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

An extension of the Lévy characterization to fractional Brownian motion