= Gaussian exponential martingale with deterministic variance
{c}
{title2=$M_t=\exp(X_t-v(t)/2)$}
Let $X_0=0$ and let $X$ be a centered <Gaussian process> whose increments are independent of the past <filtration>, with deterministic variance $v(t)$. Its displayed exponential is a positive <martingale>. Indeed, $X_t-X_s$ is a centered <Gaussian random variable> with variance $v(t)-v(s)$, so the <Gaussian moment-generating function> gives
$$
\mathbb E[M_t\mid\mathcal F_s]=M_s\mathbb E\exp(X_t-X_s-(v(t)-v(s))/2)=M_s.
$$
The sign-reversed process has the same variance and the same conclusion. Continuity of paths is unnecessary for this conditional-expectation argument.
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