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
$$
\mathbb E(X_t\mid\mathcal F_s)
=X_s+\frac{t-s}{1-s}(X_1-X_s).
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-201/4/a/solution