An extension of the L\'{e}vy characterization to fractional Brownian motion

Assume that $X$ is a continuous square integrable process with zero mean, defined on some probability space $(\Omega,\mathrm {F},\mathrm {P})$. The classical characterization due to P. L\'{e}vy says that $X$ is a Brownian motion if and only if $X$ and $X_t^2-t$, $t\ge0,$ are martingales with respect to the intrinsic filtration $\mathrm {F}^X$. We extend this result to fractional Brownian motion.