Transfer Principle for nth Order Fractional Brownian Motion with Applications to Prediction and Equivalence in Law

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The nth order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index H ∈ (n-1,n), having nth order stationary increments. We provide a transfer principle for the nth order fractional Brownian motion, i.e., we construct a Brownian motion from the nthe order fractional Brownian motion and then represent the nthe order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the nthe order fractional Brownian motion and the associated Brownian motion coincide. By using this transfer principle, we provide the prediction formula for the nthe order fractional Brownian motion and also a representation formula for all the Gaussian processes that are equivalent in law to the nth order fractional Brownian motion.