Identity of Running maximum

Question

let $X_t$ denotes a arithmetic Brownian motion process. I am wondering if the following identity is true ? $$ \mathrm{P}\left[\sup_{0 \le s \le t} \mathrm{e}^{X_t} < x\right] = \mathrm{P}\left[\sup_{0 \le s \le t} X_t < \ln{x}\right], $$ How may we show this result ?

Answer 1

Say that we work on a probability space $(\Omega, \mathcal{F},\mathbb{P})$. Define the set $A$ in $\Omega$ by $$A:=\{\omega \in \Omega | \sup_{0\leq s \leq t} e^{X_t(\omega)} < x\}.$$ Since the exponential is invertible we easily see that this is equivalent to the definition

$$ A:= \{\omega \in \Omega | \sup_{0\leq s \leq t} X_t(\omega) < \ln x\} .$$ So the probabilities of these sets must be equal.