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## EXACT FORMULAE FOR DIFFUSION AND HYDRODYNAMIC MODELS

Dear Humans,

I’m heartbroken but I’m used to it.  After a while one begins to understand that there is nothing to do but simply work on something fun and hope that the United States Disability can cover my precarious retirement with their reduction of payment so that I can’t even afford a goddam retirement on it with an ounce of dignity and am evicted from a place I rented with a legal lease for which I was paying and the fuckface greedy owners had to get richer from richer tenants breaking the law because money is parking into San Francisco real estates; this is a very San Francisco story, illegal evictions and trying to drive out the poor to make room for the rich.  In America, there are zoning laws and gentrification means white people moving into an area; so I’m fucked doubly: I’m not white and I’m poor.  I don’t care.  I’ll survive.  I’m American.  This is my country.  No one is driving me out of my city, thank you very much.  This is my home.  I’m  planning never to leave this city again if I can.  Let cultured people with money what they will.  I’m going to do what I love to do and what is that but right now work on proving to the world that modeling volatility by calm water phenomena like diffusion is not just biased because sources of data are expensive for finance generally although I got free data for stocks from Quandl, Google, and Yahoo for my analysis.

Let’s recall a bit of history.  A few weeks ago out of the wings of love I came upon some novel results in 1900 stocks.  Well maybe wings of love requires a delusional reality to comprehend but nothing less could explain how within a few days of getting the data under hobo conditions I was able to get a model of shocks for the market proxied by 1900 stocks where volatility defined without denoising, log(returns^2), for 1900 stocks were processed as follows.  I defined the noise level by standard deviation of the volatility and identified by soft thresholding of volatility those which could be considered the sharp movements of the volatility changes to be above three standard deviations as the noise level.  Then I counted per day the total number of these volatility shocks and plotted the graph of this waiting time distribution for a renewal time type model.  A Poisson model with intensity $\lambda$ would produce the expectation $EN(t) = \lambda t$ but the empirical shock distribution does not follow a linear law but fits quite closely a power law distribution $EN(t) = c t^{\alpha}$ but there is a correction to the power law that is visible.  The shock distribution and the residuals to a power-law fit in the log-log scale is below.

This work I have dedicated to my love for Natalia de Varsgaard who inspired it.  The source of the project is this poem:

YOUR HEART WOULD HAVE RESPONDED GAILY
(A poem dedicated to Talia de Varsgaard)

When invited
Beating obedient to controlling hands

High above the clouds of dust

The swirling storms of money
Crawling insects the fate of the fallen rebels

The boat responded gaily

To the hand expert at sail and oar
Arisen from a long sleep
Shattered cryogenci chambers
Love, the World exists only within

And thunders sear my heart
That create and end worlds of
The distant volatility storms
That I was meant to tame

The correction to power law is probably produced in this case directly by GORENFLO and MAINARDI’s formula (6) in wright-function-power-law-correction,  Francesco MAINARDI was I think responsible for the exact formula with GORENFLO of an exact formula for the fundamental solution of the the fractional diffusion equations which have many applications in anomalous diffusion and other areas.  Note that the waiting time distribution over daily data is a model without direct observation as we have to assume that there is some distribution during the day and this is the reason why tick-data was asked for in the recent Gorenflo-Mainardi work.

Now for my more aggressive speculative leap:  there is also an exact formula for the HYDRODYNAMIC equations in equations (20)-(24) of chaurasia-solution-fractional-navier-stokes.  The reason for this aggressive speculative leap is that it is not even theoretically possible in diffusion equations for any phase transition to turbulence.  In order to understand why this is the case just consider the well-known aspects of hydrodynamic turbulence with say the Dirichlet boundary consitions:

$\frac{\partial u}{\partial t} - \Delta u= \kappa u\nabla u$

$u(x,0) = f(x)$

This is the Navier-Stokes for incompressible fluid without pressure term and it has a phase transition from laminar flow to turbulence when $\kappa$ is large and this $\kappa$ is called the Reynold’s number.  Here it’s quite clear that if the right hand side were to be set to zero there cannot be a theoretical possibility of turbulence.

In finance modeling, the stochastic models of volatility is from some family of stochastic processes that include the Brownian motion and the fractional Brownian motion, the entire class, whatever characteristics it has, does not have the same type of phase transition to turbulence as the hydrodynamic case because the right hand side is zero and these diffusions are smoothing in fact, by the Bochner subordination theory even fractional diffusions will be subordinated to the Brownian motion and therefore form a contractive semigroup $\exp(tL)$ for the generator $L$ .  In the case of volatility there are other ‘mean-reverting’ stochastic processes used sometime as well such as the Ornstein-Uhlenbeck process but an artist looking at the markets would — and this is from actual chat with artistists with direct intuition of diffusion and fluid flow and their mapping to the volatility for whatever value that has – that there is turbulence in volatility, and cascading effects had been recorded in FX markets a couple of decades ago even in Nature, so this is not a frivolous scientific question either.  When we model a univariate volatility process as a diffusion do we not make a giant disservice to the seven billion of humanity who can, if not some relief from a world order that is based on resource control of the world, extreme inequalities in opportunities etc etc. to at least give some models in any science of finance that actually map to lived reality of that volatility and consider these financial volatility storms as far more dangerous to the lives and livelihoods of all people than actual hurricanes and storms from which most people can find shelter from?