# Geometric Brownian motion hitting time

Let $X$ be a geometric Brownian motion $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let $\tau_a$ be the first hitting time of $a$ by $X$. How can we relate the Laplace transform of the hitting time of $X$ to the one of $W$ for which we know the expression?

• It might help to note that $\tau_a$ is the hitting time by $W$ of the line $t\mapsto [\log(a/x)+(\sigma^2/2-\mu)t]/\sigma$, where $x=X_0$. Commented May 7, 2016 at 16:42
• Hi @JohnDawkins, thanks. I tried finding the corresponding hitting level for $W$ but then I was stuck. I was not aware of this result about the hitting time of a line by a Brownian motion. Just a correction, IMO, the line should be $t \mapsto [\log(a/x) - \mu t ] (2/\sigma^2)$ where $x = X_0$ Commented May 7, 2016 at 16:57
• But $X_t = X_0\exp(\sigma W_t-\sigma^2t/2+\mu t)$. Commented May 8, 2016 at 16:20
• You're right, my mistake, too much stuff on my pad. Commented May 8, 2016 at 23:11