I was inspired by love and had a chance to re-evaluate the fundamental questions capable of driving my passions, with my practical life being destroyed by a housing bubble. Probably the most significant advance I’d made is to break the century-long stranglehold on quantitative financial models by diffusions and now fractional diffusion sample paths (Gorenflo, Mainardi have been doing some of the best work I’d seen on this) but in order to appreciate the Aristotelian edifice built in finance from sample paths of diffusions, see the Chapter 3 of Robert Merton’s merton-continuous-finance. This is the representation of the MIT school of Samuelson and will tell you that sample paths of diffusions are the most basic foundation of continuous-time finance since Bachelier’s thesis Theory of Speculation around 1900, the Ph.D. thesis he did under Henri Poincare. In order to break this mould, I’ve taken on the project of considering the work of Onsager on turbulent hydrdodynamics from early 1940s where he explained the formation of new vortices in turbulent hydrodynamics by introduction of a negative temperature. I am considering the following setting:
(a) Global financial volatility is the fundamental observable of finance and exact models for this are of interest.
(b) Diffusion sample paths cannot have phase transitions to turbulence. It is well-known in hydrodynamics that phase transition to turbulence is controlled by a parameter called the Reynold’s number which is inversely proportional to the viscosity of the fluid. The hydrodynamic equation without a pressure term is:
with initial condition and for this there exists a phase transition to turbulence when with a critical threshold for phase transition. The diffusion equation results when . While this is known for the exact Navier-Stokes well since the work of Kolmogorov and Onsager (early 1940s), there has not been a serious effort to seek quantitative models in finance that consider these nonlinear turbulence-capable models for use in continuous-time finance, even though since early 1990s there have been work published in Nature and other top science journals showing that turbulent cascades are observed empirically in FX markets.
(c) When employing (fractional) partial differential equations, there needs to be a coherent space variable. Our choice for a natural space variable for financial markets is the complex network formed by the mathematical graph with nodes the number of assets traded in the markets. In contrast to the recent progress in the study of complex network following the work of Watts-Strogatz and Barbasi-Albert, who posit degree power law distribution scale-free networks, we found that the empirical market graph is far denser if determined by high correlation of volatilities and retains around 75% of the edges of the complete graph. This space variable allows us to employ the graph Laplacian for the diffusion term in an evolution equation for volatility.
Our initial approach to this problem is necessarily a simple pragmatic approach to obtain some proof-of-concept results. Turbulent dynamics is observed for global financial volatility and therefore it is natural to seek quantitative models for turbulence and perhaps break with the diffusion models.
Other projects begun during this time were originally to construct a technology-finance type company which never took off because I was too distracted by other interests and did not have the sense that out of the blue my landlard would attempt an unlawful eviction and now they are barring landlords from raising rents on the housing prices here with thousands of evictions daily that is the norm in this city with money pouring into real estate. I had some good ideas about artificial intelligence and computer graphics which are worth following through but this volatility storm project is much more serious and could have a far greater impact in making an exact science of finance realistic and able to provide some understanding of the fundamental quantitative models for global financial volatility which could help humanity manage the markets not with ideology and religious dogma on markets but without conspiracy theories deal with the issues that are analogous to hydrodynamic turbulence based on viscosity of money flowing into the market which produces a phase transition to turbulence.