A Short Note about Predictable and Optional Process

In the context of stochastic calculus, integrands have to be predictable. However, the precise definition of predictability is not very direct, therefore, usually in the introductory text we turn to require the integrand to be caglad.

The concept of predictability is not at all obscured; predictable processes are generated by adapted, caglad processes in the same way as continuous functions generate measurable functions.

Simply put, all the adapted, caglad processes generate the smallest sigma-algebra on [0,T]*\Omega; predictable processes are the processes that are measurable function on this sigma-algebra.

Similarly, optional processes are the processes generated by adapted, cadlag processes.

One question arises: how close is a general predictable to caglad processes? Or, how do we approximate a predictable process by caglad processes?

Recall a theorem in measure theory: measurable function is almost as good as a continuous function in the sense that for all \epsilon there exist a compact set K (which has less than \epsilon area difference between the domain) such that the measurable function is uniformly continuous on K. (I forgot the detail about this theorem; I might drop some conditions or even the conclusion is wrong.)

Have no idea how to connect these two things.