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## HOW TO EXTEND HESTON MODEL TO LONG MEMORY

The key to extending the Heston model to a fractional stochastic volatility model is to consider the equation for the characteristic function

$f(x,v,t) = \exp( C(t) + G(t) v + i\phi x)$

in Heston’s notation (Heston-original) but with

$G(t) = \int_0^t S_{\alpha}(s) D'(s) ds$

where $D(t)$ solves the Heston’s Riccati equation.  The difference between this solution and the Comte-Renault approach is that they fractionally integrate the volatility of Heston’s model while this approach produces directly a closed form solution as follows: solve for $D(t)$ using Heston’s Riccati equation and then integrate by $S_{\alpha}(t) = t^{\alpha-1} E_{\alpha,\alpha}(-\kappa t^\alpha)$ to obtain the solution of an exact PDE for long memory fractional stochastic volatility.

Some more details are here (fracsv)