Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable?

This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion http://www.springer.com/gb/book/9783540643258.

They define the space of integrands with respect to a continuous $L^2$-bounded martingale $M$ as the progressively measurable processes $\phi$ such that:

$$\int_0^\infty \phi^2 dM<\infty$$

So they ask $\phi$ to be progressively measurable instead of the very usual, stronger, hypothesis of $\phi$ being predictable.

${}{}$ I wonder whether this space is actually bigger than the one we get with the predictability imposition, or if this yields the same result in this case.

• I think so, although I don't know any examples off the top of my head. They might coincide under some mild assumptions though. To be honest I really don't know, but it is a great question. – Chill2Macht Jul 15 '16 at 20:41

On the other hand, if you augment your filtration to make it complete and right-continuous, as one so often does, there are plenty of easy examples of progressive processes which are not predictable. (Actually, right-continuity is all that matters in the following.) For example, define $(X_t)_{t \ge 0}$ by $X_t=0$ for $t < 1$ and $X_t=\pm 1$ with some nontrivial (say, equal) probabilities. Consider the augmented filtration generated by $X$,
$\mathcal{F}_t = \cap_{s > t} \sigma(X_s) \vee \mathcal{N}$,
where $\mathcal{N}$ is the set of null sets of the ambient probability space. Then the right-limit process $X_+=(X_{t+})_{t \ge 0}$ is progressive but not predictable, with respect to this filtration. To see this, recall that the predictable processes are (by definition) those generated by the left-continuous adapted processes. Because $X$ is constant strictly before time $1$, every adapted process must be a.s. constant and deterministic on $[0,1)$. A left-continuous process is then also a.s. constant and deterministic on $[0,1]$, and we conclude that the same is true of predictable processes. As $X_{1+}$ is random, it follows that $X_+$ is not predictable. To see that $X_+$ is progressive, just note that it is right-continuous and adapted, because $X_{1+}$ is $\mathcal{F}_1$-measurable due to the right-continuity imposed on the filtration.