Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 4 / d

The process $M$ is continuous and Gaussian. Since $A$ has finite variation,The Lévy characterization of Brownian motion makes $M$ a Brownian motion in the enlarged filtration.

Moreover,Every finite vector from $M$ is jointly Gaussian with $X_1$, so zero covariance implies independence. Thus the whole process $M$ is independent of $X_1$.