Conditional expectations and martingales in the fractional Brownian field

Abstract

Conditional expectations of a fractional Brownian motion with Hurst index H respect to the filtration of a fractional Brownian motion with Hurst index H^′, both contained in the fractional Brownian field, are studied. A stochastic integral representation of those processes is constructed from the covariance structure of the underlying fractional Brownian field. As processes, the conditional expectations contain martingale components and for dual pairs of Hurst indices the processes become pure martingales which, up to a multiplicative constant, coincide with the fundamental martingales of fractional Brownian motions.