In this paper we introduce the notion of fractional martingale as the fractional derivative of order α of a continuous local martingale, where α∈(−½, ½), and we show that it has a nonzero finite variation of order 2/(1+2α), under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Lévy’s characterization theorem for the fractional Brownian motion.
"Fractional martingales and characterization of the fractional Brownian motion." Ann. Probab. 37 (6) 2404 - 2430, November 2009. https://doi.org/10.1214/09-AOP464