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I don't quite understand what the natural filtration really is. Imagine e.g. a sequence of independent and identically distributed random $N(0,1)$ variables. What is their natural filtration, and how do I calculate, e.g, $$E(X_T | A_{t-1})$$ where $X_t$ is the $t$th variable and $A_{t-1}$ is the $t-1$th $\sigma$-algebra in the filtration?

According to the definition, the filtrations are given by $A_t = \sigma(X_t^{-1}(B), B \in A)$, but in this continuous case, I have no clue how to determine these. Are they all the same?

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  • $\begingroup$ The canonical (or natural) filtration is the smallest filtration to which $\{X_t\}$ is adapted. In this case, that is $\{\mathcal F_t^X:t\geqslant 0\}$, where $$\mathcal F_t^X = \sigma(X_s : 0\leqslant s\leqslant t\}. $$ $\endgroup$
    – Math1000
    Jun 2, 2016 at 13:58

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You need to be a bit more formal to get it. You should start with some probability space $(\Omega, \mathscr F, \mathsf P)$ and construct on it variables $X_t:\Omega\to\Bbb R$ such that they happen to be iid with a given distribution. Once you did that, it means that every measurable map $X_t$ pulls back Borel $\sigma$-algebra from $\Bbb R$ to a sub-$\sigma$-algebra $X^{-1}_t(\mathscr B(\Bbb R))\subseteq\mathscr F$. Each element of the natural filtration of $X$ is just a union of those $\sigma$-algebras (well, rather the $\sigma$-algebra generated by that union).

The conditional expectation you are talking about is $0$ since your variables are iid.

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  • $\begingroup$ Wait, is the conditional expectation zero always if there is independence? Or is it zero because that's the expectation of the underlying $X$ (AND that we have independence)? $\endgroup$
    – Makeol
    Jun 2, 2016 at 14:04
  • $\begingroup$ @Makeol: the latter, conditional expectation of a constant 1 would be 1, even though it is independent from anything. $\endgroup$
    – SBF
    Jun 2, 2016 at 14:05
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The "natural filtration" for a sequence $(X_n)_{n \in \mathbb N}$ is like this: $\mathcal A_n$ is the sigma-algebra generated by $X_1, X_2, \dots, X_n$.

But later you say the words "continuous case", so maybe you mean a stochastic process indexed by real numbers, like this: $(X_t)_{t \in [0,\infty)}$. Then you may take $\mathcal A_t$ to be the sigma-algebra generated by the set $\{X_s : s \le t\}$. That is an uncountable set of random variables, so it could be nasty; but if, for example, your process has continuous sample paths you can get the same sigma algebra with just countably many generators.

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