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Levy's characterisation of Brownian motion:
If X$X$ is a continuous martingale and X$X$ has quadratic variation process $\[ X\]_t = t$$[ X ]_t = t$ then $X$ is a standard Brownian motion.
If X is a continuous martingale and X has quadratic variation process $\[ X\]_t = t$ then $X$ is a standard Brownian motion.
If $X$ is a continuous martingale and $X$ has quadratic variation process $[ X ]_t = t$ then $X$ is a standard Brownian motion.
