# Unconditional Expectation vs. Conditional Expectation at time $0$

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!

• time 0 is usually "today" and the (stock) prices are known, therefore valid reasoning. – alexprice May 6 '19 at 21:16

An explicit reference could be helpful. It seems to me like an independence statement. For if $$Y$$ is independent of $$\mathcal{F}_{0}$$, then $$\mathbb{E}[Y|\mathcal{F}_{0}]=\mathbb{E}[Y]$$.