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## BOTH FRACTIONAL INTEGRATION AND FRACTIONAL BROWNIAN MOTION SHOULD APPEAR IN THE CORRECT MODEL OF VOLATILITY

By ‘correct’ we mean one that produces autocorrelations that fit empirical autocorrelations, i.e. Mittag-Leffer functions $(a,b,c)\rightarrow E_{a,b}(-ct^a)$.

$D_t^{\mu} x_t = (\alpha - \beta x_t) + \sigma \sqrt{x_t} dw_t /dt$

Apply $D^{1-\mu}$ to obtain

$D^1_t x_t = D^{1-\mu} (\alpha - \beta x_t) + \sigma D_t^{1-\mu}(\sqrt{x_t} dw_t/dt)$ (*)

Integrate the second quantity by parts using the fact that $D_t^\alpha = 1/\Gamma(1-\alpha) d/dt \int_0^t (t-s)^{-\alpha} (\cdot) ds$ to get

$C(\alpha) (\sqrt{x_t} - \sqrt{x_0}) dw^{1-\mu}_t/dt - 1/2 \Gamma(-\mu) \sqrt{x_t} dw^{1-\mu}_t/dt$

This then still leaves the fractional integration in the first term of (*).  So this is partly the problem with using fractional Brownian motions only in the model $dX_t = (\alpha-\beta X_t) dt + \sigma \sqrt{X_t} dw^{1-\mu}_t$ it would seem.