I apologize if this question is not placed in the right place. But I am having a hard time to figure it out. It would be greatly appreciated if some one could help me out.

Assume that there are two processes $X_1(t),X_2(t)$
. The process $X_1(t)$ could take three possible values $x_{1,1}(t),x_{1,2}(t),x_{1,3}(t)$. Likewise,
$X_2(t)$ could take three possible distinct values $x_{2,1}(t),x_{2,2}(t),x_{2,3}(t)$.
Hence $(X_1(t),X_2(t))$ could take nine possible values $(x_{1,i}(t),x_{2,j}(t))$ for $i,j=1,2,3$.
Assume that
$P((X_1(t),X_2(t))=(x_{1,i}(t),x_{2,j}(t)))=p_{i,j}(t)$ which **is known** for $i,j=1,2,3$
Now let two correlated Poisson processes $N_i(t)$'s with intensity rate $\lambda_i t>0$

(more specifically $p(N_i(t)=k)=e^{-\lambda_it}\frac{(\lambda_it)^k}{k!}$)

Define two new processes: $X'_1(t):=X_1(t)+\sum_{i=1}^{N_1(t)}Y_i(t)$

and $X'_2(t)=X_2(t)+\sum_{i=1}^{N_2(t)}Y_j'(t)$

Where $\{Y_i(t)\}$ and $\{Y_i'(t)\}$ are sequences of i.i.d $N(\mu,\delta)$, and independent of $X_i$, $N_i$.

Note that there joint distribution of $(N_1,N_2)$ can be seen here https://math.stackexchange.com/questions/244989/correlated-poisson-distribution.

**My question is** : Can we find the probabilities (or at least approximate the probabilities) of $P((X'_1(t),X'_2(t))=(x_{1,i}(t),x_{2,j}(t))$ for $i,j=1,2,3$, and $P((X'_1(t),X'_2(t))\neq (x_{1,i}(t),x_{2,j}(t))$ based on the distribution of $(X_1,X_2)$ , $Y_i$, $Y_j'$, $N_1$ and $N_2$ ?