Exponential test-function characterization of Brownian motion

Let $W$ be continuous with $W_0=0$. Ifis a martingale for every real $c$, then conditional expectations solveThus $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and distributed as $N(0,t-s)$, so $W$ is Brownian motion.

For an adapted continuous process starting at zero, the displayed conditional characteristic function identifies every increment as $N(0,t-s)$ and makes it independent of the preceding sigma-field. Multiply by the indicator of an event in that sigma-field, then use the uniqueness theorem for characteristic functions for finite measures. Thus the process is a Brownian motion in the given filtration.