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## THE NEED FOR A TIME-FRACTIONAL ITO CALCULUS

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:

$D^{\alpha}_t V(t) = \kappa(\theta-V(t)) + \sigma \sqrt{V(t)} dw^2(t)/dt$

which has the solution (fractional-stochastic-diff-eq-sakthivel-rethavi-ren-2013):
$V(t) = T_{\alpha}(t) V(0) + \int_0^t S_{\alpha}(s) \kappa\theta ds + \int_0^t \sigma S_{\alpha}(s) \sqrt{V(s)} dw(s)$

One can get the Ito $dV(t)$ from this solution.