 Solution

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 givesUniqueness 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 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.