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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?

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up vote 6 down vote accepted

The process $(A_t)_{t\geqslant1}$ is $(\mathcal F_t)_{t\geqslant0}$-predictable if and only if, for every $t\geqslant0$, $A_{t+1}$ is $\mathcal F_{t}$-measurable.

In other words, at each time $t$, one can predict the next value $A_{t+1}$ of the process using only the information available at time $t$, that is, the sigma-algebra $\mathcal F_t$.

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