Exponential test-function characterization of Brownian motion
= Exponential test-function characterization of Brownian motion
Let $W$ be continuous with $W_0=0$. If
$$
e^{cW_t}-\frac{c^2}{2}\int_0^te^{cW_s}\,ds
$$
is a martingale for every real $c$, then conditional expectations solve
$$
\mathbb E[e^{cW_t}\mid\mathcal F_s]
=e^{cW_s+c^2(t-s)/2}.
$$
Thus $W_t-W_s$ is independent of $\mathcal F_s$ and distributed as $N(0,t-s)$, so $W$ is Brownian motion.