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In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional expectation. This is on a probability space equipped with the filtration generated by the standard Wiener Process. I know that this is true when $\mathcal F_0$ is the trivial sigma-algebra, but it seems like in the finance perspective, the $\mathcal F_0$ information also includes the time $0$ asset prices, which are random variables, and so $\mathcal F_0$ doesn't seem to be trivial in these cases.

I have seen that in some books, the stock/relevant prices are considered deterministic at time $0$, and therefore $\mathcal F_0$ is trivial. Is this a valid reasoning? I don't understand how the measurability of Random Variables is consistent then, since if one were to calculate the conditional expectation of a random variable, $Y$ that is $\mathcal F_0$- measurable (but not one of these deterministic time $0$ asset prices), then assuming $\mathcal F_0$ is trivial leads to $\mathbb E[Y|\mathcal F_0] = E[Y]$ and $\mathbb E[Y|\mathcal F_0] = Y$, which makes it seem like $Y$ is deterministic. And the notation is confusing since only constants are supposed to be measurable with respect to the trivial sigma algebra. So if $\mathcal F_0$ is considered trivial it seems like all random variables at time $0$ have to be deterministic.

I am not sure what I'm missing here. Thanks in advance!

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  • $\begingroup$ time 0 is usually "today" and the (stock) prices are known, therefore valid reasoning. $\endgroup$ – alexprice May 6 '19 at 21:16
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An explicit reference could be helpful. It seems to me like an independence statement. For if $Y$ is independent of $\mathcal{F}_{0}$, then $\mathbb{E}[Y|\mathcal{F}_{0}]=\mathbb{E}[Y]$.

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