TITLE: Predictable Processes in Brownian Setting
QUESTION [1 upvotes]: Maybe it's a silly question. I've been reading Protter's book on stochastic integration. And all the integrands are required to be predictable.
But from what I can recall, in the traditional stochastic calculus in Brownian setting (i.e., integration with respect to a Brownian motion), the integrands only need to be in $H^2$ (or $L_{\text{loc}}^2$) and there does not seem to be any mentioning of predictability.
So is it that the Protter's general setting does not entirely include the traditional Brownian setting as a special case, or that the predictable $\sigma$-algebra is special in the Brownian setting (e.g., every adapted process is predictable)?
EDIT: I found a result saying that if all martingales are continuous, then the predictable sigma-algebra equals the optional sigma-algebra (which is generated by all adapted, cadlag processes). So in the case of Brownian filtration, "predictable" is equivalent to "optional" by the Martingale Representation Theorem. But can one go further?

REPLY [0 votes]: It turns out that it is specific to Brownian setting. I found that the book "Introduction to Stochastic Integration" by Chung and Williams has a chapter on this issue. In the general setting, integrands have to be predictable. But when the process we integrate against has some "nice properties" (which Brownian motion satisfies), then we can extend the class of integrands to include all locally $L^2(\mathcal{W})$ processes for some specific sub-sigma-algebra $\mathcal{W}$ of $\mathcal{B}\times\mathcal{F}$. And it can be shown that adapted processes $Z$ satisfying a.s.
$$\int_0^tZ_s^2ds<\infty\text{ for all $t$}$$
are locally $L^2(\mathcal{W})$ processes.
For reference, see Section 3.3 of Chung&Williams.
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