Is Brownian Motion increasing?

Question

Given a process $Y_t = e^{B_t}$

We know that since Brownian motion is continuous for $t \geq 0$. Since $B_t$ is a completely random motion, it is true that we cannot say whether it is monotone increasing/decreasing over time, no?

So if at some instance, $B_{n} < B_{n+1}$ but $B_{n+1} > B_{n+3}$ then $Y_t$ is not monotone at all, correct?

edit: The specific question is, is $Y_t$ monotone increasing in $t$ ?

Answer 1

We can say that $B_t$ is monotone with probability 0 (just was we can say $B_t$ is differentiable w.p. 0).

For example, if $B_t$ is now 1, then it will hit $2$ some time later w.p. 1; from here it will some time hit $1$ again w.p. 1.

It follows that $e^{B_t}$ is also monotone with zero probability.