Some Martingales from a Fractional Brownian Motion and Applications

In this paper, some continuous martingales are constructed from a fractional Brownian motion with the Hurst parameter in the interval (1/2, 1), and some applications are made. These processes are obtained using a stochastic calculus for a fractional Brownian motion. Square integrable, continuous martingales are exhibited as stochastic integrals with respect to a fractional Brownian motion and the associated increasing processes are given. These martingales are used to construct Radon-Nikodym derivatives (likelihood functions) for some measures that are absolutely continuous with respect to the measure of a fractional Brownian motion. A Radon-Nikodym derivative is used to relate a mutual information between a stochastic signal and this signal plus a fractional Gaussian noise to an estimation error.