  In this paper we introduce the notion of fractional martingale as the
fractional derivative of order $\alpha$ of a continuous local martingale, where
$\alpha\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation
of order $\frac{2}{1+2\alpha}$, under some integrability assumptions on the
quadratic variation of the local martingale. As an application we establish an
extension of L\'evy's characterization theorem for the fractional Brownian
motion.
