Example of a predictable process that isn't caglad

Question

The predictable processes are defined as the elements of the predictable sigma algebra, which is generated by the caglad processes. What kinds of extra processes do we get by extending from caglad to predictable processes? Specifically, are there predictable processes that are not left continuous? Apologies if there are lots of obvious examples (it seems like there should be, by analogy with extending from continuous to measurable functions) but it isn't obvious to me from the definition what these processes might be or how to go about constructing them.

Answer 1

Answering my own question, but here is what I was looking for. Credit goes entirely to TheBridge for guiding me to this. Let $T$ be a predictable stopping time. Then $X=(X_t)_{t\geq 0}$ defined by $X_t = \mathbb{1}_{\{t \geq T\}}$ for $t\geq 0$ is a predictable process. To prove this, let $T_n$ be an announcing sequence for $T$. Clearly the processes $X^n_t = \mathbb{1}_{\{t > T_n\}}$ are predictable (they are càglàd). Define a stochastic interval as follows \begin{equation*} [T,\infty) = \{(t,\omega) \in \mathbb{R}^+ \times \Omega : T(\omega) \leq t\}. \end{equation*} We can similarly define $(T_n,\infty)$ and observe that $(T_n,\infty) \in \sigma(X^n)$ and therefore these intervals are in $\mathcal{P}$, the predictable $\sigma$-algebra. Furthermore, \begin{equation*} [T,\infty) = \bigcap_n (T_n,\infty) \in \mathcal{P}. \end{equation*} It follows that $\sigma(X) \subset \mathcal{P}$, i.e. $X$ is predictable.