Definition of predictable stopping time

Question

I am looking for a counter-example to $$ \tau \text{ is a predictable stopping time} \iff \forall t ~~~ \lbrace \tau \leq t \rbrace \in \bigcap_{l<t} \mathcal{F}_l $$

I have already shown that the only-if is true. I found the (almost) same question here: Definition of predictable process
But I can't understand the answer, I don't see why if we have $X_t = X_{t-}$ a.e. then $X_t \in \mathcal{F_{t-}}$
As I see it, what we're trying to do here is describe the elements of a sigma-algebra/filtration, and that doesn't depend on any measurement property placed on this measurable space.

Answer 1

Because the filtration is complete, any almost impossible event is in $\mathcal{F}_{t-}$ including any event contained in the event $X_t \neq X_{t-}$ since $X_t = X_{t-}$ a.s. It then follows that $X_t \in \mathcal{F}_{t-}$ from gluing together the part where it is equal to $X_{t-}$ and the part where it isn't but has to be measurable anyway because it has probability 0.

As I see it, what we're trying to do here is describe the elements of a sigma-algebra/filtration, and that doesn't depend on any measurement property placed on this measurable space.

This is a good observation but the key here is in the completeness of the filtration. The filtration is constructed based on the measure so that properties of the measure get integrated into the elements of the filtration.

Regarding why the equivalence does not hold: note that your definition only means that one can predict the events about time $t$ at time $t-$. Predictability states essentially that random variables that are measurable at time $t+$ are measurable at time $t$ where random variables are considered measurable at time $t+$ if they are measurable at every time $t'$ strictly after $t$. The key here is that $t-$ and $t+$ are not real times themselves like $t$ but only limits so that the relation between $t$ and $t-$ can be different from the relation between $t+$ and $t$.

Answer 2

Do you mean $\sigma(\cup_{l<t}\mathcal F_l)$ rather than $\cap_{l<t}\mathcal F_l$? The latter is $\mathcal F_{0+}=\mathcal F_0$ (under the usual conditions).

Here's a counter-example of the type (I think) you're seeking. Let $(\mathcal F_t)_{t\ge 0}$ be the natural filtration (completed) of a unit-rate Poisson process $N=(N_t)_{t\ge 0}$. Let $\tau$ be the first jump time of $N$. Then $\tau$ is a totally inaccessible stopping time, so not predictable, But $P(\tau=t)=0$ for each $t\ge 0$, so $\{\tau\le t\}$ differs from $\{\tau<t\}$ by a null set. And $$ \{\tau<t\}=\cup_{q\in\Bbb Q, q<t}\{N_q=1\}\in\mathcal F_{t-}, $$ so $\{\tau\le t\}\in\mathcal F_{t-}$ as well.