Let $X$ be a fractional Brownian motion. It is known that $M_t=int m_t dX,, tge 0$, where $m_t$ is a certain kernel, defines a martingale $M$, and also that $X$ can be represented by $X_t=int x_t dM,, tge 0$, for some kernel $x_t$. We derive these results by using the spectral representation of the covariance function of $X$. A formula for the covariance between $X$ and $M$ is also given.

,
,
CWI
CWI. Probability, Networks and Algorithms [PNA]
Stochastics

Dzhaparidze, K.O, & Ferreira, J.A. (2001). A frequency domain approach to some results on fractional Brownian motion. CWI. Probability, Networks and Algorithms [PNA]. CWI.