Lévy characterization of Brownian motion
= Lévy characterization of Brownian motion
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= Lévy's characterization of Brownian motion
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A continuous local martingale $M$ with $M_0=0$ is standard Brownian motion exactly when its <quadratic variation> satisfies $[M]_t=t$.
The characterization asserts that a continuous <local martingale> starting at zero with quadratic variation $t$ is <Brownian motion>.