Brownian Motion with drift (stupid question)

Question

How do you prove that $$ \lim_{t\to +\infty} (B_t+ct)=+\infty $$ almost surely?

$(B_t)_t$ is the standard Brownian Motion starting from $0$.

Answer 1

There are several possible ways to prove it, depending on what you already know. If you know the strong law of large numbers for Brownian motion, which says that as $t \to \infty$, $\frac{B_t}{t} \to 0$ almost surely, then we have $$\frac{B_t + ct}{t} = \frac{B_t}{t} + c \to c\quad \text{a.s.}$$ Since $c > 0$ it follows that ${B_t + ct} \to +\infty$ a.s.