  The $n$th 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 $H \in (n-1,n)$, having $n$th order stationary increments. We
provide a transfer principle for the $n$th order fractional Brownian motion,
i.e., we construct a Brownian motion from the $n$the order fractional Brownian
motion and then represent the $n$the order fractional Brownian motion by using
the Brownian motion in a non-anticipative way so that the filtrations of the
$n$the order fractional Brownian motion and the associated Brownian motion
coincide. By using this transfer principle, we provide the prediction formula
for the $n$the order fractional Brownian motion and also a representation
formula for all the Gaussian processes that are equivalent in law to the $n$th
order fractional Brownian motion.
