Max of Brownian motion with drift is finite almost surely

Question

For $B_t$ Brownian Motion with drift $\mu<0$, I need to prove that the max value, $X = \max_{0<t<\infty}B_t$ is finite almost surely, ie $P(X<\infty)=1$.

Now, I know that because the mean is negative, it will go more and more negative, and it is also a supermartingale. But I don't know how to prove almost surely...

Appreciate any hints.

Answer 1

Hint: Try the strong law of large numbers. What does it say about $\lim_{t \to \infty} B_t/t$? What does this say about the sign of $B_t$ for large $t$?

Answer 2

Actually that is wrong in the sense that it is not finite in the limit. Brownian Motion with a negative drift will wander off to -∞ almost surely. It will tend to go down in a linear fashion with the slope equal to the drift parameter.