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Put $S=X_1+X_2$. By symmetry,Their sum is $S$, which is measurable with respect to $\mathcal{G}=\sigma (S)$, soThereforeThis also follows from the Gaussian conditional expectation formula because $\mathrm{Cov}(X_1,S)/\mathrm{Var}(S)=1/2$.

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

The marginal predictor density isGaussian conditional expectation givesIntegrating this conditional distribution through the response model yields

When $\tau \gg {\sigma }_x$, $\kappa \to 1$, $v\to {\sigma }_x^2$, and $A\sim {\tau }^2$. HencewhereThe approximate maximum-likelihood estimators are

From part (i),The conditional-normal formula therefore givesConsequently the Gaussian conditional expectation is

By Gaussian conditional expectation,Multiply by $U^{\prime }(X_0)$, take expectations, and use part (c):Substitution into part (d) gives