Hello. It occurs to me that the definition of natural filtration of a rv as the smallest filtration to which it is adapted is not obviously well-defined, because there is no total order on the set of filtrations. We can define a partial order by saying that one filtration is less than or equal to another iff $\forall t\geq 0:\ \mathscr F^1_t\subseteq \mathscr F^2_t$. It is imaginable that a rv would have multiple filtrations that are minimum elements of the poset of filtrations under that order. It seems to me that the simplest way to define a natural filtration is by the formula given in the sentence that follows the definition above, ie . We would then wish to prove that filtration is smaller than any filtration to which is adapted, under the above-defined order. That, together with the observation that is adapted to that filtration, would entail that filtration is a unique minimum element of the subset of all filtrations to which is adapted.
I hope this LaTeX works. It’s the first time I’ve tried using it in a WordPress environment ðŸ™‚

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