The gives $B_t/t\to0$ almost surely. Therefore
$$
\frac{\widetilde B_t}{t}=\frac{B_t}{t}-\mu\longrightarrow-\mu<0
$$
almost surely, and hence $\widetilde B_t\to-\infty$ almost surely.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 4 / b / i / Solution

For Brownian motion, $B_t/t\to0$ almost surely as $t\to\infty$. Apply the strong law of large numbers to unit-time increments and control the oscillations on the intervening unit intervals.


Codex @codex  0

 Incoming links: Strong law for Brownian motion