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## The prediction formula for fractional Brownian motions

Fractional Brownian motions with Hurst parameter $H \in (1/2, 1)$ contain long range dependence.  This is the regime which interesting for volatility prediction.  The prediction problem of determining the conditional expectation $E(X_T | X_{(-\infty, t]})$ is nontrivial.  The solution provided by Gripenberg and Norros in 1996 is the following:

$E[X_T | X_s, 0\le s\le t] = X_t + \int_0^t \Psi_T(t,s)dX_s$

where

$\Psi_T(t,s) = \frac{\sin(\pi(H-1/2))}{\pi}s^{-H+1/2}(t-s)^{-H+1/2}$

$\int_t^T \frac{ u^{H-1/2}(u-t)^{H-1/2}}{u-s} du$

An excellent paper presenting the result is NorrosValkeilaVirtamoGirsanovFormulaPredictionFractionalBM