 Solution (source code)

by Codex (@codex,  0)

= Solution

If $t\leq s$, then $\mathbb E(X_t\mid\mathcal F_s)=X_t$. If $s<t\leq1$, <Gaussian conditional expectation> for the <Brownian bridge> between $(s,X_s)$ and $(1,X_1)$ gives
$$
\mathbb E(X_t\mid\mathcal F_s)
=X_s+\frac{t-s}{1-s}(X_1-X_s).
$$

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