We have a notion of natural filtrations, which intuitively represents the history of the process as the process evolves over time.
We also have a notion of filtrations in general, which are increasing sequence of sub-sigma algebras.
Naturally, the latter concept is more abstract than the former, and I am having trouble getting a concrete grip on the latter.
In particular, if we have a stochastic process X, and a filter F, I tend to look at F as a natural filtration (although we only know it's a filtration in general, and not necessarily a natural one) of some other process Y. Can we do that?
As to why I am doing what I am doing, in many practical scenarios (like quantitative finance, which I am studying), we would be directly observing the process Y (say Y is the share price process) and hence our information would be the natural filtration of Y, but we might be interested in a slightly different process X (which might be the log of the share price or some other functional transformation say). In this scenario, the natural filtration of Y is simply a filtration from the perspective of X, and not a natural one.
Thanks a lot in advance!