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## FIRST ATTEMPT AT LONG-MEMORY FOR STEIN-STEIN-SCHOBEL-ZHU MODEL

The Stein-Stein model (stein-stein-1991) came before Heston, in 1991 and had the problem (from the scientific point of view) that there was no correlation between the brownian motion driving price and volatility which was fixed by Schobel-Zhu in 1999 (Schobel-Zhu-1999).  I included long memory in Heston model relatively easily.  In the case of Stein-Stein-Schobel-Zhu it is more complicated.  Essentially, Stein-Stein-Schobel-Zhu boils down to solving the three ordinary differential equations

$D_t = -\sigma^2 D^2 + 2k D + 2s_1$

$B_t = (k-\sigma^2 D) B - k\theta D + s_2$

$C_t = -\frac{1}{2} \sigma^2 B^2 - k\theta B - \frac{1}{2} \sigma^2 D$

These can be solved explicitly.

In order to add long memory with Mittag-Leffler waiting times, we replace

$\sigma \rightarrow \sigma S_\alpha(t)$ and the same with $\rho$ where

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