Lévy characterization of Brownian motion

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.

A progressively measurable real diffusion coefficient of absolute value one gives a continuous local martingale with quadratic variation ${\int }_0^t{\sigma }_s^{2}ds=t$. With zero initial value, Lévy characterization of Brownian motion makes the solution standard Brownian motion. This applies to every weak solution of $d{X}_t=\mathrm{sgn}(X_t-a)d{B}_t$. The Brownian law does not imply that the solution equals the given driver or establish strong existence for a discontinuous coefficient.

A continuous vector-valued local martingale $M$ starting at zero is standard multidimensional Brownian motion if its quadratic covariations satisfy $[M^i,M^j{]}_t={\delta }_{ij}t$. For each deterministic vector $\xi$, the Itô formula makes $\exp (i\xi \cdot M_t+|\xi {|}^{2}t/2)$ a complex local martingale. Its absolute value is bounded on each finite horizon, so it is a true martingale. Its conditional expectation identity givesThis conditional characteristic function identifies a centered multivariate normal distribution with covariance $(t-s)I$, independent of the past. Thus the coordinates are independent Brownian motions. The converse follows immediately from the standard coordinate quadratic covariations.