Predictable Processes in Brownian Setting 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?

It is a bit subtile but in the particular case of the Brownian motion, you doen't need to assume the integrand is predicable. However it is only because you consider a particular $L^2$ space where your adapted process is a.s. equal to a predicable one. Thus, the loss of generality is an illusion.
