Let \(M\) be a Gaussian martingale with bracket \(\langle Mangle\). Prove that the process \(\langle Mangle\) is deterministic.

The Gaussian property implies that, for \(t>s\), the r.v. \(M_{t}-M_{s}\) is independent of \(\mathcal{F}_{s}^{M}\), hence

\[\mathbb{E}\left(\left(M_{t}-M_{s}\right)^{2} \mid \mathcal{F}_{s}^{M}\right)=\mathbb{E}\left(\left(M_{t}-M_{s}\right)^{2}\right)=A(t)-A(s)\]

with \(A(t)=\mathbb{E}\left(M_{t}^{2}\right)\) which is deterministic.