Are fractional Brownian motions predictable?

TL;DR

This paper investigates the predictability of fractional Brownian motions based on their Hurst index, introducing a local predictor concept that distinguishes between predictable and non-predictable cases.

Contribution

It introduces the local predictor, extending the predictable compensator, and characterizes fractional Brownian motions' predictability based on the Hurst index.

Findings

fBm with H > 1/2 is predictable via the local predictor

fBm with H < 1/2 is not predictable using the local predictor

Brownian motion has a trivial local predictor of zero

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.