I have two Ornstein-Uhlenbeck processes (mean-reverting) with jumps:
$$dX_{1,t} = (\alpha_1 - \kappa_1 X_{1,t})\,dt + \sigma_1 \,dW_{1,t} + J(\mu_{1,J},\sigma_{1,J})\,d\Pi(\lambda_1)$$ $$dX_{2,t} = (\alpha_2 - \kappa_2 X_{2,t})\,dt + \sigma_2 \,dW_{2,t} + J(\mu_{2,J},\sigma_{2,J})\,d\Pi(\lambda_2)$$
Each process is discretized in the most simple way producing: $$X_t = \alpha \,\Delta t + \phi X_{t-1} + \sigma \chi $$ with probability $1- \lambda \Delta t$ and $$X_t = \alpha \,\Delta t + \phi X_{t-1} + \sigma \chi + \mu_J + \sigma_J \chi_J$$ with probability $\lambda \,\Delta t$, where $\chi$ and $\chi_J$ are independendent standard normal random variables.
Obtaining the density for each of the two processes independently is straightforward and leads to: $$(\lambda \,\Delta t) N(\alpha \,\Delta t + \phi X_{t-1} + \mu_J, \sigma^2 + \sigma^2_J ) + (1-\lambda \,\Delta t) N(\alpha \,\Delta t + \phi X_{t-1}, \sigma^2) $$
I would like now to model correlation for those two processes. For instance I could assume either $\operatorname{corr}(\chi_1,\chi_2) = \rho_1$ or $\operatorname{corr}(\chi_{1,J},\chi_{2,J}) = \rho_2$ or even both.
Is it possible to obtain the joint probability distribution of the two correlated variables (in closed form)?