= Solution
The one-dimensional <Lévy characterization of Brownian motion> states: a continuous adapted process $L$, with $L_0=0$, is standard <Brownian motion> relative to its filtration if and only if it is a <local martingale> and $[L]_t=t$ for all $t$ almost surely.
For necessity, the centered independent Gaussian increments of Brownian motion give both the martingale property and the martingale $L_t^2-t$. The uniqueness characterization of <quadratic variation> then gives $[L]_t=t$.
For sufficiency fix $\theta\in\mathbb R$. The <Itô formula> gives
$$
E_t=\exp(i\theta L_t+\tfrac12\theta^2t),\qquad dE_t=i\theta E_t\,dL_t.
$$
The second-order term cancels the time derivative because $d[L]_t=dt$. Thus $E$ is a complex local martingale. Its modulus is the deterministic quantity $e^{\theta^2t/2}$, bounded on every finite horizon, so its real and imaginary parts are true martingales by the <bounded local martingale criterion>. For $s<t$,
$$
\mathbb E[e^{i\theta(L_t-L_s)}\mid\mathcal F_s]=e^{-\theta^2(t-s)/2}.
$$
Uniqueness of <characteristic functions> says that the increment has <normal distribution> $N(0,t-s)$ independently of $\mathcal F_s$. Indeed multiplying the identity by any bounded $\mathcal F_s$-measurable variable gives the product characteristic-function identity expressing independence. Iterating over deterministic times gives independent increments. Together with continuity and $L_0=0$, these are precisely the defining properties of standard Brownian motion. If the starting value is a fixed $x$, the same argument identifies $L-x$ as standard Brownian motion.
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