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I'd be astounded if this isn't a duplicate, but I've yet to find anything equivalent. Whenever I've seen the natural filtration of a Brownian Motion stated explicitly, it's been in the form of $\mathcal{F}_t=\sigma\{W_s:s\leq t\}$ for any $t\geq0$. In other words, the natural filtration of a Brownian Motion up to some point $t$ is the sigma-field generated by all of the Brownian Motions from every time not after $t$.

My question is, why is that last bit necessary? Why do we not just have $\mathcal{F}_t=\sigma\{W_t\}$ for any $t\geq0$? By definition $W_t$ is $\sigma\{W_t\}$-measurable and $W_t$ can clearly take all of the values that any $W_s$ (with $s\leq t$) can, so I can't see what is gained by adding the fields generated by the "earlier" random variables. Even in cases where we want to talk about some point $b\leq t$, I can't see why we would use $\mathcal{F}_b=\sigma\{W_a:a\leq b\}$ when we could use $\mathcal{F}_b=\sigma\{W_b\}$. Indeed, I can't see why these two fields would not be identical. Is there an obvious element that I'm missing?

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  • $\begingroup$ Yes, that's a filtration, but it loses information compared to the "natural" one. Keep in mind that events are evaluated with respect to a sigma-algebra, so losing the past means all you know is where the process is right now. $\endgroup$ – nomen Feb 25 at 17:06
  • $\begingroup$ @nomen Isn't that a contradiction? You've just said that my proposed filtration is a filtration that is smaller than the natural one. $\endgroup$ – J. Mini Feb 25 at 17:08
  • $\begingroup$ There are lots of filtrations... the constantly empty filtration is the smallest. The smallest one that captures the "past" is called the natural filtration. $\endgroup$ – nomen Feb 25 at 17:11
  • $\begingroup$ @nomen Ahh, so you're saying that the process is not adapted to the filtration that I've proposed? $\endgroup$ – J. Mini Feb 25 at 17:13
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I think I've got it, or at least something along the right lines.

The proposed filtration is not a filtration at all. If it was, then we would have $\mathcal{F}_s\subseteq\mathcal{F}_t$ for all $s\leq t$. However, this is clearly not the case. Consider $\{W_0=0\}\subseteq\mathcal{F}_0$ where $\mathcal{F}_0=\sigma\{W_0\}$. Clearly this set is in $\mathcal{F}_1$ if we take $\mathcal{F}_1=\sigma\{W_s:s\leq 1\}$ (and therefore we have $\mathcal{F}_0\subseteq\mathcal{F}_1$), but this is not the case if we take $\mathcal{F}_1=\sigma\{W_1\}$ (and therefore we have $\mathcal{F}_0\not\subseteq\mathcal{F}_1$), meaning that the proposed filtration is not a filtration.

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