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 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.