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

Abstract

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.