Suppose, it is time $t$, and you have a time based random variable $x_{t+2}$ and two events $A_{t+1},A_{t+2}$ that relate to the random variable $x$ which updates every period. The events relate to whether it crosses a non-negativity threshold $A_{t+2}=\{x_{t+2} \geq 0 \}$ and $A_{t+1}=\{x_{t+1} \geq 0 \}$
Here is the twist, event $A_{t+2}$ cannot happen if $A_{t+1}$ does not occur. Hence the fact $A_{t+2}$ is a possibility implies $A_{t+1}$ occurred. Moreover $A_{t+2}$ contains all the information $A_{t+1} $ has, in other words, $A_{t+1} \subset A_{t+2}$ .
My Goal: I want to take the conditional expectation below
$$E_t[x_{t+2}|A_{t+2},A_{t+1}]$$
In order to do it, can I make the following simplifying assumption?
$$E_t[x_{t+2}|A_{t+2},A_{t+1}]=E_t[x_{t+2}|A_{t+2}]$$
Is this correct? Thank you!
(This is a survival model)
Your setup seems inconsistent. From what you write about $A_{t+1}$ and $A_{t+2}$, we have $A_{t+2}\subseteq A_{t+1}$, not the other way around.
If that's what you meant, then indeed
$$E_t[x_{t+2}\mid A_{t+2},A_{t+1}]=E_t[x_{t+2}\mid A_{t+2}\cap A_{t+1}]=E_t[x_{t+2}\mid A_{t+2}]\;.$$