 Solution

By the usual definition of a Lévy process, $X_0=0$ almost surely. The two martingale assumptions giveFor a continuous Lévy process, the jump measure in the Lévy–Khintchine formula is zero, and the process has the form of a Brownian Gaussian component plus a deterministic drift. Thus its characteristic function isThe first moment makes $m=0$, and the second moment makes ${\sigma }^2=1$. ConsequentlyEquivalently, continuity and the martingale identity $X_t^2-t$ identify the quadratic variation as $t$, so the Lévy characterization of Brownian motion gives the same conclusion. A continuous nonzero drift is excluded by the mean-zero martingale assumption.