As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is.
However, since I don't have any idea of what the difference between the two is, I can't remember a definition, right or wrong for either of them.
Would it be possible to give the difference in terms of "simple" adapted and predictable processes (i.e. the composition of a discrete-time process with an a.s. increasing sequence of stopping times) as is done here for predictable processes only? (i.e. what about simple adapted processes?)
Moreover, I know that an adapted process is a well-defined concept for discrete time; is the same true for a predictable process? If so, what is the difference between their definitions in discrete time?
I would be happy to understand even just this, because then I could take the intuition behind the Ito Isometry to just be "it's true for discrete time, now take the limit in probability and somehow it works magically in continuous time".
Proof that the predictable sigma algebra is also generated by continuous and adapted processes
Is a predictable process adapted?
What is the definition of a "predictable process"?
If the above two questions give the answer to my question, I don't understand how, soI would be very grateful for a definition.