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The key issue is how to squeeze the solution of a constant coefficient Riccati equation.  For the Heston model the key point for getting an explicit solution for the probability density is by solving a Riccati equation $y' = ay^2 + by + c$.  For the time-fractional case you want to solve instead $(y/S)' = a y^2 + by + c$ where $S= S_{\alpha}(t)$ is the waiting time of the fractional Poisson process.  The trick is to replace the time variable $t$ with $\int_0^t S_{\alpha}(s) ds$.  So you have to insert this expression into the explicit formula for the Heston probability density in DragulescuYakovenko2002 eq (18) say.