= Solution
{c}
If $X$ is <Brownian motion> with respect to the given <filtration>, independent Gaussian increments give the identity immediately.
Conversely, fix $s<t$ and let $\Delta=X_t-X_s$. Taking expectations gives its <characteristic function> $e^{-\xi^2(t-s)/2}$, so $\Delta\sim N(0,t-s)$. To prove <independence>, for any $A\in\mathcal F_s$ define the <finite measure> $\nu_A(E)=\mathbb P(A\cap\{\Delta\in E\})$. The conditional identity gives
$$
\int e^{i\xi u}\nu_A(du)
=\mathbb P(A)e^{-\xi^2(t-s)/2}.
$$
Uniqueness of <Fourier transforms> of <finite measures> implies $\nu_A=\mathbb P(A)N(0,t-s)$. This is <independence> of $\Delta$ from $\mathcal F_s$. Iterating over times proves <independent increments>. Together with the stated <continuity> and $X_0=0$, this gives \b[standard <Brownian motion>].
Here a process with the stated <filtration> is understood to be <adapted>, as required when saying it is Brownian with respect to that <filtration>. If adaptedness is not implicit, it must be added: an ordinary <Brownian motion> with a trivial <filtration> satisfies the displayed <conditional characteristic function> but is not <adapted> to that <filtration>.
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