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## THE EXACT ANALOGY OF VOLATILITY LONG MEMORY TO QUEUE PROCESSES

Rigorous results are known for large deviations for fractional Poisson process due to Beghin-Macci here (Beghin-Macci-LargeDeviationsFPP)

Now I think the right way to think about the volatility is by an analogy to queuing processes where large deviations were used before for example (LargeDeviations-LMQueue-1998).  The key issue is that the LDP gives a rate function that tells us about the size of volatility jumps directly.

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.