 Gaussian characteristic-function martingale from covariance loss

If a continuous mean martingale $M$ and a nonnegative finite-variation process $V$ satisfy $d⟨M⟩=-dV$, then the Itô formula shows $\exp (iM-V/2)$ is a bounded complex martingale. Its terminal expectation establishes an entire Gaussian distribution with the initial mean and variance. This is stronger than matching only the first two moments in a random-domain field construction.