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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

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  • $\begingroup$ This was still interesting regarding my question but I didn't understand in the comment of the link why "An event $𝐴∈\mathcal{F}_𝑡$ has the property that if $𝜔∈𝐴$ and if $𝜔′∈Ω$ is such that $𝜔′(𝑠)=𝜔(𝑠)$ for all $𝑠∈[0,𝑡]$ then $𝜔′∈𝐴$ as well" $\endgroup$
    – roi_saumon
    Oct 5, 2021 at 21:37

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