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## SOLUTION OF FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATION FOR SQUARE ROOT PROCESS

The square root process with fractional derivatives is

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

$= -\kappa V(t) + \kappa\theta + \nu\sqrt{V(t)} dw/dt$

The solution is given by

$V(t) = E_{\alpha}(-\kappa t^\alpha) + \int_0^t \kappa\theta S_{\alpha}(t-s) ds + \int_0^t S_{\alpha}(t-s) \nu \sqrt{V(s)} dw(s)$

where $S_{\alpha}(t) = t^{\alpha-1} E_{\alpha,\alpha}(-\kappa t^{\alpha})$.

The existence and uniqueness theorem for this equation is given in Theorem 3.2 of fractional-stochastic-diff-eq-sakthivel-rethavi-ren-2013.

Writing $f(t) = E_{\alpha}(-\kappa t^{\alpha})$ with some manipulation with the Ito formula we get an ordinary differential equation with white noise for $\sqrt{V(t)}$:

$d\sqrt{V(s)} = \frac{ f(s)}{\nu S_{\alpha}(t-s)\sqrt{V(s)} } - \frac{\nu}{2} \frac{S_{\alpha}(t-s)}{\sqrt{V(s)}} + dw(s)$

So we could solve this equation to obtain the drift term for the square root process variance without worrying about fractional Brownian motions.  This model is likely to be more exact fit to the empirical autocorrelations of volatility which I have shown are fit extremely well by Mittag-Leffler functions.  To remind readers, here are some of the empirical fits.

The solution of the ode without noise is thus

$V(t) = \int_0^t \frac{2 E_{\alpha}(-\kappa t^{\alpha})}{\nu S_\alpha(t-s)} - \nu S_{\alpha}(t-s) ds$

The issue here is not closed form option pricing but rather some exact stochastic volatility models; the analysis of the Heston model provides deep theoretical insights which we could hope would be carried over to an exact model fitting empirical autocorrelations.