The deterministic bound
$$
0\leq\sqrt{\epsilon^2+x^2}-|x|
=\frac{\epsilon^2}{\sqrt{\epsilon^2+x^2}+|x|}
\leq\epsilon
$$
holds for every <real number> $x$. Therefore
$$
\sup_{t\leq T}|f_\epsilon(B_t)-|B_t||\leq\epsilon
$$
for every $T$, so $f_\epsilon(B)$ converges to $|B|$ uniformly on every compact time interval, and hence ucp.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-202/1/c/solution