What is the difference between an adapted process and a predictable process?

Question

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".

Seemingly related:
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.

Answer 1

In discrete time at least, the definitions I'm familiar with are fairly straightforward.

Given an increasing filtration $\{\mathcal{F}_n\}_{n=0}^{\infty}$, a process $\{X_n\}_{n=0}^{\infty}$ is adapted if each $X_n$ is $\mathcal{F}_n$-measurable.

For predictable processes, the random variables are measurable with respect to slightly smaller $\sigma-$algebras.

Given an increasing filtration $\{\mathcal{F}_n\}_{n=0}^{\infty}$, a process $\{Y_n\}_{n=1}^{\infty}$ is predictable if each $Y_n$ is $\mathcal{F}_{n-1}$-measurable.

I remember these definitions by an analogy with gambling:

An adapted process $X_n$ represents the cumulative gain or loss after $n$ turns, while a predictable process represents a betting strategy. It stands to reason that your betting strategy at the $n$th step can depend on the outcome of the previous $n-1$ steps, but not on the $n$th step itself.