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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?

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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.

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  • $\begingroup$ Could you elaborate on the statement "your adapted process is a.s. equal to a predicable one"? $\endgroup$ – user69818 Feb 23 '15 at 1:55
  • $\begingroup$ You probably noticed that you consider some Hilbert space. It means that you are identifying the processes a.s. equal to 0 for instance. Here it is the same. Constructing the predicable process for a given adapted process is a bit heavy if you want to follow all details, but it is a very old result. You will find references in classical lectures on stochastic calculus. I think to remember that a short proof of this uses the Lebesgue derivation theorem $\endgroup$ – Null Feb 23 '15 at 2:19
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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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