orthogonal basis expansion on the graph using eigenvectors of Laplacian to deal with the F(u,..) term in our expected NS replaceement, since we can deal with the heat equation and know that we’ll get smooth solutions for heat flow with many different interpretations like random walks. (d/dtLaplacian)u == 0 can be considered understood and then we model the excess on data seeking F(u,…) with eigenvector bases with perturbation theory just as physicists have been doing for ages and seek phase transition to chaos based on these parameters. If we succeed we have a scientific model rather than ‘irrational exhuberance’ and other supertitions that plague the nonscience of financial economics. 2 Attachments Preview attachment Screenshot from 20150528 11:48:23.png Screenshot from 20150528 11:48:23.png
Preview attachment Physica1985_wolf_LyapunovExpo.pdf Physica1985_wolf_LyapunovExpo.pdf
I’ve been unsuccessful thus far in producing any finite values for the Lyapunov exponent calculators for time series on our volatility series but this is not really priority — packages exist with functions for these in R: tseriesChaos, fractal, fNonlinear all have functions for published algorithms to calculate the Lyapunov exponents which when positive indicate a system in chaos. So now that we have some idea of what sort of phase we might be looking for, let us review the basic picture of what we’re trying to reach: (a) CHAOTIC versus nonchaotic phase description quantitatively as a function of some small set of parameters which should mimick the perturbation parameter for the right hand side object of (d/dt – Laplacian)u = F(u,…). Theory should tell us that there should be a transition to chaos when the parameters cross some thresholds. The data should allow us to find the laws with the parameters that determine transitions to chaos. Lyapunov exponent code should guide us in this project. Then we will have addressed the problem of a real science of finance that is able to give us a law of some sort that is a differential equation and an indication of whether the threshold has been crossed. Finally the task is to try to manage these volatility storms that rage when there is any chaotic dynamics by examining the threshold in the actual world.
If one’s goal is only to find if a system is in chaotic phase, and this is not clear from the heat map of the volatility attached, then one also has some good methods. The attached papers provides one to calculate the highest Lyapunov exponent which seems to be the quantitative manner in which to decide on the chaos phase — checking this is positive. In fact the differential delay equation the paper mentions, d/dt X = F(X) with delay terms is exactly the form of the model I’m proposing as dynamics for volatility with noise. In fact since Laplacian contains graph geometry information, this is the form we’d expect any dynamics. We do have a theoretically interesting issue of subtleties of approximating a large graph by a smaller one statistically and analytically. This is the problem that we have data now for 188 assets not a million. Suppose that the Lovasz limit of the graph with Wolfe theory or Diaconis theory were applied successfully to model the market. Now on the large graph we’d be dealing with dynamics of a much higher dimension, and our approximation is underestimating a great deal of complexity and interaction with edges with many vertices of the larger graph not present in the 188 vertex graph. So in the context of Wolfe theory, we’d like to understand classical dynamical systems which each have classical delays where in theory the largest Lyapunov exponent can be computed by a packaged algorithm and therefore we’s have easy check for chaos or not. The issue then becomes of whether we can talk about limits of dynamical systems in some rigorous fashion. The real question is generalizability. It would be very nice to get phase transition to chaos for infinite vertex graphs of particular characteristics like fixed spectrum of Laplacian. Then we’d have developed interesting new mathematics to solve a great problem.

Spiralling slowly toward a new science of finance
May 28, 2015 by zulfahmed
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