Assume that X is a continuous square integrable process with zero mean, defined on some probability space (Ω, F, P). The classical characterization due to P. Lévy says that X is a Brownian motion if and only if X and Xt2 − t, t ≥ 0, are martingales with respect to the intrinsic filtration FX. We extend this result to fractional Brownian motion.
"An extension of the Lévy characterization to fractional Brownian motion." Ann. Probab. 39 (2) 439 - 470, March 2011. https://doi.org/10.1214/10-AOP555