Are fractional Brownian motions predictable?

Abstract

We provide a device, called the local predictor, which extends the idea of the predictable compensator. It is shown that a fBm with the Hurst index greater than 1/2 coincides with its local predictor while fBm with the Hurst index smaller than 1/2 does not admit any local predictor. The local predictor of a martingale (in particular: Brownian motion) trivially exists and equals 0.

7 pages, to appear in proceedings of the Sixth Seminar on Stochastic Analysis, Random Fields and Applications, Ascona 2008