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There has never been a science of finance except through the leisure time of aristocratic gamblers from its origins with Laplace to the foundation of its mathematics with Louis Bachelier whose 1900 paper defined Brownian motion.  The mathematical theory of finance began with a mathematical student of Henri Poincare, Henri Bachelier at the turn of the century.  Bachelier’s basic model was the continuous stochastic process the Brownian motion.  Norbert Wiener was responsible for giving a mathematically rigorous definition of Brownian motion as a measure on the space of continuous paths (although a standard topic in courses is to prove that the typical Brownian path is almost nowhere differentiable).  This fit in well with an institution of economics and finance that lived in dogma rather than truth.  They had decided a priori that markets are unpredictable and therefore — yes yes their lack of logic is breathtaking — unpredictable and therefore it should be modelled as INDEPENDENT and GAUSSIAN.  Note that Gauss would have turned in his grave at the idea that his methods carefully applied to scientific observational data in the nineteenth century, and here at least we are dealing with repeated controlled measurements to declare mean and standard deviation, were to be applied as an arbitrary shock of a complex system that is affecting the lives and livelihoods of seven billion of our hapless race and claim that they were doing science.  Imagine the nerve of these people.  They take speculative parlour games and gambling and put together a little trick like card counting in a casino and then put Laplace on a banner and declare to claim territory in science.  Valiant efforts had gone against these atrocities to truth and science.  Benoit Mandelbrot in late 1960s even gave his model of fractional Brownian motion and fractional noise and even analyzed cotton prices.  He wrote books on the misbehavior of markets but too many Nobel prizes had been given to the Markov cultists of the world, people who would put the convenience of calculation for Markov processes — I mean what probability theorist does not get a little joy in most of the trailing memory disappearing from his conditional probabilities from one line to the next when doing some Bayesian calculation?  This little convenience is nice but it can no longer hold the wretched of the earth hostage to their dogmatism.  Markovian finance must fall, and if it does not fall we shall burn it down to the ground!!

Apologies, Louis Bachelier, not Henri (Freudian swap of Poincare’s first name perhaps).  Now let’s backtrack for a moment and rest with Benoit Mandelbrot’s 1963 paper on cotton prices, which is attached.  He introduced the fractional Brownian motion five years later and showed that the fractional noises pervade financial assets.  A couple of years ago I showed that he’s right universally in the sense that stochastic volatility always has Hurst exponent H>0.5 where the Brownian motion type model is exactly H=0.5.  The fractional Brownian motion models are not Markovian and a stochastic calculus for these are being developed by David Nualart and coworkers and there is now an Ito formula for them.

All this pertains to a SINGLE NODE of a grand graph that we want to call financial graph, the x-variable of volatility dynamics.  If we want to take fluid dynamics and chaos as our major inspiration for a science of finance, we want differential equations and in order to gain the benefits of partial differential equations thinking of several centuries by mathematicians and physicists we NEED an x-variable.  I have given the world the SPACE variable in financial volatility and this is a giant graph where assets are nodes and we do not consider the volatility of an asset at a moment to be an independent object at all.  In fact it is a piece of the volatility monster and its dynamics must be understood in the context of the graph.

Now the volatility graph has millions of nodes for which we began with 188 nodes to get a sense of what this space is like.  By David Donoho’s suggestion we have come upon Patrick Wolfe’s work, which contains very clear explanations for the statistical models of graphs with n nodes.  The edge existence probabilities are modeled using n parameters (alpha_1,…,alpha_n) using the following model and it’s generalizations:

$logit(p_ij) = alpha_i + alpha_j$

This is a logistic regression model.  Perry and Wolfe prove some theorems giving precise bounds on maximum likelihood estimation.  One interesting feature of their analysis is the use of sharp bounds for the Newton-Kantorovich theorem.

PRACTICAL STATISTICAL MODELS FOR WORLD FINANCIAL GRAPH

Practically, of course all the careful analysis is not necessary to fit these models.  In R, the glm() function with family=binomial(logit) will fit the model on data.