Suppose we have the following, $P_i(Z_t^1=j)=P(X_t^1=j|X_{t-1}^1=i)$ and $P_i(Z_t^2=j)=P(X_t^2=j|X_{t-1}^2=i)$ where $\{X_t^1\}_{t \geq1}$ and $\{X_t^2\}_{t \geq1}$ are 2 discrete-time homogeneous Markov processes and $i,j\in\Omega$ with $\Omega$ countable and finite. We also assume under measure $P_i$ that processes $Z_t^1$ and $Z_t^2$ are conditionally independent given a discrete random variable $Y$, i.e. $P_i(Z_t^1, Z_t^2|Y) = P_i(Z_t^1|Y) P_i(Z_t^2|Y)$, $\forall$ t.
The goal would be to get an explicit expression of the following joint probability
$P(X_t^1=j_1,X_t^2=j_2) = ?$
Both processes $\{X_t^1\}_{t \geq1}$ and $\{X_t^2\}_{t \geq1}$ are obviously dependent from each other, so how could I compute this probability using the above hypothesis on conditional independence for $Z_t^1$ and $Z_t^2$ ?
Thanks in advance