What would be an example of a simple/useful stochastic process $(X_t)_{t\in T}$ for $T=\mathbb{N}$ or $T=\mathbb{R}$ where it is useful to consider a filtration different from the natural filtration $\mathcal{F}^X_t:=\sigma\{X_s :s\le t\}$?
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Not exactly what you were looking for, but perhaps still interesting. Let $T = \mathbb{R}^+$, take $X$ to be Brownian motion. Then the natural filtration is not right-continuous. So instead, it's often preferred to augment the filtration to be right-continuous.
To see that the natural filtration is not right-continuous: Proving that the natural filtration of Brownian motion (not augmented) is not right-continuous
Dasherman
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This was still interesting regarding my question but I didn't understand in the comment of the link why "An event $∈\mathcal{F}_$ has the property that if $∈$ and if $′∈Ω$ is such that $′()=()$ for all $∈[0,]$ then $′∈$ as well" – roi_saumon Oct 05 '21 at 21:37