the breakup of determinism and stochasticity in an exact science
June 11, 2015 by zulfahmed
The division between noise and determinism in sciences has always given preference to determinism not merely out of respect for tradition that Newton and Laplace and others have founded on the differential equations formalism. For this reason, a true science of finance that models global volatility as its primary object would presumably choose a FRACTIONAL DIFFERENTIAL system with white noise — in which case optimal denoising of white noise is guaranteed already by Donoho-Johnstone wavelet thresholding to leave a deterministic rather than stochastic system if this is the truth. It is thus incumbent upon me to spend time on fractional differential equations and their perturbations as deterministic systems and fit those to daily and other scale volatility data which we already have collected with some preliminary results and understandings. We know that a fractional Poisson process does describe volatility shock distribution if we go the stochastic route. But I suspect that after ‘fractionality’ is detected after a theoretically justified denoising step, we can do better than any stochastic model and found a true science of finance in an overoptimistic situation. The tremendous amount mathematical progress on fractional differential equations suggest that an exact global volatility model is timely. It is most likely done in the industry with the wrong view of profit-motive diminishing such work as pure science for the benefit of quelling the volatility storms which is a great task that I’m proud to expend my efforts.
This is apparently far from your expertise, sir, but I can reduce the problem to one that is within precisely: wavelet denoising optimality for underlying signal that is deterministic and Besov (by some theorem that fractional differential equations have unique deterministic solution that is Besov say) is optimally minimax recovered and what denoising does here is recover a trajectory of a fractional deterministic equation that is an exact model for volatility. That exact model can be studied as the true science of global volatility.
An example of technically similar but obviously different application area is image denoising using fractional PDE. Recovery of solutions of deterministic fractional differential equations from white noise seems very good first step to an exact scientific model of volatility globally. Then we can take the ergodic theory of deterministic systems.
This seems quite ambitious but since people have worked out the theories in multiple directions, it’s simply an exercise in collating and interpreting results and coding them up in R or Octave.
I apologize for my long monologue. Now I will be much more rigorous in the next steps with a firm understanding of a good direction in research.