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There is a weak relationship between volume and volatility that is well-known.  A natural conjecture is that Mandelbrot’s insights apply to financial markets by an exact analogy to the Hurst work on water levels of Nile in the sense that volatility is a natural function of new trade orders arriving to the market that is being processed by the market.  The fractional Poisson process $\nu \sim 1.2$ in this case and the large deviation rate function is given in the Beghin-Macci paper; now the exact analogy to long memory queues is interesting because it could tell us something about the actual sizes of jumps.