An extension of the Lévy characterization to fractional Brownian motion

Details An extension of the Lévy characterization to fractional Brownian motion An extension of the Lévy characterization to fractional Brownian motion

2006-11-29

arxiv.org/abs/math/0611913v4

Annals of Probability 2011, Vol. 39, No. 2, 439-470

Mathematics

Probability

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

Scientific paper

10.1214/10-AOP555

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.

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