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

Yuliya Mishura·Esko Valkeila
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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.

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Yuliya Mishura, Esko Valkeila. 2006-11-29. An extension of the L\'{e}vy characterization to fractional Brownian motion. https://doi.org/10.1214/10-aop555

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