An extension of the Lévy 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. Lévy says that $X$ is a Brownian motion if and only if $X$ and $X_t^2-t$, $t\ge 0,$ are martingales with respect to the intrinsic filtration ${\mathrm{F}}^X$. We extend this result to fractional Brownian motion.

Published in at http://dx.doi.org/10.1214/10-AOP555 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)