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Our most significant result on search for law underlying volatility as a global object found a slightly nonlinear parametric shape for the integrated over time total number of shocks. The result was at odds with an exact linear law expected for a Poisson renewal theory. We know renewal theory generally will fail to give us a nonlinear EN(t) for any shock model because N(t)/t converges to a constant as t tends to infinity for renewal theory quite generally as both doob and feller’s attached papers will reinforce. But in order to gain any significant insight into the type of assumptions that could produce the attached empirical integrated shock count.
I will generally ask the following question next. For renewal type processes as analogies of Brownian motion, what would be the process with EN(t) ~ a t^b, where a,b are real? I suspect that a power law does characterize the integrated shocks in actual volatility shocks over the entire global financial network defined in our manner using latent stochastic volatility correlations.
As a complete stab in the dark, for Poisson one considers exp(lambda*t) and I would consider functions exp( – a*t^b ) as a family of probabilities of interest. Regardless of whether correct, I will play around with these distributions to try to understand a better shock distribution for volatility.
Preview attachment fellerintegralequationrenewaltheory.pdf
Preview attachment doobrenewaltheory1948.pdf
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