Let $X_s$ be a $(\mu,\sigma)$ geometric Brownian motion with $X_0 = x$. For some positive numbers $c < x < d$ and time $t$, what is the probability $X_s \in [c,d]$ for all $s \in [0,t]$?
In particular, is there a closed-form expression for this probability? This problem is not too bad with an arithmetic Brownian motion, but I am struggling to find a clean solution for the geometric case.