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FRACTIONAL DIFFUSION STOCHASTIC VOLATILITY MODEL

The model is the following:

$\log(|r_t|^2) = h_t + log(\epsilon_t^2)$

and $h_t$ is a sample path of a fractional diffusion with white noise

$D^\alpha u = -\lambda \Delta u + A u$

so $dh_t = A I^{1-\alpha} h_t dt - \lambda dz^{1-\alpha}_t$ (*)

where $Corr(\epsilon_t,\xi_t) = \rho$ and $D^\alpha$ is the Caputo derivative so that the product rule holds.  For the fractional diffusion equation have the solution $v(x,t) = E_{\alpha}((At)^\alpha) u(x,t)$ where $D^\alpha_t u(x,t) = -\lambda \Delta u(x,t)$.

For denoising correlated fractional noise we can apply the thresholds discovered by Wang in wang-1996-thresholding-long-memory

We can approximately decorrelate the fractional Brownian motion by applying a wavelet transform $W_b$: let $Y_t = W_b h_t$ so that (*) implies

$dY_t = A I^{1-\alpha} Y_t dt - \lambda dw_t$

Then we have various options to estimate $A,\alpha,\lambda$ from the wavelet transform.