Let $d(x,A)$ be the distance to the closed set $A$. Continuity gives
$$
\{T_A\leq t\}
=\left\{\inf_{s\in[0,t]}d(X_s,A)=0\right\}
=\left\{\inf_{q\in\mathbb Q\cap[0,t]}d(X_q,A)=0\right\}.
$$
Every variable in the countable infimum is $\mathcal F_t$-measurable by adaptedness, so the event belongs to $\mathcal F_t$. Hence $T_A$ is a <stopping time>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 6 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-201/6/b/solution