To clarify better the notions of predictable and optional processes, I am looking for a simple example of a process that is optional, but not predictable. I found out something useful here, however, I didn't completely understand the discussion so I am going to reformulate it here in a slightly different form. Consider a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})$ and the process $$ X_t = \begin{cases} 0 & 0\leq t<1\\ \xi & t\geq 1 \end{cases} $$ where $\xi$ is, for example, a binary random variable $\mathbb{P}[\xi=\pm 1]=p\in(0,1)$. If we consider the filtration $\{F_t^{X}\}_{t\geq 0}$ generated by $X$ we have that any $F_t^{X}$-adapted process must be constant in $[0,1)$ because $F_t^X=\{\emptyset,\Omega\}$ for $0\leq t<1$. If the process is, in addition, also left-continuous, it must be constant in $[0,1]$. Accordingly, any predictable process must be constant in $[0,1]$ and so $X$ is not predictable, but it is optional because it is (trivially) $F_t^X$-adapted and right-continuous with left-limit.
Is this reasoning correct?
Finally, again from the answer to this post, I didn't get why it is necessary to consider $X_{t^{+}}$. Besides, it is said, if I understood correctly, that the notions of predictability and of progressive measurability coincide if the filtration considered is the raw one, i.e. it is not assumed to be right-continuous. It would be nice to see a reference on this claim.
