Usually in my probability theory class, we define a filtered probability space in the background $\left(\Omega, F, \left\lbrace F_t \right\rbrace P\right)$ and do all of our work on that space. I'm having a difficult time understanding what the space looks like for specific examples.
For example, suppose we have a 1-D random walk starting at $0$, define $S_n = \sum_{i=1}^{n} \xi_i$, where $\xi_i \in \left\lbrace0,1\right\rbrace$, all i.i.d with $P(\xi = +1) = \frac{1}{2} = P(\xi = -1)$. Suppose further we stop the process when either $S_n = -a$ or $S_n = b$ for some $-\infty < -a < 0 < b <\infty$.
How would we define $\Omega, F,$ and $\left\lbrace F_t \right\rbrace$? I would guess that $\Omega = \left\lbrace -a, -a+1, ..., 0, 1, 2, ..., b\right\rbrace$? but I'm unsure about how $F$ and the filtration would be written explicitly if they can be? I assume that whatever they look like in discrete time would extend to continuous time fairly easily.