In many of the courses I am taking I come across expectations conditional on the filtration of a process, where the filtration is most often informally defined as the history of a process up until some time t. Unfortunately the concept is not made rigorous to us(students) as we have not been taught measure theory.

For some reason the concept of an expectation conditional on a filtration has been really confusing to me. I will try to illustrate my current thoughts on it now by means of an example:

Let $Z_t$ be a random variable with mean zero and variance 1. Further let $X_t=\sigma_tZ_t$ with $\sigma^2_{t}=a+bX^2_{t-1}$ such that $\{X_t\}_{t \in Z}$ is effectively an ARCH(1) process (under conditions such and such). Then let $\mathcal{F_t}$ denote the filtration (history) of the process $\{X_t\}_{t \in Z}$ up to time t.

1) Now take $E[\sigma^2_{t}|\mathcal{F_{t-1}}]$. As of now I would say this can be interpreted as $E[\sigma^2_{t}|X_{t-1}]$ such that $E[\sigma^2_{t}|\mathcal{F_{t-1}}]=\sigma^2_{t}=a+bX^2_{t-1}$ where $X_{t-1}$ is still a random variable and thus $\sigma^2_t$ is still a random variable. This in direct contrast to $E[\sigma^2_{t}|X_{t-1}=x]=a+bx^2$ where the conditional expectation would be a real number.

2) Suppose $Z_t \sim N(0,1)$ then $X_t|\mathcal{F_{t-1}} \sim N(0,\sigma^2_t)$ such that the distribution of $X_t$ given the filtration, $\mathcal{F_{t-1}}$, is normal with random parameter $\sigma^2_t$.

Can anyone confirm these two thoughts or provide some extra intuition on this matter.


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