Lévy's characterization of Brownian motion

Remarks

The second hypothesis says that the quadratic variation of $X$ is $[X]_t = t$. For a fixed $u \in \R$, Itô's formula shows that $Z_t = \exp(i u X_t + u^2 t / 2)$ is a local martingale; it is bounded on bounded time intervals, hence a martingale. So $E[\exp(i u (X_t - X_s)) \mid \mathcal{F}_s] = \exp(-u^2 (t - s)/2)$, which identifies the conditional law of the increment as normal with mean $0$ and variance $t - s$. Continuity is necessary: for a Poisson process $N$ of rate $1$, both $N_t - t$ and $(N_t - t)^2 - t$ are martingales, but $N_t - t$ is not a Wiener process. In dimension $d$, a continuous process $X$ with $X_0 = 0$ is a $d$-dimensional Wiener process if each $X^i$ and each $X^i X^j - \delta_{ij} t$ is a local martingale.

Going deeper

For a student

The theorem is the standard way to recognise a Brownian motion inside stochastic calculus. It gives the Dambis–Dubins–Schwarz theorem: a continuous local martingale $M$ with $[M]_\infty = \infty$ is a time-changed Brownian motion, $M_t = B_{[M]_t}$. It is also a step in the proof of the Girsanov theorem, and it shows that $\int_0^t \sgn(W_s) \, dW_s$ is a Wiener process, which is used with Tanaka's formula.