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I am reading the book Stochastic Calculus for Finance II. I have a problem about Filtration. The author said that: 'A filtration tells us the information we will have at future times. More precisely, when we get to time t, we will know for each set in $\mathcal{F}(t)$ whether the true $\omega$ lies in that set'.

If $X(t)$ is the price at time $t$, so what are contained in $\mathcal{F}(t)$? It is very helpful if there is an real example.

user2693571
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  • You might want to take a look at this question (.. and the questions which you find "linked" there, on the right-hand side). – saz Oct 30 '20 at 10:55
  • Thank you, I read your suggestion question before I asked my question, but I can't imagine how $\mathcal{F}(t)$ look like? I found some example bout $\mathcal{F}(t)$ with coin toss, it is easy to understand because it is discrete. – user2693571 Oct 30 '20 at 12:18
  • Well, but what do you expect? In general, a $\sigma$-algebra is huge (in the sense that it contains many sets). Unless you are in a very simple setting (e.g. coin toss) you cannot write down $\sigma(X)$ explicitly, even for a single random variable $X$, let alone a whole stochastic process. – saz Oct 30 '20 at 12:44
  • Thank you for your patience, I don't need the full value on $\mathcal{F}(t)$, just the type of data in $\mathcal{F}(t)$, set of $t$ or set of $X(t)$ or something else? – user2693571 Oct 30 '20 at 14:17

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