Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $I$ be an index set
- $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$
- $X=(X_t)_{t\in I}$ be a stochastic process on $(\Omega,\mathcal A,\operatorname P)$
If $I=\mathbb{N}_0$, then $X$ is called $\mathbb F$-predictable $:\Leftrightarrow$ $X_0$ is a constant and $$X_n\text{ is }\mathcal F_{n-1}\text{-measurable}\;\;\;\text{ for all }n\in\mathbb N\;.\tag{1}$$ If one thinks about $\mathcal F_n$ as being the information known about $X$ until time $n$, $(1)$ means that at time $n-1$ we already know how $X_n$ will behave.
If $I=[0,\infty)$, then $X$ is called $\mathbb F$-predictable $:\Leftrightarrow$ $X$ is measurable with respect to $$\sigma\left(\left\{ \left\{0\right\}\times F:F\in\mathcal F_0\right\}\cup\bigcup_{0\le s<t}\left\{(s,t]\times F:F\in\mathcal F_s\right\}\right)\;.$$ Let $t>0$. Now, the intuition is much harder to find. There is no single number $s$ immediately before $t$. And clearly, $s<t$ doesn't imply the $\mathcal F_s$-measurability of $X_t$.
So, what is the intuition? Maybe we can show that $X_t$ is measurable with respect to $$\bigcup_{0\le s<t}\mathcal F_s\tag {2}\;.$$ I would be very happy if this would be true and someone could provide a proof.ic
Parallel to the discrete-time case, in continuous time a measurable process $X$ is predictable if and only if the random variable $X_T$ is $\mathcal F_{T-}$ measurable for each bounded predictable stopping time $T$.