Suppose I have a continuous-time stochastic process $\{X(t)\}$ defined on a filtered probability space $(\Omega, \mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$ and that I know the distribution of a stopping time $\tau$, can I say something about the distribution (not just the moments) of the random variable $X(\tau)$?
I guess in general the answer is no, but something could be said if the stochastic process $\{X(t)\}$ has some kind of structure (e.g. Geometric Brownian Motion, Lévy processes, martingales, ...). Any help would be kindly appreciated.
In general, no, not without knowing the joint distribution of $(\tau, X)$. We can have two processes $X$ and $Y$ with the same distribution, but $X(\tau)$ has a different distribution than $Y(\tau)$.
For example, let $B$ and $W$ be independent Brownian motions and $\tau := \inf\{t : B(t) = 1\}$. Even though $B$ and $W$ have the same structure and distribution, we have $B(\tau) = 1$ but $W(\tau)$ is not identically $1$. One can show that $W(\tau)$ has a Cauchy distribution.