2
$\begingroup$

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!

$\endgroup$
1
  • $\begingroup$ time 0 is usually "today" and the (stock) prices are known, therefore valid reasoning. $\endgroup$
    – alexprice
    May 6, 2019 at 21:16

1 Answer 1

1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.