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

If $t\le s$, then $\mathbb{E}(X_t\mid {\mathcal{F}}_s)=X_t$. If $s<t\le 1$, Gaussian conditional expectation for the Brownian bridge between $(s,X_s)$ and $(1,X_1)$ gives

Since $X_1-X_s$ is centered Gaussian with variance $1-s$,Fubini's theorem therefore shows that the defining integral for $A_1$ is absolutely finite almost surely.

For $s<r<1$, part a givesConditional Fubini then yieldsHence $\mathbb{E}(M_t\mid {\mathcal{F}}_s)=M_s$, so $M$ is a martingale.

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 $X_1$.