I am reading a book on financial mathematics, and frequently encounter the phrase "predictable process", which I haven't seen definition of, and cannot find the definition online.
At first I thought that this was referring to a process which is known exactly at $t = 0$, but that is not the case, because then I see decomposition of an $(\mathcal F_t)_{t\in\mathbb N}$-adapted process $(X_t)$ into a martingale process $(M_t)$ and a predictable process $(A_t)$ where
$ \begin{eqnarray} M_0 = 0, &\hspace{10mm} &\Delta M_t = M_t - M_{t-1} = X_t - E(X_t|\mathcal F_{t-1}) \\ A_0 = 0, &\hspace{10mm} &\Delta A_t = A_t - A_{t-1} = E(X_t|\mathcal F_{t-1}) - X_{t-1} \end{eqnarray} $
so it seems like $A_t$ can only be predicted at $t-1$. Is that what a "predictable process" is?