Assume that we have the following equation and the following assumption. The scope is to solve for some particular variables expressed later.
Update
$$E_{t}\left[ b(A_{t+1})^{1-\gamma} *R_{t+1}^{-\gamma}*[r_k(A_{t+1},a)-r_h(A_{t+1})+\eta)] \right]=0$$
where $R=(1-\theta_1-\theta_2)*r^f + \theta_1*r_k(A,a)+\theta_2 *[r_h(A)+\eta]$
Also consider that the number $b(A)$ is pinned down recursively by the following equation:
$$b_t = G[E_t(b_{t+1}*R_{t+1}^{1-\gamma})]$$, where $G(\cdot)$ is a known function
Main Assumptions
$A_t$ is a random variable that follows a Markov process, and we know its value at time $t$, but not at time $t+1$.
$a$ is an $IID$ random variable lognormally distributed with some mean $\mu_a$ and variance $\sigma^2_a$.
$\eta$ is another $IID$ random variable normally distributed with some mean $\mu_{\eta}$ and variance $\sigma^2_\eta$.
The RVs are independent of each other. That is, $a$ is independent of $\eta$, while these two can be correlated with $A$, thought are still independently drawn from $A$.
$b(A)$ is a number that depends on $A$ and I do not know their functional relationship.
$r^f$ is a known number.
by $r_k(A,a) $ and respectively $r_h(A)$ I mean a functional relationship (which I know) between these two that depend on $A, a$ and $A$ respectively.
To sum up, the expectations operator is about the unknowns in the next period $A,a$ and $\eta$. To fix the ideas. Let's suppose a simple case, where $a,\eta$ take two values and $A$ follows a two-state Markov process and also take two values. Let's also suppose that at time $t$ we are in one of these states. In that case, the Expectations operator is defined over the next possible state for $A$, and the possible values for the $IID$ RVs.
THE COMPUTATIONAL TASK: Solve for $\theta_1$ and $\theta_2$
Apparently, the solution for each $\theta$ will depend on the possible values of $A$, so I expect to be a vector.
Any guidance on how to solve this equation will be appreciated.