Buoyed by the Delbaen-Schachermayer 1994 result that price processes must be a semimartingale, Ito calculus is sometimes seen as sufficient for finance. However, Comte-Renault (comte-renault-2003-affine-sv) have compellingly shown that stochastic volatility needs to be a long memory process in order to explain the term structure of option smiles in maturity. What Comte-Renault do is simply fractionally integrate a short memory square-root diffusion, which in my opinion is not the right model for long memory. Instead, the right model for volatility should be:
which has the solution (fractional-stochastic-diff-eq-sakthivel-rethavi-ren-2013):
One can get the Ito from this solution.