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

Abstract

The n-th order fractional Brownian motion was introduced by Perrin et al. [13]. It is the (up to a multiplicative constant) unique self-similar Gaussian process with the Hurst index H∈(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-th order fractional Brownian motion and then represent the n-th order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the n-th order fractional Brownian motion and the associated Brownian motion coincide. By using this transfer principle, we provide the prediction formula for the n-th order fractional Brownian motion and also a representation formula for all Gaussian processes that are equivalent in law to the n-th order fractional Brownian motion.