Let us consider a measurable space $(\Omega, \mathcal{F})$, with a filtration $(\mathcal{F}_t)_t$ of sub $\sigma$-algebras of $\mathcal{F}$.
The predictable $\sigma$-algebra $\mathcal{P}$ is the $\sigma$-algebra generated by all processes of the form
$$
X : [0,\infty) \times \Omega \rightarrow \mathbb{R}
$$
such that
-
$X(t,\cdot)$ is $\mathcal{F}_t$-measurable for all $t>0$ (i.e. $X$ is adapted to the filtration $(\mathcal{F}_t)_t$),
-
$X(\cdot, \omega)$ is left-continuous for all $\omega \in \Omega$.
My question is:
Is every $\mathcal{P}$-measurable process necessarily adapted? If so, is it necessarily adapted to the coarser filtration $(\mathcal{F}_{t-})_t$?
Best Answer
There is also the concept of an optional process:
It is possible to show that $P\subseteq O$ ($P$ denotes the predictable $\sigma$-algebra). Thus, every predictable process is optional. For optional process, we have the following result:
Applying this to the stopping time $T\equiv t$ for $t\geq 0$ gives $X_t=X_T 1_{T<\infty}$ is $\mathcal{F}_T$-measurable. Since $\mathcal{F}_T=\mathcal{F}_t$, we see that a predictable process is adapted to the filtration $(\mathcal{F}_t)_{t\geq 0}$
Regarding your second question, we have the following statement:
Applying this to the stopping time $T\equiv t$ yields $X_t=X_t 1_{t<\infty}$ is $\mathcal{F}_{t-}$ measurable for all $t\geq 0$ and a predictable process $X$.
The proof for theorem 3 can be found in the first edition of "Foundations of modern probability theory" by Olav Kallenberg in lemma 22.3.